* Added reviews and initial responses
authorjcheney@inf.ed.ac.uk
Fri, 01 Mar 2013 18:04:49 +0000
changeset 5812 28cc7b3a41a9
parent 5811 48e4610fe4e3
child 5813 5f362ce79bcf
* Added reviews and initial responses
* Moved review copy into folder
semantics/prov-sem-review-20130226.html
semantics/prov-sem.html
semantics/review/ISSUE-630-khalid.txt
semantics/review/ISSUE-630-luc.txt
semantics/review/ISSUE-630-paolo.txt
semantics/review/ISSUE-630-satya.txt
semantics/review/ISSUE-630-simon.txt
semantics/review/prov-sem-review-20130226.html
--- a/semantics/prov-sem-review-20130226.html	Fri Mar 01 17:58:46 2013 +0000
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-  <body><div class="head">
-  <p>
-    
-      <a href="http://www.w3.org/"><img src="https://www.w3.org/Icons/w3c_home" alt="W3C" height="48" width="72"></a>
-    
-  </p>
-  <h1 class="title" id="title">Semantics of the PROV Data Model</h1>
-  
-  <h2 id="w3c-editor-s-draft-25-february-2013"><abbr title="World Wide Web Consortium">W3C</abbr> Editor's Draft 25 February 2013</h2>
-  <dl>
-    
-      <dt>This version:</dt>
-      <dd><a href="http://dvcs.w3.org/hg/prov/raw-file/default/semantics/prov-sem.html">http://dvcs.w3.org/hg/prov/raw-file/default/semantics/prov-sem.html</a></dd>
-      <dt>Latest published version:</dt>
-      <dd><a href="http://www.w3.org/TR/prov-sem/">http://www.w3.org/TR/prov-sem/</a></dd>
-    
-    
-      <dt>Latest editor's draft:</dt>
-      <dd><a href="http://dvcs.w3.org/hg/prov/raw-file/default/semantics/prov-sem.html">http://dvcs.w3.org/hg/prov/raw-file/default/semantics/prov-sem.html</a></dd>
-    
-    
-    
-    
-      
-    
-    
-    
-    <dt>Editor:</dt>
-    <dd><a href="http://homepages.inf.ed.ac.uk/jcheney">James Cheney</a>, University of Edinburgh</dd>
-
-    
-  </dl>
-  
-  
-  
-  
-    
-      <p class="copyright">
-        <a href="http://www.w3.org/Consortium/Legal/ipr-notice#Copyright">Copyright</a> © 
-        2012-2013
-        
-        <a href="http://www.w3.org/"><abbr title="World Wide Web Consortium">W3C</abbr></a><sup>®</sup> 
-        (<a href="http://www.csail.mit.edu/"><abbr title="Massachusetts Institute of Technology">MIT</abbr></a>,
-        <a href="http://www.ercim.eu/"><abbr title="European Research Consortium for Informatics and Mathematics">ERCIM</abbr></a>,
-        <a href="http://www.keio.ac.jp/">Keio</a>, <a href="http://ev.buaa.edu.cn/">Beihang</a>), All Rights Reserved.
-        <abbr title="World Wide Web Consortium">W3C</abbr> <a href="http://www.w3.org/Consortium/Legal/ipr-notice#Legal_Disclaimer">liability</a>,
-        <a href="http://www.w3.org/Consortium/Legal/ipr-notice#W3C_Trademarks">trademark</a> and
-        <a href="http://www.w3.org/Consortium/Legal/copyright-documents">document use</a> rules apply.
-      </p>
-    
-  
-  <hr>
-</div><div style="visibility: hidden; overflow: hidden; position: absolute; top: 0px; height: 1px; width: auto; padding: 0px; border: 0px none; margin: 0px; text-align: left; text-indent: 0px; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal;"><div id="MathJax_Hidden"></div></div><div style="display: none;" id="MathJax_Message"></div> 
-
-    <section class="introductory" id="abstract"><h2>Abstract</h2>
-<p>
-Provenance is information about entities, activities, and people
-involved in producing a piece of data or thing, which can be used to
-form assessments about its quality, reliability or
-trustworthiness. PROV-DM is the conceptual data model that forms a
-basis for the <abbr title="World Wide Web Consortium">W3C</abbr> provenance (PROV) family of specifications.
-</p>
-
-
-<p> This document presents a model-theoretic semantics for the PROV
-data model (called the <dfn id="dfn-reference-semantics">reference semantics</dfn>), viewing
-PROV-DM statements as atomic formulas in the sense of first-order
-logic, and viewing the constraints and inferences specified in
-PROV-CONSTRAINTS as a first-order theory. It is shown that the
-first-order theory is sound with respect to the reference semantics.
-This information may be useful to researchers or users of PROV to
-understand the intended meaning and use of PROV for modeling
-information about the actual history, derivation or evolution of Web
-resources.  It may also be useful for development of additional
-constraints or inferences for reasoning about PROV or integration of
-PROV with other Semantic Web vocabularies.  It is <b>not</b> proposed
-as a canonical or required semantics of PROV and does not place any
-constraints on use of PROV.</p>
-
-<p>The PROV Document Overview describes the overall state of PROV, and should be read before other PROV documents.</p>
-</section><section id="sotd" class="introductory"><h2>Status of This Document</h2>
-  
-    
-      
-        <p>
-          <em>This section describes the status of this document at the time of its publication. Other
-          documents may supersede this document. A list of current <abbr title="World Wide Web Consortium">W3C</abbr> publications and the latest revision
-          of this technical report can be found in the <a href="http://www.w3.org/TR/"><abbr title="World Wide Web Consortium">W3C</abbr> technical reports
-          index</a> at http://www.w3.org/TR/.</em>
-        </p>
-        
-<h4 id="prov-family-of-documents">PROV Family of Documents</h4>
-This document is part of the PROV family of documents, a set of documents defining various aspects that are necessary to achieve the vision of inter-operable
-interchange of provenance information in heterogeneous environments such as the Web.  These documents are listed below. Please consult the [<cite><a href="#bib-PROV-OVERVIEW" class="bibref">PROV-OVERVIEW</a></cite>] for a guide to reading these documents. 
-<ul>
-<li> <a href="http://www.w3.org/TR/2013/WD-prov-overview-20130312/">PROV-OVERVIEW</a> (To be published as Note), an overview of the PROV family of documents [<cite><a href="#bib-PROV-OVERVIEW" class="bibref">PROV-OVERVIEW</a></cite>];</li>
-<li> <a href="http://www.w3.org/TR/2013/WD-prov-primer-20130312/">PROV-PRIMER</a> (To be published as Note), a primer for the PROV data model [<cite><a href="#bib-PROV-PRIMER" class="bibref">PROV-PRIMER</a></cite>];</li>
-<li> <a href="http://www.w3.org/TR/2013/PR-prov-o-20130312/">PROV-O</a> (Proposed Recommendation), the PROV ontology, an OWL2 ontology allowing the mapping of PROV to RDF [<cite><a href="#bib-PROV-O" class="bibref">PROV-O</a></cite>];</li>
-<li> <a href="http://www.w3.org/TR/2013/PR-prov-dm-20130312/">PROV-DM</a> (Proposed Recommendation), the PROV data model for provenance [<cite><a href="#bib-PROV-DM" class="bibref">PROV-DM</a></cite>];</li>
-<li> <a href="http://www.w3.org/TR/2013/PR-prov-n-20130312/">PROV-N</a> (Proposed Recommendation), a notation for provenance aimed at human consumption [<cite><a href="#bib-PROV-N" class="bibref">PROV-N</a></cite>];</li>
-<li> <a href="http://www.w3.org/TR/2013/PR-prov-constraints-20130312/">PROV-CONSTRAINTS</a> (Proposed Recommendation), a set of constraints applying to the PROV data model  (this document);</li>
-<li> <a href="http://www.w3.org/TR/2013/WD-prov-xml-20130312/">PROV-XML</a> (To be published as Note),  an XML schema for the PROV data model [<cite><a href="#bib-PROV-XML" class="bibref">PROV-XML</a></cite>];</li>
-<li> <a href="http://www.w3.org/TR/2013/WD-prov-aq-20130312/">PROV-AQ</a> (To be published as Note), the mechanisms for accessing and querying provenance [<cite><a href="#bib-PROV-AQ" class="bibref">PROV-AQ</a></cite>]; </li>
-<li> <a href="http://www.w3.org/TR/2013/WD-prov-dictionary-20130312/">PROV-DICTIONARY</a> (To be published as Note) introduces a specific type of collection, consisting of key-entity pairs [<cite><a href="#bib-PROV-DICTIONARY" class="bibref">PROV-DICTIONARY</a></cite>];</li>
-<li> <a href="http://www.w3.org/TR/2013/WD-prov-dc-20130312/">PROV-DC</a> (To be published as Note) provides a mapping between PROV and Dublic Core Terms [<cite><a href="#bib-PROV-DC" class="bibref">PROV-DC</a></cite>];</li>
-<li> <a href="http://www.w3.org/TR/2013/WD-prov-sem-20130312/">PROV-SEM</a> (To be published as Note), a declarative specification in terms of first-order logic of the PROV data model [<cite><a href="#bib-PROV-SEM" class="bibref">PROV-SEM</a></cite>];</li>
-<li> <a href="http://www.w3.org/TR/2013/WD-prov-links-20130312/">PROV-LINKS</a> (To be published as Note) introduces a mechanism to link across bundles [<cite><a href="#bib-PROV-LINKS" class="bibref">PROV-LINKS</a></cite>].</li>
-</ul>
-
-
-        <p>
-          This document was published by the <a href="http://www.w3.org/2011/prov/">Provenance Working Group</a> as an Editor's Draft.
-          
-          
-          If you wish to make comments regarding this document, please send them to 
-          <a href="mailto:[email protected]">[email protected]</a> 
-          (<a href="mailto:[email protected]?subject=subscribe">subscribe</a>,
-          <a href="http://lists.w3.org/Archives/Public/public-prov-comments/">archives</a>).
-          
-          
-          
-          
-        All comments are welcome.
-        
-        
-          </p><p>
-            Publication as an Editor's Draft does not imply endorsement by the <abbr title="World Wide Web Consortium">W3C</abbr> Membership.
-            This is a draft document and may be updated, replaced or obsoleted by other documents at 
-            any time. It is inappropriate to cite this document as other than work in progress.
-          </p>
-        
-        
-        <p>
-          
-            This document was produced by a group operating under the 
-            <a href="http://www.w3.org/Consortium/Patent-Policy-20040205/">5 February 2004 <abbr title="World Wide Web Consortium">W3C</abbr> Patent Policy</a>.
-          
-          
-          
-            
-              <abbr title="World Wide Web Consortium">W3C</abbr> maintains a <a href="http://www.w3.org/2004/01/pp-impl/46974/status" rel="disclosure">public list of any patent disclosures</a> 
-            
-            made in connection with the deliverables of the group; that page also includes instructions for 
-            disclosing a patent. An individual who has actual knowledge of a patent which the individual believes contains
-            <a href="http://www.w3.org/Consortium/Patent-Policy-20040205/#def-essential">Essential Claim(s)</a> must disclose the
-            information in accordance with <a href="http://www.w3.org/Consortium/Patent-Policy-20040205/#sec-Disclosure">section
-            6 of the <abbr title="World Wide Web Consortium">W3C</abbr> Patent Policy</a>.
-          
-          
-        </p>
-        
-      
-    
-  
-</section><section id="toc"><h2 class="introductory">Table of Contents</h2><ul class="toc"><li class="tocline"><a class="tocxref" href="#introduction"><span class="secno">1. </span>Introduction</a><ul class="toc"><li class="tocline"><a class="tocxref" href="#purpose"><span class="secno">1.1 </span>Purpose of this document</a></li><li class="tocline"><a class="tocxref" href="#structure-of-this-document"><span class="secno">1.2 </span>Structure of this document</a></li><li class="tocline"><a class="tocxref" href="#audience"><span class="secno">1.3 </span> Audience </a></li></ul></li><li class="tocline"><a class="tocxref" href="#basics"><span class="secno">2. </span> Basics </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#identifiers"><span class="secno">2.1 </span> Identifiers </a></li><li class="tocline"><a class="tocxref" href="#times-and-intervals"><span class="secno">2.2 </span> Times and Intervals </a></li><li class="tocline"><a class="tocxref" href="#attributes-and-values"><span class="secno">2.3 </span> Attributes and Values </a></li><li class="tocline"><a class="tocxref" href="#formulas"><span class="secno">2.4 </span>Atomic Formulas</a></li><li class="tocline"><a class="tocxref" href="#first-order-formulas"><span class="secno">2.5 </span>First-Order Formulas</a></li></ul></li><li class="tocline"><a class="tocxref" href="#structures"><span class="secno">3. </span> Structures and Interpretations </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#things-1"><span class="secno">3.1 </span> Things </a></li><li class="tocline"><a class="tocxref" href="#objects-1"><span class="secno">3.2 </span> Objects </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#entities-1"><span class="secno">3.2.1 </span> Entities </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#plans-1"><span class="secno">3.2.1.1 </span> Plans </a></li></ul></li><li class="tocline"><a class="tocxref" href="#actvities"><span class="secno">3.2.2 </span> Actvities </a></li><li class="tocline"><a class="tocxref" href="#agents-1"><span class="secno">3.2.3 </span> Agents </a></li><li class="tocline"><a class="tocxref" href="#interactions-1"><span class="secno">3.2.4 </span> Interactions </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#events-1"><span class="secno">3.2.4.1 </span> Events </a></li><li class="tocline"><a class="tocxref" href="#associations-1"><span class="secno">3.2.4.2 </span> Associations </a></li><li class="tocline"><a class="tocxref" href="#communications"><span class="secno">3.2.4.3 </span>Communications</a></li><li class="tocline"><a class="tocxref" href="#delegations"><span class="secno">3.2.4.4 </span>Delegations</a></li><li class="tocline"><a class="tocxref" href="#derivations-1"><span class="secno">3.2.4.5 </span> Derivations </a></li></ul></li></ul></li><li class="tocline"><a class="tocxref" href="#relations"><span class="secno">3.3 </span> Relations </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#derivation-paths-and-derivedfrom"><span class="secno">3.3.1 </span> Derivation paths and DerivedFrom </a></li></ul></li><li class="tocline"><a class="tocxref" href="#putting-it-all-together"><span class="secno">3.4 </span> Putting it all together </a></li><li class="tocline"><a class="tocxref" href="#interpretations"><span class="secno">3.5 </span> Interpretations </a></li></ul></li><li class="tocline"><a class="tocxref" href="#semantics"><span class="secno">4. </span> Semantics </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#satisfaction"><span class="secno">4.1 </span> Satisfaction </a></li><li class="tocline"><a class="tocxref" href="#attribute-matching"><span class="secno">4.2 </span> Attribute matching </a></li><li class="tocline"><a class="tocxref" href="#semantics-of-element-formulas"><span class="secno">4.3 </span> Semantics of Element Formulas </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#entity"><span class="secno">4.3.1 </span> Entity </a></li><li class="tocline"><a class="tocxref" href="#activity"><span class="secno">4.3.2 </span> Activity </a></li><li class="tocline"><a class="tocxref" href="#agent"><span class="secno">4.3.3 </span> Agent </a></li></ul></li><li class="tocline"><a class="tocxref" href="#semantics-of-relations"><span class="secno">4.4 </span> Semantics of Relations </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#generation"><span class="secno">4.4.1 </span> Generation </a></li><li class="tocline"><a class="tocxref" href="#use"><span class="secno">4.4.2 </span> Use </a></li><li class="tocline"><a class="tocxref" href="#invalidation"><span class="secno">4.4.3 </span> Invalidation </a></li><li class="tocline"><a class="tocxref" href="#association"><span class="secno">4.4.4 </span> Association </a></li><li class="tocline"><a class="tocxref" href="#start-formulas"><span class="secno">4.4.5 </span> Start Formulas </a></li><li class="tocline"><a class="tocxref" href="#end"><span class="secno">4.4.6 </span> End </a></li><li class="tocline"><a class="tocxref" href="#attribution"><span class="secno">4.4.7 </span> Attribution </a></li><li class="tocline"><a class="tocxref" href="#communication"><span class="secno">4.4.8 </span>Communication</a></li><li class="tocline"><a class="tocxref" href="#responsibility"><span class="secno">4.4.9 </span> Responsibility </a></li><li class="tocline"><a class="tocxref" href="#derivation"><span class="secno">4.4.10 </span> Derivation </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#precise"><span class="secno">4.4.10.1 </span> Precise </a></li><li class="tocline"><a class="tocxref" href="#imprecise"><span class="secno">4.4.10.2 </span> Imprecise </a></li></ul></li><li class="tocline"><a class="tocxref" href="#influence"><span class="secno">4.4.11 </span>Influence</a></li><li class="tocline"><a class="tocxref" href="#specialization"><span class="secno">4.4.12 </span> Specialization </a></li><li class="tocline"><a class="tocxref" href="#alternate"><span class="secno">4.4.13 </span> Alternate </a></li><li class="tocline"><a class="tocxref" href="#membership"><span class="secno">4.4.14 </span> Membership </a></li></ul></li><li class="tocline"><a class="tocxref" href="#auxiliary-formulas"><span class="secno">4.5 </span>Auxiliary formulas</a><ul class="toc"><li class="tocline"><a class="tocxref" href="#equals"><span class="secno">4.5.1 </span>Equals</a></li><li class="tocline"><a class="tocxref" href="#precedes-and-strictly-precedes"><span class="secno">4.5.2 </span>Precedes and Strictly Precedes</a></li><li class="tocline"><a class="tocxref" href="#notnull"><span class="secno">4.5.3 </span>notNull</a></li><li class="tocline"><a class="tocxref" href="#typeof"><span class="secno">4.5.4 </span>typeOf</a></li></ul></li></ul></li><li class="tocline"><a class="tocxref" href="#theory"><span class="secno">5. </span> Inferences and Constraints </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#inferences"><span class="secno">5.1 </span>Inferences</a></li><li class="tocline"><a class="tocxref" href="#constraints"><span class="secno">5.2 </span>Constraints</a><ul class="toc"><li class="tocline"><a class="tocxref" href="#uniqueness-constraints"><span class="secno">5.2.1 </span>Uniqueness constraints</a></li><li class="tocline"><a class="tocxref" href="#ordering-constraints"><span class="secno">5.2.2 </span>Ordering constraints</a></li><li class="tocline"><a class="tocxref" href="#typing-constraints"><span class="secno">5.2.3 </span>Typing constraints</a></li><li class="tocline"><a class="tocxref" href="#impossibility-constraints"><span class="secno">5.2.4 </span>Impossibility constraints</a></li></ul></li></ul></li><li class="tocline"><a class="tocxref" href="#soundness"><span class="secno">6. </span>Soundness</a></li><li class="tocline"><a class="tocxref" href="#acknowledgements"><span class="secno">A. </span>Acknowledgements</a></li><li class="tocline"><a class="tocxref" href="#references"><span class="secno">B. </span>References</a><ul class="toc"><li class="tocline"><a class="tocxref" href="#informative-references"><span class="secno">B.1 </span>Informative references</a></li></ul></li></ul></section>
-
-
-
-
-
-
-    <section id="introduction"> 
-      <!--OddPage--><h2><span class="secno">1. </span>Introduction</h2> 
-
-<p>
-Provenance is a record that describes the people, institutions, entities, and activities involved in producing, influencing, or delivering a piece of data or a thing.
-This document complements
-  the PROV-DM specification [<cite><a href="#bib-PROV-DM" class="bibref">PROV-DM</a></cite>] that defines a data model for
-  provenance on the Web, and the PROV-CONSTRAINTS specification
-[<cite><a href="#bib-PROV-CONSTRAINTS" class="bibref">PROV-CONSTRAINTS</a></cite>] that
-specifies definitions, inferences, and constraints that can be used to
-reason about PROV documents, or determine their validity.  This document
-provides a reference formal semantics of PROV, providing a formal
-counterpart to the informal descriptions and motivations given
-elsewhere in PROV specifications.</p>
-
-
-
-<!--
-<section id="conventions"> 
-<h3>Conventions</h3>
-
-
-<p>In this document, logical formulas contain variables written as
-    lower-case identifiers.  Some of these variables are written
-    beginning with the underscore character <span
-    class="name">_</span>, by convention, to indicate that they
-    appear only once in the formula.  Such variables are
-    provided merely as an aid to the reader.  </p>
-
-</section>
--->
-
-<section id="purpose">
-
-<h3><span class="secno">1.1 </span>Purpose of this document</h3>
-
-</section>
-<p>The PROV-DM and PROV-CONSTRAINTS give motivating examples that
-provide an intuition about the meaning of the constructs.  For some
-concepts, such as use, start, end, generation/invalidation, and
-derivation, the meaning is either obvious or situation-dependent.
-However, during the development of PROV, the importance of additional
-concepts became evident, but the intuitive meaning or correct use of
-these concepts were not clear.  For example, the <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-1-Frame" class="MathJax"><nobr><span id="MathJax-Span-1" class="math"><span style="display: inline-block; position: relative; width: 83px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.509em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-5" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-6" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-7" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-8" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-9" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-10" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-11" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-12" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-13" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-1" type="math/tex">alternateOf</script> and
-<span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-2-Frame" class="MathJax"><nobr><span id="MathJax-Span-14" class="math"><span style="display: inline-block; position: relative; width: 115px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.51em); top: -2.57em; left: 0em;"><span id="MathJax-Span-15" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-16" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-17" class="mi">p</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-18" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-19" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-20" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-21" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-22" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-23" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-24" class="mi">z</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-25" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-26" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-27" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-28" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-29" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-30" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-31" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-2" type="math/tex">specializationOf</script> relations are used in PROV to relate different
-entities that present aspects of "the same thing".  
-Over time the working group came to a
-consensus about these concepts and how they are to be used, but this
-understanding is based on abstract notions that are not explicit in
-PROV documents; instead, some of their properties are captured
-formally through certain constraints and inferences, while others are
-not captured in PROV specifications at all.
-</p>
-
-<p>The purpose of this document is to present the working group's
-consensus view of the semantics of PROV, using tools from mathematical
-logic, principally model theory (though our use of these tools is lightweight).  This information may be useful to users for understanding the
-intent behind certain features of PROV, to researchers investigating
-richer forms of reasoning over provenance, or to future efforts
-building upon PROV.  It is intended as an exploration of a <b>reference</b>
-semantics for PROV, not a definitive specification of the  <b>only</b>
-semantics of PROV.  </p>
-
-<p> Nevertheless, the reference semantics does have appealing
-properties.  Specifically, it provides a declarative counterpart to
-the operational definition of validity taken in PROV-CONSTRAINTS.  In
-the specification, validity is defined via a normalization process
-followed by constraint checking on the normal form.  This approach was adopted
-to keep the specification closer to implementations, although other
-implementations are possible and allowed.  In addition to providing a
-reference semantics, this document shows that the operational
-presentation of PROV validity checking is sound with respect to
-declarative presentation adopted here.  This could help justify
-alternative approaches to validity checking.</p>
-
-<section id="structure-of-this-document"> <h3><span class="secno">1.2 </span>Structure of this document</h3> <p>
-
-</p>
-
-<ul>
-  <li><a href="#basics">Section 2</a> summarizes the basic concepts
-  from mathematical logic used in the semantics,  recapitulates how
-  PROV statements can be viewed as atomic formulas, and introduces
-  some auxiliary formulas.</li>
-  <li><a href="#structures">Section 3</a> presents the mathematical
-  structures used for situations that PROV statements can
-  describe.</li>
-  <li><a href="#semantics">Section 4</a> defines the semantics of PROV
-  statements and auxiliary formulas, indicating when a given formula
-  is satisfied in a structure.</li>
-  <li><a href="#theory">Section 5</a> presents the inferences and
-  constraints from PROV-CONSTRAINTS as first-order formulas, and gives
-  brief justifications for their soundness.</li>
-  <li><a href="#soundness">Section 6</a> relates the PROV notion of
-  validity to the logical notion of consistency.  A PROV instance is
-  valid if and only if its corresponding first-order theory is
-  satisfiable.</li>
-</ul>
-<div class="note"><div class="note-title"><span>Note</span></div><div class="">
-  Completeness direction above doesn't hold for current semantics,
-  because attributes assumed to be single-valued; for
-  example, <span class="MathJax_Preview">entity(id,[a=1,a=2])</span><script id="MathJax-Element-3" type="math/tex">entity(id,[a=1,a=2])</script> is valid but doesn't have a
-  reference model.
-  </div></div>
-
-</section>
-<section id="audience">
-<h3><span class="secno">1.3 </span> Audience </h3>
-
-
-<p>This document assumes familiarity with [<cite><a href="#bib-PROV-DM" class="bibref">PROV-DM</a></cite>] and
-  [<cite><a href="#bib-PROV-CONSTRAINTS" class="bibref">PROV-CONSTRAINTS</a></cite>] and employs (a
-  simplified form of) 
-[<cite><a href="#bib-PROV-N" class="bibref">PROV-N</a></cite>] notation.  In particular it assumes familiarity with the concepts
-  from logic, and the relationship between PROV statements and
-  instances and first-order formulas and theories, respectively,
-  presented in <a href="http://www.w3.org/TR/2013/PR-prov-constraints-20130312/#overview">Section 2.5</a> of PROV-CONSTRAINTS.
-</p>
-
-  <p>This document may be useful to users of PROV who have a formal
-  background and are interested in the rationale for some of the
-  constructs of PROV; for resaerchers investigating extensions of PROV
-  or alternative approaches to reasoning about PROV; or for future
-  efforts on provenance standardization.  </p>
-
-  </section>
-
-  </section>
-
-
-  <!--
-  
-<section id="overview">
-<h2> Overview </h2>
-
-
-<p>The idea of this document is to sketch what aspects of the provenance model can be formalized and how they can be formalized, as a first step towards establishing a consensus on the (intended) meaning of the components of the model and the consistency constraints or inferences that can be applied to the model to distinguish valid from invalid provenance records.</p>
-
-<p>The <a href="http://www.w3.org/TR/prov-constraints/">PROV-CONSTRAINTS</a> document contains formal content specifying a notion of validity (approximately, logical consistency) for PROV documents.  The formal semantics, PROV-SEM, is planned for release as a W3C Note that will complement the procedural specification in PROV-CONSTRAINTS with a declarative specification formulated in terms of first-order logic.   The formal semantics is <b>work in progress</b>.  The drafts below are intermediate stages and some of them are out of date.  
-</p>
-
-<h3> Status </h3>
-
-<p>
-This is work in progress.  The semantics is being updated to be consistent with the Candidate Recommendation of PROV.  The plan is to release the semantics as a Note over the next few months.  At that point, the wiki pages containing drafts of the semantics will be superseded.
-</p>
-
-
-<h3> Idea of the semantics </h3>
-
-<p>As a starting point, I will assume that we intend the statements made in a PROV-DM instance to be intended to describe one, consistent state of the world, much like a logical formula is said to be satisfied in a mathematical model.  That is, I propose an approach similar to that taken in model theory, where the PROV-DM instance corresponds to a formula or theory of a logic, and the semantics corresponds to what logicians call a model.</p>
-
-<p>For example, the formula $\forall x. P(x) \Rightarrow Q(x)$ is satisfied in a mathematical model where the relation $P$ denotes a set of elements that is contained in that denoted by $Q$.  Here, the goal is to come up with a plausible "intended model" for interpreting PROV-DM instances, where the formulas are statements in PROV-DM and the individuals are things and agents.  This is complicated by the fact that many statements about provenance involve talking about objects that change over time.</p>
-
-<p>The word "world" is used in PROV-DM to talk about the actual state
-of affairs that the PROV-DM instance describes, which is what I would
-usually call a "model".  The word "model" is used in PROV-DM mainly in
-the sense of "data model", that is, to talk about what I would
-otherwise call the syntax of PROV-DM.  To avoid confusion with the
-uses of terms in PROV-DM, I will use "structure" to describe the
-mathematical structure that corresponds to actual state of affairs.
-We use the term "model" to describe a structure that satisfies a given
-theory; this sense of "model" does not mean "data model".</p>
-
-<h3> Axiomatization and relationship to PROV-CONSTRAINTS </h3>
-
-<p>One goal of the semantics is to link the procedural specification of validity and equivalence with traditional notions of logical consistency and equivalence of theories, for example in first-order logic.  A first-order axiomatization that corresponds to the formal constraints and is sound for reasoning about the models described below is in progress at the end of the document.
-</p>
-</section>
--->
-
-<section id="basics">
-<!--OddPage--><h2><span class="secno">2. </span> Basics </h2>
-
-
-
-
-
-<section id="identifiers">
-<h3><span class="secno">2.1 </span> Identifiers </h3>
-
-<p>A lowercase symbol <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-4-Frame" class="MathJax"><nobr><span id="MathJax-Span-54" class="math"><span style="display: inline-block; position: relative; width: 49px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.951em, -0.553em); top: -2.57em; left: 0em;"><span id="MathJax-Span-55" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-56" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-57" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-58" class="mi">y</span><span style="font-family: STIXGeneral;" id="MathJax-Span-59" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-60" class="mo">.</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-61" class="mo">.</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-62" class="mo">.</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.817em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-4" type="math/tex">x,y,...</script> on its own denotes an identifier.
-Identifiers may or may not be URIs.  Identifiers are viewed as variables in logic (or blank nodes in RDF): just because we have two different identifiers <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-5-Frame" class="MathJax"><nobr><span id="MathJax-Span-63" class="math"><span style="display: inline-block; position: relative; width: 8px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.553em); top: -2.57em; left: 0em;"><span id="MathJax-Span-64" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-65" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.609em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-5" type="math/tex">x</script> and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-6-Frame" class="MathJax"><nobr><span id="MathJax-Span-66" class="math"><span style="display: inline-block; position: relative; width: 8px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.951em, -0.55em); top: -2.57em; left: 0em;"><span id="MathJax-Span-67" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-68" class="mi">y</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.817em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-6" type="math/tex">y</script> doesn't tell us that they denote different things, since we could discover that they are actually the same later.  We write <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-7-Frame" class="MathJax"><nobr><span id="MathJax-Span-69" class="math"><span style="display: inline-block; position: relative; width: 75px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-70" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-71" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-72" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-73" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-74" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-75" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-76" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-77" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-78" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-79" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-80" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-81" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-7" type="math/tex">Identifiers</script> for the set of identifiers of interest in a given situation (typically, the set of identifiers present in the PROV instance of interest).
-</p>
-</section>
-
-<section id="times-and-intervals">
-<h3><span class="secno">2.2 </span> Times and Intervals </h3>
-
-<p>We assume a linearly ordered set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-8-Frame" class="MathJax"><nobr><span id="MathJax-Span-82" class="math"><span style="display: inline-block; position: relative; width: 73px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.719em, 1000em, 2.922em, -0.478em); top: -2.57em; left: 0em;"><span id="MathJax-Span-83" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-84" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-85" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-86" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-87" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-88" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-89" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-90" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-91" class="mo">≤</span><span style="font-family: STIXGeneral;" id="MathJax-Span-92" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.038em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-8" type="math/tex">(Times,\leq)</script> of time instants.  For convenience we assume the order is total or linear order, corresponding to a linear timeline; however, PROV does not assume that time is linear and events could be partially ordered and not necessarily reconciled to a single global clock.  
-</p>
-<p>We also consider a set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-9-Frame" class="MathJax"><nobr><span id="MathJax-Span-93" class="math"><span style="display: inline-block; position: relative; width: 62px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.763em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-94" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-95" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-96" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-97" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-98" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-99" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-100" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-101" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-102" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-103" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.875em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-9" type="math/tex">Intervals</script> of closed intervals of the form <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-10-Frame" class="MathJax"><nobr><span id="MathJax-Span-104" class="math"><span style="display: inline-block; position: relative; width: 108px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.705em, 1000em, 2.934em, -0.426em); top: -2.57em; left: 0em;"><span id="MathJax-Span-105" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-106" class="mo">{</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-107" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-108" class="mo">∣</span><span style="padding-left: 0.313em;" id="MathJax-Span-109" class="msubsup"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-110" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.274em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-111" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-112" class="mo">≤</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-113" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-114" class="mo">≤</span><span style="padding-left: 0.313em;" id="MathJax-Span-115" class="msubsup"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-116" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.274em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-117" class="mn">2</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-118" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.066em; vertical-align: -0.265em;"></span></span></nobr></span><script id="MathJax-Element-10" type="math/tex">\{t \mid t_1 \leq t \leq t_2\}</script>.
-</p>
-</section>
-
-<section id="attributes-and-values">
-<h3><span class="secno">2.3 </span> Attributes and Values </h3>
-
-<p>We assume a set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-11-Frame" class="MathJax"><nobr><span id="MathJax-Span-119" class="math"><span style="display: inline-block; position: relative; width: 69px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.758em, -0.577em); top: -2.57em; left: 0em;"><span id="MathJax-Span-120" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-121" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-122" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-123" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-124" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-125" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-126" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-127" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-128" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-129" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-130" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.87em; vertical-align: -0.076em;"></span></span></nobr></span><script id="MathJax-Element-11" type="math/tex">Attributes</script> of attribute labels and a set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-12-Frame" class="MathJax"><nobr><span id="MathJax-Span-131" class="math"><span style="display: inline-block; position: relative; width: 48px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.763em, -0.45em); top: -2.57em; left: 0em;"><span id="MathJax-Span-132" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-133" class="mi">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-134" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-135" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-136" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-137" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-138" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.875em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-12" type="math/tex">Values</script>
-of possible values of attributes.  To allow for the fact that some
-attributes can have undefined or multiple values, we sometimes use the set
-<span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-13-Frame" class="MathJax"><nobr><span id="MathJax-Span-139" class="math"><span style="display: inline-block; position: relative; width: 63px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.922em, -0.526em); top: -2.57em; left: 0em;"><span id="MathJax-Span-140" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-141" class="mi">P</span><span style="font-family: STIXGeneral;" id="MathJax-Span-142" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-143" class="mi">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-144" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-145" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-146" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-147" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-148" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.045em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-13" type="math/tex">P(Value)</script>, that is, the set of sets of values.
-</p> </section>
-
-<section id="formulas">
-<h3><span class="secno">2.4 </span>Atomic Formulas</h3>
-
-<p>The following atomic formulas correspond to the statements of PROV-DM.  We assume that definitions 1-4 of PROV-CONSTRAINTS have been applied in order to expand all optional parameters; thus, we use uniform notation <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-14-Frame" class="MathJax"><nobr><span id="MathJax-Span-149" class="math"><span style="display: inline-block; position: relative; width: 106px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.922em, -0.481em); top: -2.57em; left: 0em;"><span id="MathJax-Span-150" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-151" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-152" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-153" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-154" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-155" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-156" class="msubsup"><span style="display: inline-block; position: relative; width: 0.951em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-157" class="mi">a</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.526em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-158" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-159" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-160" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-161" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-162" class="msubsup"><span style="display: inline-block; position: relative; width: 0.951em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-163" class="mi">a</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.526em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-164" class="mi">n</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-165" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.045em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-14" type="math/tex">r(id,a_1,\ldots,a_n)</script> instead of the semicolon notation <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-15-Frame" class="MathJax"><nobr><span id="MathJax-Span-166" class="math"><span style="display: inline-block; position: relative; width: 107px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.922em, -0.481em); top: -2.57em; left: 0em;"><span id="MathJax-Span-167" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-168" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-169" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-170" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-171" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-172" class="mo">;</span><span style="padding-left: 0.188em;" id="MathJax-Span-173" class="msubsup"><span style="display: inline-block; position: relative; width: 0.951em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-174" class="mi">a</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.526em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-175" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-176" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-177" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-178" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-179" class="msubsup"><span style="display: inline-block; position: relative; width: 0.951em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-180" class="mi">a</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.526em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-181" class="mi">n</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-182" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.045em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-15" type="math/tex">r(id;a_1,\ldots,a_n)</script>.
-</p>
-<p>Each parameter is either an identifier, a constant (e.g. a time or other literal value in an attribute list), or a null symbol "-".  Null symbols can only appear in the specified arguments in <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-16-Frame" class="MathJax"><nobr><span id="MathJax-Span-183" class="math"><span style="display: inline-block; position: relative; width: 135px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.763em, -0.511em); top: -2.57em; left: 0em;"><span id="MathJax-Span-184" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-185" class="mi">w</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-186" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-187" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-188" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-189" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-190" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-191" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-192" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-193" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-194" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-195" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-196" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-197" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-198" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-199" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-200" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-201" class="mi">h</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.875em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-16" type="math/tex">wasAssociatedWith</script> and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-17-Frame" class="MathJax"><nobr><span id="MathJax-Span-202" class="math"><span style="display: inline-block; position: relative; width: 121px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.763em, -0.511em); top: -2.57em; left: 0em;"><span id="MathJax-Span-203" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-204" class="mi">w</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-205" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-206" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-207" class="mi">D</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-208" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-209" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-210" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-211" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-212" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-213" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-214" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-215" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-216" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-217" class="mi">m</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.875em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-17" type="math/tex">wasDerivedFrom</script>, as shown in the grammar below.
-</p>
-<span class="MathJax_Preview"></span><div style="text-align: center;" aria-readonly="true" role="textbox" class="MathJax_Display"><span style="" id="MathJax-Element-18-Frame" class="MathJax"><nobr><span id="MathJax-Span-218" class="math"><span style="display: inline-block; position: relative; width: 487px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(-20.206em, 1000em, 23.795em, -0.359em); top: -2.044em; left: 0em;"><span id="MathJax-Span-219" class="mrow"><span style="padding-right: 0.167em; padding-left: 0.167em;" id="MathJax-Span-220" class="mtable"><span style="display: inline-block; position: relative; width: 28.126em; height: 0px;"><span style="position: absolute; clip: rect(22.201em, 1000em, 65.879em, -0.526em); top: -44.334em; left: 0em;"><span style="display: inline-block; position: relative; width: 7.477em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.509em); top: -23.845em; right: 0em;"><span id="MathJax-Span-221" class="mtd"><span id="MathJax-Span-222" class="mrow"><span style="font-family: STIXGeneral; 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font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1048" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1049" class="mo">)</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.954em, 1000em, 2.951em, -0.553em); top: 5.57em; left: 0em;"><span id="MathJax-Span-1074" class="mtd"><span id="MathJax-Span-1075" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1076" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1077" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1078" class="mi">y</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.863em, 1000em, 2.951em, -0.553em); 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padding-left: 0.313em;" id="MathJax-Span-1101" class="mo">⪯</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1102" class="mi">y</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.712em, 1000em, 2.922em, -0.512em); top: 9.574em; left: 0em;"><span id="MathJax-Span-1110" class="mtd"><span id="MathJax-Span-1111" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1112" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1113" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1114" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1115" class="mi">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.06em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1116" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1117" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1118" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1119" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1120" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1121" class="mo">)</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; 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font-style: italic;" id="MathJax-Span-1138" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1139" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1140" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1141" class="mi">y</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1142" class="mo">)</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.733em, 1000em, 2.902em, -0.438em); top: 12.198em; left: 0em;"><span id="MathJax-Span-1153" class="mtd"><span id="MathJax-Span-1154" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1155" class="mo">[</span><span style="font-family: STIXGeneral; 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left: 0.386em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1161" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1162" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1163" class="msubsup"><span style="display: inline-block; position: relative; width: 0.893em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.763em, -0.506em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1164" class="mi">v</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.467em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1165" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1166" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1167" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-1168" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1169" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1170" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1171" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span id="MathJax-Span-1172" class="msubsup"><span style="display: inline-block; position: relative; width: 0.811em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.745em, -0.481em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1173" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.386em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1174" class="mi">n</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1175" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-1176" class="msubsup"><span style="display: inline-block; position: relative; width: 0.893em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.763em, -0.506em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1177" class="mi">v</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.467em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1178" class="mi">n</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1179" class="mo">]</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.741em, 1000em, 2.951em, -0.495em); top: 13.458em; left: 0em;"><span id="MathJax-Span-1189" class="mtd"><span id="MathJax-Span-1190" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1191" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1192" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1193" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1194" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1195" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1196" class="mi">y</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.741em, 1000em, 2.951em, -0.509em); top: 14.79em; left: 0em;"><span id="MathJax-Span-1204" class="mtd"><span id="MathJax-Span-1205" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1206" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1207" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1208" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1209" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1210" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1211" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1212" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1213" class="mi">y</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.849em, 1000em, 2.951em, -0.509em); top: 16.122em; left: 0em;"><span id="MathJax-Span-1221" class="mtd"><span id="MathJax-Span-1222" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1223" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1224" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1225" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1226" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1227" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.601em); top: 17.461em; left: 0em;"><span id="MathJax-Span-1235" class="mtd"><span id="MathJax-Span-1236" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1237" class="mi">p</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1238" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1239" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1240" class="mi">v</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1241" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1242" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1243" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1244" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1245" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1246" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1247" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1248" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1249" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1250" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1251" class="mi">n</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.601em); top: 18.799em; left: 0em;"><span id="MathJax-Span-1259" class="mtd"><span id="MathJax-Span-1260" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1261" class="mi">p</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1262" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1263" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1264" class="mi">v</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1265" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1266" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1267" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1268" class="mi">p</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1269" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1270" class="mi">y</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1271" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1272" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1273" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1274" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1275" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1276" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1277" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1278" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1279" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1280" class="mi">n</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="display: inline-block; width: 0px; height: 44.509em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.044em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 46.831em; vertical-align: -23.148em;"></span></span></nobr></span></div><script id="MathJax-Element-18" type="math/tex; mode=display">
-\begin{array}{rcl}
-  atomic\_formula & {::=}& element\_formula\\
-          & | & relation\_formula\\
-          & | & auxiliary\_formula\\ 
-  element\_formula
-          &{::=} &entity(id,attrs) \\
-          & |&  activity(id,st,et,attrs)\\
-          & |&  agent(id,attrs)\\
-  relation\_formula
-          &{::=}& wasGeneratedBy(id,e,a,t,attrs)\\
-          & |&  used(id,e,a,t,attrs)\\
-          & |&  wasInvalidatedBy(id,e,a,t,attrs)\\
-          & |&  wasStartedBy(id,a_2,e,a_1,attrs)\\
-          & |&  wasEndedBy(id,a_2,e,a_1,attrs)\\
-          & |&  wasAssociatedWith(id,ag,act,pl,attrs)\\
-          & |&  wasAssociatedWith(id,ag,act,-,attrs)\\
-          & |&  wasAttributedTo(id,e,ag,attrs)\\
-          & |&  actedOnBehalfOf(if,ag_2,ag_1,act,attrs)\\
-          & |& wasInformedBy(id,a_2,a_1,attrs)\\
-          & |&  wasDerivedFrom(id,e_2,e_1,act,g,u,attrs)\\
-          & |&  wasDerivedFrom(id,e_2,e_1,-,-,-,attrs)\\
-& | & wasInfluencedBy(id,x,y,attrs)\\
-          & |&  alternateOf(e_1,e_2)\\
-          & |&  specializationOf(e_1,e_2)\\
- & | & hadMember(c,e)\\
-  auxiliary\_formula
-          &{::=}& x = y\\
-          & | & x \prec y\\
-          & | & x \preceq y\\
-          & | & notNull(x)\\
-          & | & typeOf(x,ty)\\
-attrs &::=& [attr_1 = v_1, \ldots,attr_n = v_n]\\
- ty &{::=}& entity \\
-&|& activity\\
-&|& agent\\
-&|& prov:Collection\\
-&|& prov:EmptyCollection
-\end{array}
-</script>
-
-</section>
-
-<section id="first-order-formulas">
-<h3><span class="secno">2.5 </span>First-Order Formulas</h3>
-
-<p>We also consider the usual connectives and quantifiers of
-first-order logic [<cite><a href="#bib-Logic" class="bibref">Logic</a></cite>].
-<span class="MathJax_Preview"></span><div style="text-align: center;" aria-readonly="true" role="textbox" class="MathJax_Display"><span style="" id="MathJax-Element-19-Frame" class="MathJax"><nobr><span id="MathJax-Span-1281" class="math"><span style="display: inline-block; position: relative; width: 190px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(-4.148em, 1000em, 7.737em, -0.359em); top: -2.044em; left: 0em;"><span id="MathJax-Span-1282" class="mrow"><span style="padding-right: 0.167em; padding-left: 0.167em;" id="MathJax-Span-1283" class="mtable"><span style="display: inline-block; position: relative; width: 10.778em; height: 0px;"><span style="position: absolute; clip: rect(6.133em, 1000em, 17.695em, -0.526em); top: -12.208em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.643em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -7.787em; right: 0em;"><span id="MathJax-Span-1284" class="mtd"><span id="MathJax-Span-1285" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1286" class="mi">ϕ</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(3.797em, 1000em, 4.147em, -0.526em); top: -7.856em; right: 0em;"><span id="MathJax-Span-1308" class="mtd"><span id="MathJax-Span-1309" class="mrow"></span></span><span style="display: inline-block; width: 0px; height: 3.972em;"></span></span><span style="position: absolute; clip: rect(3.797em, 1000em, 4.147em, -0.526em); top: -6.656em; right: 0em;"><span id="MathJax-Span-1321" class="mtd"><span id="MathJax-Span-1322" class="mrow"></span></span><span style="display: inline-block; width: 0px; height: 3.972em;"></span></span><span style="position: absolute; clip: rect(3.797em, 1000em, 4.147em, -0.526em); top: -5.456em; right: 0em;"><span id="MathJax-Span-1335" class="mtd"><span id="MathJax-Span-1336" class="mrow"></span></span><span style="display: inline-block; width: 0px; height: 3.972em;"></span></span><span style="position: absolute; clip: rect(3.797em, 1000em, 4.147em, -0.526em); top: -4.118em; right: 0em;"><span id="MathJax-Span-1347" class="mtd"><span id="MathJax-Span-1348" class="mrow"></span></span><span style="display: inline-block; width: 0px; height: 3.972em;"></span></span><span style="position: absolute; clip: rect(3.797em, 1000em, 4.147em, -0.526em); top: -2.745em; right: 0em;"><span id="MathJax-Span-1363" class="mtd"><span id="MathJax-Span-1364" class="mrow"></span></span><span style="display: inline-block; width: 0px; height: 3.972em;"></span></span><span style="position: absolute; clip: rect(3.797em, 1000em, 4.147em, -0.526em); top: -1.371em; right: 0em;"><span id="MathJax-Span-1379" class="mtd"><span id="MathJax-Span-1380" class="mrow"></span></span><span style="display: inline-block; width: 0px; height: 3.972em;"></span></span><span style="position: absolute; clip: rect(3.797em, 1000em, 4.147em, -0.526em); top: 0.002em; right: 0em;"><span id="MathJax-Span-1395" class="mtd"><span id="MathJax-Span-1396" class="mrow"></span></span><span style="display: inline-block; width: 0px; height: 3.972em;"></span></span><span style="position: absolute; clip: rect(3.797em, 1000em, 4.147em, -0.526em); top: 1.34em; right: 0em;"><span id="MathJax-Span-1408" class="mtd"><span id="MathJax-Span-1409" class="mrow"></span></span><span style="display: inline-block; width: 0px; height: 3.972em;"></span></span></span><span style="display: inline-block; width: 0px; height: 12.208em;"></span></span><span style="position: absolute; clip: rect(6.123em, 1000em, 17.476em, -0.459em); top: -11.974em; left: 1.768em;"><span style="display: inline-block; position: relative; width: 1.285em; height: 0px;"><span style="position: absolute; clip: rect(1.936em, 1000em, 2.756em, -0.445em); top: -7.787em; left: 50%; margin-left: -0.643em;"><span id="MathJax-Span-1287" class="mtd"><span id="MathJax-Span-1288" class="mrow"><span id="MathJax-Span-1289" class="texatom"><span id="MathJax-Span-1290" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1291" class="mo">:<span style="font-family: STIXGeneral; font-style: normal; font-weight: normal;">:</span><span style="font-family: STIXGeneral; font-style: normal; font-weight: normal;">=</span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.719em, 1000em, 2.759em, -0.459em); top: -6.454em; left: 50%; margin-left: -0.146em;"><span id="MathJax-Span-1310" class="mtd"><span id="MathJax-Span-1311" class="mrow"><span id="MathJax-Span-1312" class="texatom"><span id="MathJax-Span-1313" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1314" class="mo">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.719em, 1000em, 2.759em, -0.459em); top: -5.254em; left: 50%; margin-left: -0.146em;"><span id="MathJax-Span-1323" class="mtd"><span id="MathJax-Span-1324" class="mrow"><span id="MathJax-Span-1325" class="texatom"><span id="MathJax-Span-1326" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1327" class="mo">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.719em, 1000em, 2.759em, -0.459em); top: -4.054em; left: 50%; margin-left: -0.146em;"><span id="MathJax-Span-1337" class="mtd"><span id="MathJax-Span-1338" class="mrow"><span id="MathJax-Span-1339" class="texatom"><span id="MathJax-Span-1340" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1341" class="mo">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.719em, 1000em, 2.759em, -0.459em); top: -2.716em; left: 50%; margin-left: -0.146em;"><span id="MathJax-Span-1349" class="mtd"><span id="MathJax-Span-1350" class="mrow"><span id="MathJax-Span-1351" class="texatom"><span id="MathJax-Span-1352" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1353" class="mo">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.719em, 1000em, 2.759em, -0.459em); top: -1.343em; left: 50%; margin-left: -0.146em;"><span id="MathJax-Span-1365" class="mtd"><span id="MathJax-Span-1366" class="mrow"><span id="MathJax-Span-1367" class="texatom"><span id="MathJax-Span-1368" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1369" class="mo">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.719em, 1000em, 2.759em, -0.459em); top: 0.031em; left: 50%; margin-left: -0.146em;"><span id="MathJax-Span-1381" class="mtd"><span id="MathJax-Span-1382" class="mrow"><span id="MathJax-Span-1383" class="texatom"><span id="MathJax-Span-1384" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1385" class="mo">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.719em, 1000em, 2.759em, -0.459em); top: 1.404em; left: 50%; margin-left: -0.146em;"><span id="MathJax-Span-1397" class="mtd"><span id="MathJax-Span-1398" class="mrow"><span id="MathJax-Span-1399" class="texatom"><span id="MathJax-Span-1400" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1401" class="mo">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.719em, 1000em, 2.759em, -0.459em); top: 2.742em; left: 50%; margin-left: -0.146em;"><span id="MathJax-Span-1410" class="mtd"><span id="MathJax-Span-1411" class="mrow"><span id="MathJax-Span-1412" class="texatom"><span id="MathJax-Span-1413" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1414" class="mo">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="display: inline-block; width: 0px; height: 11.974em;"></span></span><span style="position: absolute; clip: rect(6.366em, 1000em, 18.134em, -0.524em); top: -12.442em; left: 4.178em;"><span style="display: inline-block; position: relative; width: 6.6em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.509em); top: -7.787em; left: 0em;"><span id="MathJax-Span-1292" class="mtd"><span id="MathJax-Span-1293" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1294" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1295" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1296" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1297" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1298" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1299" class="mi">c</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1300" class="mi">_</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1301" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1302" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1303" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1304" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1305" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1306" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1307" class="mi">a</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.742em, 1000em, 2.756em, -0.467em); top: -6.454em; left: 0em;"><span id="MathJax-Span-1315" class="mtd"><span id="MathJax-Span-1316" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1317" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1318" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1319" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1320" class="mi">e</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.712em, 1000em, 2.758em, -0.518em); top: -5.254em; left: 0em;"><span id="MathJax-Span-1328" class="mtd"><span id="MathJax-Span-1329" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1330" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1331" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1332" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1333" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1334" class="mi">e</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.478em); top: -4.054em; left: 0em;"><span id="MathJax-Span-1342" class="mtd"><span id="MathJax-Span-1343" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1344" class="mi">¬</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1345" class="mtext">&nbsp;</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1346" class="mi">ϕ</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.712em, 1000em, 2.986em, -0.499em); top: -2.716em; left: 0em;"><span id="MathJax-Span-1354" class="mtd"><span id="MathJax-Span-1355" class="mrow"><span id="MathJax-Span-1356" class="msubsup"><span style="display: inline-block; position: relative; width: 1.068em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1357" class="mi">ϕ</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.862em; left: 0.643em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1358" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1359" class="mo">∧</span><span style="padding-left: 0.25em;" id="MathJax-Span-1360" class="msubsup"><span style="display: inline-block; position: relative; width: 1.068em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1361" class="mi">ϕ</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.862em; left: 0.643em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1362" class="mn">2</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.712em, 1000em, 2.986em, -0.499em); top: -1.343em; left: 0em;"><span id="MathJax-Span-1370" class="mtd"><span id="MathJax-Span-1371" class="mrow"><span id="MathJax-Span-1372" class="msubsup"><span style="display: inline-block; position: relative; width: 1.068em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1373" class="mi">ϕ</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.862em; left: 0.643em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1374" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1375" class="mo">∨</span><span style="padding-left: 0.25em;" id="MathJax-Span-1376" class="msubsup"><span style="display: inline-block; position: relative; width: 1.068em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1377" class="mi">ϕ</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.862em; left: 0.643em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1378" class="mn">2</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.712em, 1000em, 2.986em, -0.499em); top: 0.031em; left: 0em;"><span id="MathJax-Span-1386" class="mtd"><span id="MathJax-Span-1387" class="mrow"><span id="MathJax-Span-1388" class="msubsup"><span style="display: inline-block; position: relative; width: 1.068em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1389" class="mi">ϕ</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.862em; left: 0.643em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1390" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1391" class="mo">⇒</span><span style="padding-left: 0.313em;" id="MathJax-Span-1392" class="msubsup"><span style="display: inline-block; position: relative; width: 1.068em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1393" class="mi">ϕ</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.862em; left: 0.643em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1394" class="mn">2</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.524em); top: 1.404em; left: 0em;"><span id="MathJax-Span-1402" class="mtd"><span id="MathJax-Span-1403" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1404" class="mi">∀</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1405" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1406" class="mo">.</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1407" class="mi">ϕ</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.453em); top: 2.742em; left: 0em;"><span id="MathJax-Span-1415" class="mtd"><span id="MathJax-Span-1416" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1417" class="mi">∃</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1418" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1419" class="mo">.</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1420" class="mi">ϕ</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="display: inline-block; width: 0px; height: 12.442em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.044em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 12.466em; vertical-align: -5.966em;"></span></span></nobr></span></div><script id="MathJax-Element-19" type="math/tex; mode=display">
-\begin{array}{rcl}
-  \phi &{::=}& atomic\_formula\\
-& | & True\\
-& | & False\\
-&|& \neg~\phi\\
- &|& \phi_1 \wedge \phi_2\\
- &|& \phi_1 \vee \phi_2\\
- &|& \phi_1 \Rightarrow \phi_2\\
-&|& \forall x. \phi\\
-&|& \exists x. \phi\\
-\end{array}
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-
-
-</p></section>
-<!--
-<section>
-<h3>Instances as Theories</h3>
-<p>A PROV-DM
-instance can be viewed as a set of atomic formulas
-$\phi_1$...$\phi_n$, often called a <em>theory</em>.  A theory is
-viewed as a conjunction, that is, the overall instance is considered
-to hold in a given structure if each atomic formula in it holds.</p>
-</section>
--->
-
-
-</section>
-
-
-<section id="structures">
-<!--OddPage--><h2><span class="secno">3. </span> Structures and Interpretations </h2>
-
-<section id="things-1">
-
-<h3><span class="secno">3.1 </span> Things </h3> 
-
-<div class="note"><div class="note-title"><span>Note</span></div><div class="">TODO: Containment of things / collections? (for hadMember).</div></div>
-<p><em>Things</em> are things in the world.  Each thing has a lifetime during which it exists and attributes whose values can change over time.
-</p>
-<p>To model this, a structure <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-20-Frame" class="MathJax"><nobr><span id="MathJax-Span-1421" class="math"><span style="display: inline-block; position: relative; width: 16px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.763em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1422" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1423" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.843em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-20" type="math/tex">W</script> includes:
-</p>
-<div data-title="Component 1 (things)" data-count="1" class="component" id="things"><div class="ruleTitle"><a href="#things" class="internalDFN">Component 1 (things)</a></div><ol>
-  <li> a set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-21-Frame" class="MathJax"><nobr><span id="MathJax-Span-1424" class="math"><span style="display: inline-block; position: relative; width: 48px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.467em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1425" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1426" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1427" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1428" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1429" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1430" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1431" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-21" type="math/tex">Things</script> of things</li>
-  <li> a function <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-22-Frame" class="MathJax"><nobr><span id="MathJax-Span-1432" class="math"><span style="display: inline-block; position: relative; width: 207px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.485em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1433" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1434" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1435" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1436" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1437" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1438" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1439" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1440" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1441" class="mi">e</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1442" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1443" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1444" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1445" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1446" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1447" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1448" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1449" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1450" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1451" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1452" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1453" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1454" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1455" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1456" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1457" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1458" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-22" type="math/tex">lifetime : Things \to Intervals</script> from objects to time intervals</li>
-  <li>a function <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-23-Frame" class="MathJax"><nobr><span id="MathJax-Span-1459" class="math"><span style="display: inline-block; position: relative; width: 349px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1460" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1461" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1462" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1463" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1464" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1465" class="mi">e</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1466" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1467" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1468" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1469" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1470" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1471" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1472" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1473" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1474" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1475" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1476" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1477" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1478" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1479" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1480" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1481" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1482" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1483" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1484" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1485" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1486" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1487" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1488" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1489" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1490" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1491" class="mi">P</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1492" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1493" class="mi">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1494" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1495" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1496" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1497" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1498" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1499" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-23" type="math/tex">value : Things \times Attributes \times Times \to P(Values)</script>
-</li>
-</ol>
-</div>
-<p>
-The range of <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-24-Frame" class="MathJax"><nobr><span id="MathJax-Span-1500" class="math"><span style="display: inline-block; position: relative; width: 37px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.763em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1501" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1502" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1503" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1504" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1505" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1506" class="mi">e</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.875em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-24" type="math/tex">value</script> is the set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-25-Frame" class="MathJax"><nobr><span id="MathJax-Span-1507" class="math"><span style="display: inline-block; position: relative; width: 70px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.922em, -0.526em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1508" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1509" class="mi">P</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1510" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1511" class="mi">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1512" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1513" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1514" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1515" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1516" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1517" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.045em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-25" type="math/tex">P(Values)</script>, indicating that <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-26-Frame" class="MathJax"><nobr><span id="MathJax-Span-1518" class="math"><span style="display: inline-block; position: relative; width: 37px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.763em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1519" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1520" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1521" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1522" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1523" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1524" class="mi">e</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.875em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-26" type="math/tex">value</script>
-is essentially a multi-valued that returns a set of values (possibly empty).    When <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-27-Frame" class="MathJax"><nobr><span id="MathJax-Span-1525" class="math"><span style="display: inline-block; position: relative; width: 116px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.666em, 1000em, 2.922em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1526" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1527" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1528" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1529" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1530" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1531" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1532" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1533" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1534" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1535" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1536" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1537" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1538" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1539" class="mo">=</span><span style="font-family: STIXVariants; padding-left: 0.313em;" id="MathJax-Span-1540" class="mi">∅</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.094em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-27" type="math/tex">value(x,a,t) =
-\emptyset</script>, we say that attribute <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-28-Frame" class="MathJax"><nobr><span id="MathJax-Span-1541" class="math"><span style="display: inline-block; position: relative; width: 9px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1542" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1543" class="mi">a</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.609em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-28" type="math/tex">a</script> is undefined for <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-29-Frame" class="MathJax"><nobr><span id="MathJax-Span-1544" class="math"><span style="display: inline-block; position: relative; width: 8px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.553em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1545" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1546" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.609em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-29" type="math/tex">x</script> at time <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-30-Frame" class="MathJax"><nobr><span id="MathJax-Span-1547" class="math"><span style="display: inline-block; position: relative; width: 5px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1548" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1549" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.721em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-30" type="math/tex">t</script>.</p>
-
-<p>Note that this description does not say what the structure of a
-<span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-31-Frame" class="MathJax"><nobr><span id="MathJax-Span-1550" class="math"><span style="display: inline-block; position: relative; width: 41px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.467em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1551" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1552" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1553" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1554" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1555" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1556" class="mi">g</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-31" type="math/tex">Thing</script> is, only how it may be described in terms of its time interval
-and attribute values.  An object could just be a record of fixed
-attribute values; it could be a bear; it could be the Royal Society;
-it could be a transcendental number like <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-32-Frame" class="MathJax"><nobr><span id="MathJax-Span-1557" class="math"><span style="display: inline-block; position: relative; width: 9px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.967em, 1000em, 2.763em, -0.507em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1558" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1559" class="mi">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.602em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-32" type="math/tex">\pi</script>.  All that matters from
-our point of view is that we know how to map the <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-33-Frame" class="MathJax"><nobr><span id="MathJax-Span-1560" class="math"><span style="display: inline-block; position: relative; width: 41px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.467em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1561" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1562" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1563" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1564" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1565" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1566" class="mi">g</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-33" type="math/tex">Thing</script> to its time interval and attribute mapping.
-</p>
-
-
-<p>It is possible for two Things to be indistinguishable by their attribute values and lifetime, but have different identity.
-</p>
-
-
-</section>
-
-<section id="objects-1">
-<h3><span class="secno">3.2 </span> Objects </h3>
-
-<p>
-An <em>Object</em> is described by a time interval and attributes with unchanging values.  Objects encompass entities, interactions, and activities.
-To model this, a structure includes:
-</p>
-
-<div data-title="Component 2 (objects)" data-count="2" class="component" id="objects"><div class="ruleTitle"><a href="#objects" class="internalDFN">Component 2 (objects)</a></div>
-  <ol><li> a set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-34-Frame" class="MathJax"><nobr><span id="MathJax-Span-1567" class="math"><span style="display: inline-block; position: relative; width: 53px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.466em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1568" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1569" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1570" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1571" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1572" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1573" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1574" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1575" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-34" type="math/tex">Objects</script> 
-</li><li> a function <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-35-Frame" class="MathJax"><nobr><span id="MathJax-Span-1576" class="math"><span style="display: inline-block; position: relative; width: 212px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.485em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1577" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1578" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1579" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1580" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1581" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1582" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1583" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1584" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1585" class="mi">e</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1586" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1587" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1588" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1589" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1590" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1591" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1592" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1593" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1594" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1595" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1596" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1597" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1598" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1599" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1600" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1601" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1602" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1603" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-35" type="math/tex">lifetime : Objects \to Intervals</script> from objects to time intervals
-</li><li> a function <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-36-Frame" class="MathJax"><nobr><span id="MathJax-Span-1604" class="math"><span style="display: inline-block; position: relative; width: 291px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1605" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1606" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1607" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1608" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1609" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1610" class="mi">e</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1611" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1612" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1613" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1614" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1615" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1616" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1617" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1618" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1619" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1620" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1621" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1622" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1623" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1624" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1625" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1626" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1627" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1628" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1629" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1630" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1631" class="mi">P</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1632" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1633" class="mi">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1634" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1635" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1636" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1637" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1638" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1639" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-36" type="math/tex">value : Objects \times Attributes \to P(Values)</script>
-</li></ol>
-</div>
-
-<p>Intuitively, <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-37-Frame" class="MathJax"><nobr><span id="MathJax-Span-1640" class="math"><span style="display: inline-block; position: relative; width: 73px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.485em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1641" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1642" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1643" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1644" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1645" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1646" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1647" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1648" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1649" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1650" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1651" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1652" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-37" type="math/tex">lifetime(e)</script> is the time interval during which object
-<span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-38-Frame" class="MathJax"><nobr><span id="MathJax-Span-1653" class="math"><span style="display: inline-block; position: relative; width: 8px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.495em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1654" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1655" class="mi">e</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.609em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-38" type="math/tex">e</script> exists.  The set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-39-Frame" class="MathJax"><nobr><span id="MathJax-Span-1656" class="math"><span style="display: inline-block; position: relative; width: 72px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.922em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1657" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1658" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1659" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1660" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1661" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1662" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1663" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1664" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1665" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1666" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1667" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.045em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-39" type="math/tex">value(e,a)</script> is the set of values of attribute <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-40-Frame" class="MathJax"><nobr><span id="MathJax-Span-1668" class="math"><span style="display: inline-block; position: relative; width: 9px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1669" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1670" class="mi">a</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.609em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-40" type="math/tex">a</script> during the object's lifetime.
-</p>
-
-<p>As with <em>Things</em>, the range of <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-41-Frame" class="MathJax"><nobr><span id="MathJax-Span-1671" class="math"><span style="display: inline-block; position: relative; width: 37px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.763em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1672" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1673" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1674" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1675" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1676" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1677" class="mi">e</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.875em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-41" type="math/tex">value</script> is sets of values,
-making <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-42-Frame" class="MathJax"><nobr><span id="MathJax-Span-1678" class="math"><span style="display: inline-block; position: relative; width: 37px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.763em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1679" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1680" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1681" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1682" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1683" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1684" class="mi">e</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.875em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-42" type="math/tex">value</script> effectively a multivalued function.  It is also possible to have two different objects that are indistinguishable by their attributes and time intervals.  Objects are not things, and the sets of <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-43-Frame" class="MathJax"><nobr><span id="MathJax-Span-1685" class="math"><span style="display: inline-block; position: relative; width: 53px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.466em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1686" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1687" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1688" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1689" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1690" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1691" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1692" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1693" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-43" type="math/tex">Objects</script> and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-44-Frame" class="MathJax"><nobr><span id="MathJax-Span-1694" class="math"><span style="display: inline-block; position: relative; width: 48px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.467em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1695" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1696" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1697" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1698" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1699" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1700" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1701" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-44" type="math/tex">Things</script> are disjoint; however, certain objects, namely entities, are linked to things.
-
-</p>
-<section id="entities-1">
-<h4><span class="secno">3.2.1 </span> Entities </h4>
-
-An <em>entity</em> is a kind of object that describes a time-slice of a thing, during which some of the thing's attributes are fixed. We assume:
-
-<div data-title="Component 3 (entities)" data-count="3" class="component" id="entities"><div class="ruleTitle"><a href="#entities" class="internalDFN">Component 3 (entities)</a></div>
-  <ol><li> a set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-45-Frame" class="MathJax"><nobr><span id="MathJax-Span-1702" class="math"><span style="display: inline-block; position: relative; width: 128px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.527em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1703" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1704" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1705" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1706" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1707" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1708" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1709" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1710" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1711" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1712" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1713" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1714" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1715" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1716" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1717" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1718" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1719" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-45" type="math/tex">Entities \subseteq Objects</script> of entities, disjoint from <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-46-Frame" class="MathJax"><nobr><span id="MathJax-Span-1720" class="math"><span style="display: inline-block; position: relative; width: 64px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.727em, 1000em, 2.763em, -0.577em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1721" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1722" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1723" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1724" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1725" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1726" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1727" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1728" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1729" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1730" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1731" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.859em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-46" type="math/tex">Activities</script> and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-47-Frame" class="MathJax"><nobr><span id="MathJax-Span-1732" class="math"><span style="display: inline-block; position: relative; width: 46px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.763em, -0.527em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1733" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1734" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1735" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1736" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1737" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1738" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1739" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.843em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-47" type="math/tex">Events</script> below.
-</li><li> a function <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-48-Frame" class="MathJax"><nobr><span id="MathJax-Span-1740" class="math"><span style="display: inline-block; position: relative; width: 199px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1741" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1742" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1743" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1744" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1745" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1746" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1747" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1748" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1749" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1750" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1751" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1752" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1753" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1754" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1755" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1756" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1757" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1758" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1759" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1760" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1761" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1762" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1763" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1764" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-48" type="math/tex">thingOf : Entities \to Things</script> that associates each <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-49-Frame" class="MathJax"><nobr><span id="MathJax-Span-1765" class="math"><span style="display: inline-block; position: relative; width: 42px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.741em, 1000em, 2.951em, -0.527em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1766" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1767" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1768" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1769" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1770" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1771" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1772" class="mi">y</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.045em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-49" type="math/tex">Entity</script> with a <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-50-Frame" class="MathJax"><nobr><span id="MathJax-Span-1773" class="math"><span style="display: inline-block; position: relative; width: 41px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.467em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1774" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1775" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1776" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1777" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1778" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1779" class="mi">g</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-50" type="math/tex">Thing</script>, such that for each <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-51-Frame" class="MathJax"><nobr><span id="MathJax-Span-1780" class="math"><span style="display: inline-block; position: relative; width: 115px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1781" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1782" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1783" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1784" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1785" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1786" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1787" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1788" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1789" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1790" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1791" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1792" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1793" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1794" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1795" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1796" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-51" type="math/tex">t \in lifetime(obj)</script>, and for each attribute <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-52-Frame" class="MathJax"><nobr><span id="MathJax-Span-1797" class="math"><span style="display: inline-block; position: relative; width: 9px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1798" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1799" class="mi">a</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.609em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-52" type="math/tex">a</script> we have <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-53-Frame" class="MathJax"><nobr><span id="MathJax-Span-1800" class="math"><span style="display: inline-block; position: relative; width: 274px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1801" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1802" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1803" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1804" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1805" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1806" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1807" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1808" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1809" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1810" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1811" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1812" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1813" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1814" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1815" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1816" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1817" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1818" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1819" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1820" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1821" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1822" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1823" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1824" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1825" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1826" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1827" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1828" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1829" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1830" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1831" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1832" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1833" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1834" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1835" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1836" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1837" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-53" type="math/tex">value(obj,a) \subseteq value(thingOf(obj),a,t)</script>.
-</li><li> <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-54-Frame" class="MathJax"><nobr><span id="MathJax-Span-1838" class="math"><span style="display: inline-block; position: relative; width: 166px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.485em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1839" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1840" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1841" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1842" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1843" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1844" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1845" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1846" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1847" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1848" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1849" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1850" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1851" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1852" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1853" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1854" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1855" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1856" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1857" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1858" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1859" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1860" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1861" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1862" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-54" type="math/tex">lifetime(e) \subseteq lifetime(t)</script>.
-</li></ol>
-</div>
-
-<div class="remark"><p> Although both entities and things can have
-  undefined or multiple attribute values, their meaning is slightly
-  different: for a thing, <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-55-Frame" class="MathJax"><nobr><span id="MathJax-Span-1863" class="math"><span style="display: inline-block; position: relative; width: 116px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.666em, 1000em, 2.922em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1864" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1865" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1866" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1867" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1868" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1869" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1870" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1871" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1872" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1873" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1874" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1875" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1876" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1877" class="mo">=</span><span style="font-family: STIXVariants; padding-left: 0.313em;" id="MathJax-Span-1878" class="mi">∅</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.094em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-55" type="math/tex">value(x,a,t) = \emptyset</script> means that the
-  attribute <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-56-Frame" class="MathJax"><nobr><span id="MathJax-Span-1879" class="math"><span style="display: inline-block; position: relative; width: 9px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1880" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1881" class="mi">a</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.609em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-56" type="math/tex">a</script> has no value at time <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-57-Frame" class="MathJax"><nobr><span id="MathJax-Span-1882" class="math"><span style="display: inline-block; position: relative; width: 5px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1883" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1884" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.721em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-57" type="math/tex">t</script>, whereas for an entity,
-  <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-58-Frame" class="MathJax"><nobr><span id="MathJax-Span-1885" class="math"><span style="display: inline-block; position: relative; width: 103px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.666em, 1000em, 2.922em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1886" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1887" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1888" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1889" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1890" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1891" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1892" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1893" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1894" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1895" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1896" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1897" class="mo">=</span><span style="font-family: STIXVariants; padding-left: 0.313em;" id="MathJax-Span-1898" class="mi">∅</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.094em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-58" type="math/tex">value(x,a) = \emptyset</script> only means that the entity does not record
-  a fixed value for <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-59-Frame" class="MathJax"><nobr><span id="MathJax-Span-1899" class="math"><span style="display: inline-block; position: relative; width: 9px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1900" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1901" class="mi">a</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.609em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-59" type="math/tex">a</script> forthe associated thing.  This does not imply
-  that <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-60-Frame" class="MathJax"><nobr><span id="MathJax-Span-1902" class="math"><span style="display: inline-block; position: relative; width: 182px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.666em, 1000em, 2.952em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1903" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1904" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1905" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1906" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1907" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1908" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1909" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1910" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1911" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1912" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1913" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1914" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1915" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1916" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1917" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1918" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1919" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1920" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1921" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1922" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1923" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1924" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1925" class="mo">=</span><span style="font-family: STIXVariants; padding-left: 0.313em;" id="MathJax-Span-1926" class="mi">∅</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.127em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-60" type="math/tex">value(thingOf(e),a,t) = \emptyset</script> when <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-61-Frame" class="MathJax"><nobr><span id="MathJax-Span-1927" class="math"><span style="display: inline-block; position: relative; width: 100px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1928" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1929" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1930" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1931" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1932" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1933" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1934" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1935" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1936" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1937" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1938" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1939" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1940" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1941" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-61" type="math/tex">t \in lifetime(e)</script>.
-</p>
-  <p>Furthermore, all of the attribute values of the entity must be
-  present in the associated thing throughout the lifetime of the entity.
-For example, suppose <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-62-Frame" class="MathJax"><nobr><span id="MathJax-Span-1942" class="math"><span style="display: inline-block; position: relative; width: 151px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1943" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1944" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1945" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1946" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1947" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1948" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1949" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1950" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1951" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1952" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1953" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1954" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1955" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1956" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1957" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1958" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1959" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1960" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1961" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1962" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1963" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1964" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-62" type="math/tex">value(thingOf(e),a,t)</script> is <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-63-Frame" class="MathJax"><nobr><span id="MathJax-Span-1965" class="math"><span style="display: inline-block; position: relative; width: 22px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.715em, 1000em, 2.926em, -0.426em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1966" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1967" class="mo">{</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1968" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1969" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.046em; vertical-align: -0.256em;"></span></span></nobr></span><script id="MathJax-Element-63" type="math/tex">\{1\}</script> at some
-time in <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-64-Frame" class="MathJax"><nobr><span id="MathJax-Span-1970" class="math"><span style="display: inline-block; position: relative; width: 73px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.485em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1971" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1972" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1973" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1974" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1975" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1976" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1977" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1978" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1979" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1980" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1981" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1982" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-64" type="math/tex">lifetime(e)</script> and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-65-Frame" class="MathJax"><nobr><span id="MathJax-Span-1983" class="math"><span style="display: inline-block; position: relative; width: 202px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.632em, 1000em, 2.952em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1984" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1985" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1986" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1987" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1988" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1989" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1990" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1991" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1992" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1993" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1994" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1995" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1996" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1997" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1998" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1999" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2000" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2001" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2002" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2003" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-2004" class="msup"><span style="display: inline-block; position: relative; width: 0.656em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2005" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -2.466em; left: 0.347em;"><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-2006" class="mo">′</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2007" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2008" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2009" class="mo">{</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2010" class="mn">2</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2011" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.162em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-65" type="math/tex">value(thingOf(e),a,t') = \{2\}</script> at some
-  other time <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-66-Frame" class="MathJax"><nobr><span id="MathJax-Span-2012" class="math"><span style="display: inline-block; position: relative; width: 11px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.107em, 1000em, 2.231em, -0.488em); top: -2.044em; left: 0em;"><span id="MathJax-Span-2013" class="mrow"><span id="MathJax-Span-2014" class="msup"><span style="display: inline-block; position: relative; width: 0.656em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2015" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -2.466em; left: 0.347em;"><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-2016" class="mo">′</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.044em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.953em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-66" type="math/tex">t'</script>.  Then <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-67-Frame" class="MathJax"><nobr><span id="MathJax-Span-2017" class="math"><span style="display: inline-block; position: relative; width: 72px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.922em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2018" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2019" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2020" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2021" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2022" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2023" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2024" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2025" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2026" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2027" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2028" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.045em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-67" type="math/tex">value(e,a)</script> must be <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-68-Frame" class="MathJax"><nobr><span id="MathJax-Span-2029" class="math"><span style="display: inline-block; position: relative; width: 9px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.666em, 1000em, 2.819em, -0.498em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2030" class="mrow"><span style="font-family: STIXVariants;" id="MathJax-Span-2031" class="mi">∅</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.984em; vertical-align: -0.142em;"></span></span></nobr></span><script id="MathJax-Element-68" type="math/tex">\emptyset</script> because
-  there is no other set of values that is simultaneously contained in
-  both <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-69-Frame" class="MathJax"><nobr><span id="MathJax-Span-2032" class="math"><span style="display: inline-block; position: relative; width: 22px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.715em, 1000em, 2.926em, -0.426em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2033" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2034" class="mo">{</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2035" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2036" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.046em; vertical-align: -0.256em;"></span></span></nobr></span><script id="MathJax-Element-69" type="math/tex">\{1\}</script> and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-70-Frame" class="MathJax"><nobr><span id="MathJax-Span-2037" class="math"><span style="display: inline-block; position: relative; width: 22px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.715em, 1000em, 2.926em, -0.426em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2038" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2039" class="mo">{</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2040" class="mn">2</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2041" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.046em; vertical-align: -0.256em;"></span></span></nobr></span><script id="MathJax-Element-70" type="math/tex">\{2\}</script>.
-</p>
-  </div>
-
-<section id="plans-1">  
-<h5><span class="secno">3.2.1.1 </span> Plans </h5>
-<p>We identify a specific subset of the entities called
-  <em>plans</em>:
-</p><div data-title="Component 4 (plans)" data-count="4" class="component" id="plans"><div class="ruleTitle"><a href="#plans" class="internalDFN">Component 4 (plans)</a></div>
- <p> A set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-71-Frame" class="MathJax"><nobr><span id="MathJax-Span-2042" class="math"><span style="display: inline-block; position: relative; width: 115px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.848em, -0.526em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2043" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2044" class="mi">P</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2045" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2046" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2047" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2048" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2049" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2050" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2051" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2052" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2053" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2054" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2055" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2056" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2057" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.966em; vertical-align: -0.173em;"></span></span></nobr></span><script id="MathJax-Element-71" type="math/tex">Plans \subseteq Entities</script> of plans.</p>
-  </div>
-<p></p></section>
-
-  </section>
-
-    <section id="actvities">
-<h4><span class="secno">3.2.2 </span> Actvities </h4>
-
-
-<p>An <em>activity</em> is an object that encompasses a set of events.  We introduce
-</p>
-<div data-title="Component 5 (activities)" data-count="5" class="component" id="activities"><div class="ruleTitle"><a href="#activities" class="internalDFN">Component 5 (activities)</a></div>
-  <ol><li>A set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-72-Frame" class="MathJax"><nobr><span id="MathJax-Span-2058" class="math"><span style="display: inline-block; position: relative; width: 139px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.577em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2059" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2060" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2061" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2062" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2063" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2064" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2065" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2066" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2067" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2068" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2069" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2070" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2071" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2072" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2073" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2074" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2075" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2076" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2077" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-72" type="math/tex">Activities \subseteq Objects</script> of activities.</li>
-  <li> Activities are disjoint from Entities: <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-73-Frame" class="MathJax"><nobr><span id="MathJax-Span-2078" class="math"><span style="display: inline-block; position: relative; width: 168px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.666em, 1000em, 2.819em, -0.527em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2079" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2080" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2081" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2082" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2083" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2084" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2085" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2086" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2087" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2088" class="mo">∩</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-2089" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2090" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2091" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2092" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2093" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2094" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2095" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2096" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2097" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2098" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2099" class="mo">=</span><span style="font-family: STIXVariants; padding-left: 0.313em;" id="MathJax-Span-2100" class="mi">∅</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.984em; vertical-align: -0.142em;"></span></span></nobr></span><script id="MathJax-Element-73" type="math/tex">Entities\cap Activities
-  = \emptyset</script>.</li>
-  </ol>
-</div></section>
-  
-  <section id="agents-1">
-<h4><span class="secno">3.2.3 </span> Agents </h4>
-
-<p>An agent is an object that can act, by controlling, starting,
-  ending, or participating in activities.  Agents can act on behalf of
-  other agents. An agent can be an entity, an activity, or neither; an
-  agent cannot be both entity and activity because the sets of
-  entities and activities are disjoint.  We introduce:
-</p>
-<div data-title="Component 6 (agents)" data-count="6" class="component" id="agents"><div class="ruleTitle"><a href="#agents" class="internalDFN">Component 6 (agents)</a></div>
-  <p>A set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-74-Frame" class="MathJax"><nobr><span id="MathJax-Span-2101" class="math"><span style="display: inline-block; position: relative; width: 121px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.577em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2102" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2103" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2104" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2105" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2106" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2107" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2108" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2109" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2110" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2111" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2112" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2113" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2114" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2115" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2116" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-74" type="math/tex">Agents \subseteq Objects</script> of agents.</p>
-  </div>
-</section>
-
-
-<section id="interactions-1">
-<h4><span class="secno">3.2.4 </span> Interactions </h4>
-
-<p>We consider a set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-75-Frame" class="MathJax"><nobr><span id="MathJax-Span-2117" class="math"><span style="display: inline-block; position: relative; width: 160px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2118" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2119" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2120" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2121" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2122" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2123" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2124" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2125" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2126" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2127" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2128" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2129" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2130" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2131" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2132" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2133" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2134" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2135" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2136" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2137" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2138" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-75" type="math/tex">Interactions \subseteq Objects</script> which are split into
-  <em>Events</em> connecting entities and activities,
-  <em>Associations</em> between agents and activities,
-  <em>Communications</em> between pairs of activities,
-  <em>Delegations</em> between pairs of agents, and <em>Derivations</em> that describe chains of generation and usage steps.  (The first two sets may overlap.)  Interactions are disjoint from entities, activities and agents.
-</p>
-<div data-title="Component 7 (interactions)" data-count="7" class="component" id="interactions"><div class="ruleTitle"><a href="#interactions" class="internalDFN">Component 7 (interactions)</a></div>
-  <ol><li> A set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-76-Frame" class="MathJax"><nobr><span id="MathJax-Span-2139" class="math"><span style="display: inline-block; position: relative; width: 672px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2140" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2141" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2142" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2143" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2144" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2145" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2146" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2147" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2148" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2149" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2150" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2151" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2152" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2153" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2154" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2155" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2156" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2157" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2158" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2159" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2160" class="mo">∪</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-2161" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2162" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2163" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2164" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2165" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2166" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2167" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2168" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2169" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2170" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2171" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2172" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2173" class="mo">∪</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-2174" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2175" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2176" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2177" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2178" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2179" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2180" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2181" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2182" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2183" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2184" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2185" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2186" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2187" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2188" class="mo">∪</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-2189" class="mi">D</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2190" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2191" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2192" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2193" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2194" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2195" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2196" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2197" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2198" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2199" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2200" class="mo">∪</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-2201" class="mi">D</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2202" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2203" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2204" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2205" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2206" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2207" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2208" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2209" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2210" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2211" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2212" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2213" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2214" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2215" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2216" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2217" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2218" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2219" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-76" type="math/tex">Interactions = Events \cup Associations \cup
-  Communications \cup Delegations \cup Derivations \subseteq Objects</script>
-</li>
-<li> A function <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-77-Frame" class="MathJax"><nobr><span id="MathJax-Span-2220" class="math"><span style="display: inline-block; position: relative; width: 939px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.954em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2221" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2222" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2223" class="mi">y</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2224" class="mi">p</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2225" class="mi">e</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2226" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2227" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2228" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2229" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2230" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2231" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2232" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2233" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2234" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2235" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2236" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2237" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2238" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2239" class="mo">→</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2240" class="mo">{</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2241" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2242" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2243" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2244" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2245" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2246" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2247" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2248" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2249" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2250" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2251" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2252" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2253" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2254" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2255" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2256" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2257" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2258" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2259" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2260" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2261" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2262" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2263" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2264" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2265" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2266" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2267" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2268" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2269" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2270" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2271" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2272" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2273" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2274" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2275" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2276" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2277" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2278" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2279" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2280" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2281" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2282" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2283" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2284" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2285" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2286" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2287" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2288" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2289" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2290" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2291" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2292" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2293" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2294" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2295" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2296" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2297" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2298" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2299" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2300" class="mi">q</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2301" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2302" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2303" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2304" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2305" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2306" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2307" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2308" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2309" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2310" class="mi">p</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2311" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2312" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2313" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2314" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2315" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2316" class="mi">y</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2317" class="mi">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.008em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2318" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2319" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2320" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2321" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2322" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2323" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2324" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2325" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2326" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2327" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2328" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2329" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2330" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2331" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2332" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2333" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2334" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2335" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2336" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2337" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2338" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2339" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2340" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2341" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2342" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2343" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2344" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2345" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2346" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.079em; vertical-align: -0.286em;"></span></span></nobr></span><script id="MathJax-Element-77" type="math/tex">type: Interactions \to
-  \{start,end,usage,generation,invalidation,derivation,version,quotation,primarySource,attribution,delegation\}</script>.
-</li>
-<li> The sets <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-78-Frame" class="MathJax"><nobr><span id="MathJax-Span-2347" class="math"><span style="display: inline-block; position: relative; width: 46px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.763em, -0.527em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2348" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2349" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2350" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2351" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2352" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2353" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2354" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.843em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-78" type="math/tex">Events</script>, <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-79-Frame" class="MathJax"><nobr><span id="MathJax-Span-2355" class="math"><span style="display: inline-block; position: relative; width: 87px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.727em, 1000em, 2.758em, -0.577em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2356" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2357" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2358" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2359" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2360" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2361" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2362" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2363" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2364" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2365" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2366" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2367" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2368" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.854em; vertical-align: -0.076em;"></span></span></nobr></span><script id="MathJax-Element-79" type="math/tex">Associations</script>, <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-80-Frame" class="MathJax"><nobr><span id="MathJax-Span-2369" class="math"><span style="display: inline-block; position: relative; width: 117px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.729em, 1000em, 2.763em, -0.46em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2370" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2371" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2372" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2373" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2374" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2375" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2376" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2377" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2378" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2379" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2380" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2381" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2382" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2383" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2384" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.857em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-80" type="math/tex">Communications</script>, <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-81-Frame" class="MathJax"><nobr><span id="MathJax-Span-2385" class="math"><span style="display: inline-block; position: relative; width: 83px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2386" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2387" class="mi">D</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2388" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2389" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2390" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2391" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2392" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2393" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2394" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2395" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2396" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2397" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-81" type="math/tex">Delegations</script>
-  and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-82-Frame" class="MathJax"><nobr><span id="MathJax-Span-2398" class="math"><span style="display: inline-block; position: relative; width: 81px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.741em, 1000em, 2.763em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2399" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2400" class="mi">D</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2401" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2402" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2403" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2404" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2405" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2406" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2407" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2408" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2409" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2410" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.844em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-82" type="math/tex">Derivations</script> are all disjoint.
-</li><li> Interactions are disjoint from entities, agents and
-activities:  <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-83-Frame" class="MathJax"><nobr><span id="MathJax-Span-2411" class="math"><span style="display: inline-block; position: relative; width: 348px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.666em, 1000em, 2.951em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2412" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2413" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2414" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2415" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2416" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2417" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2418" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2419" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2420" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2421" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2422" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2423" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2424" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2425" class="mo">∩</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2426" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2427" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2428" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2429" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2430" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2431" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2432" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2433" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2434" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2435" class="mo">∪</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-2436" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2437" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2438" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2439" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2440" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2441" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2442" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2443" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2444" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2445" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2446" class="mo">∪</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-2447" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2448" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2449" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2450" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2451" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2452" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2453" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2454" class="mo">=</span><span style="font-family: STIXVariants; padding-left: 0.313em;" id="MathJax-Span-2455" class="mi">∅</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.125em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-83" type="math/tex">Interactions \cap (Entities \cup Activities \cup Agents) = \emptyset</script>
-</li></ol>
-</div>
-
-<section id="events-1">
-<h5><span class="secno">3.2.4.1 </span> Events </h5>
-
-<p>An <em>Event</em> is an interaction whose lifetime is a single time
-instant, and relates an activity to an entity (which could be an
-agent).  Events have types including usage, generation, invalidation, starting and ending.  Events are instantaneous.  We introduce:
-</p>
-<div data-title="Component 8 (events)" data-count="8" class="component" id="events"><div class="ruleTitle"><a href="#events" class="internalDFN">Component 8 (events)</a></div>
-<ol><li> A set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-84-Frame" class="MathJax"><nobr><span id="MathJax-Span-2456" class="math"><span style="display: inline-block; position: relative; width: 153px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.741em, 1000em, 2.848em, -0.527em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2457" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2458" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2459" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2460" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2461" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2462" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2463" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2464" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2465" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2466" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2467" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2468" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2469" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2470" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2471" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2472" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2473" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2474" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2475" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2476" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.935em; vertical-align: -0.173em;"></span></span></nobr></span><script id="MathJax-Element-84" type="math/tex">Events \subseteq Interactions</script> of events, such that
-  <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-85-Frame" class="MathJax"><nobr><span id="MathJax-Span-2477" class="math"><span style="display: inline-block; position: relative; width: 381px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2478" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2479" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2480" class="mi">y</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2481" class="mi">p</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2482" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2483" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2484" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2485" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2486" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2487" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2488" class="mo">∈</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2489" class="mo">{</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2490" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2491" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2492" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2493" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2494" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2495" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2496" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2497" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2498" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2499" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2500" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2501" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2502" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2503" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2504" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2505" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2506" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2507" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2508" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2509" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2510" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2511" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2512" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2513" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2514" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2515" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2516" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2517" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2518" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2519" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2520" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2521" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2522" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2523" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2524" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2525" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2526" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2527" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2528" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2529" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-85" type="math/tex">type(evt) \in \{start,end,generation,usage,invalidation\}</script> if and
-  only of <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-86-Frame" class="MathJax"><nobr><span id="MathJax-Span-2530" class="math"><span style="display: inline-block; position: relative; width: 89px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.772em, -0.495em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2531" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2532" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2533" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2534" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2535" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2536" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2537" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2538" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2539" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2540" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2541" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.853em; vertical-align: -0.091em;"></span></span></nobr></span><script id="MathJax-Element-86" type="math/tex">evt \in Events</script>.
-</li><li> A function <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-87-Frame" class="MathJax"><nobr><span id="MathJax-Span-2542" class="math"><span style="display: inline-block; position: relative; width: 161px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.741em, 1000em, 2.763em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2543" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2544" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2545" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2546" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2547" class="mi">e</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2548" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2549" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2550" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2551" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2552" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2553" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2554" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2555" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2556" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2557" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2558" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2559" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2560" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.844em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-87" type="math/tex">time : Events \to Times</script> giving the time of each event; i.e. <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-88-Frame" class="MathJax"><nobr><span id="MathJax-Span-2561" class="math"><span style="display: inline-block; position: relative; width: 168px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.485em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2562" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2563" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2564" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2565" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2566" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2567" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2568" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2569" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2570" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2571" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2572" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2573" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2574" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2575" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2576" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2577" class="mo">{</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2578" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2579" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2580" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2581" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2582" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2583" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2584" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2585" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-88" type="math/tex">lifetime(evt) = \{time(t)\}</script>.
-</li><li> The derived ordering on events given by <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-89-Frame" class="MathJax"><nobr><span id="MathJax-Span-2586" class="math"><span style="display: inline-block; position: relative; width: 284px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.719em, 1000em, 2.922em, -0.495em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2587" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2588" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2589" class="mi">v</span><span id="MathJax-Span-2590" class="msubsup"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2591" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.274em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2592" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2593" class="mo">≤</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2594" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2595" class="mi">v</span><span id="MathJax-Span-2596" class="msubsup"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2597" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.274em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2598" class="mn">2</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="height: 0em; vertical-align: 0em; width: 0.313em; display: inline-block; overflow: hidden;" id="MathJax-Span-2599" class="mspace"></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2600" class="mo">⟺</span><span style="height: 0em; vertical-align: 0em; width: 0.313em; display: inline-block; overflow: hidden;" id="MathJax-Span-2601" class="mspace"></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2602" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2603" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2604" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2605" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2606" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2607" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2608" class="mi">v</span><span id="MathJax-Span-2609" class="msubsup"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2610" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.274em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2611" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2612" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2613" class="mo">≤</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2614" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2615" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2616" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2617" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2618" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2619" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2620" class="mi">v</span><span id="MathJax-Span-2621" class="msubsup"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2622" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.274em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-2623" class="mn">2</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2624" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.038em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-89" type="math/tex">evt_1 \leq evt_2 \iff time(evt_1) \leq time(evt_2)</script>
-</li></ol>
-</div>
-</section>
-<section id="associations-1">
-
-<h5><span class="secno">3.2.4.2 </span> Associations </h5>
-
-<p>An <em>Association</em> is an interaction relating an agent to an activity.  To model associations, we introduce:
-</p>
-<div data-title="Component 9 (associations)" data-count="9" class="component" id="associations"><div class="ruleTitle"><a href="#associations" class="internalDFN">Component 9 (associations)</a></div>
-  <p>A set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-90-Frame" class="MathJax"><nobr><span id="MathJax-Span-2625" class="math"><span style="display: inline-block; position: relative; width: 194px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.727em, 1000em, 2.848em, -0.577em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2626" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2627" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2628" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2629" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2630" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2631" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2632" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2633" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2634" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2635" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2636" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2637" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2638" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2639" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2640" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2641" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2642" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2643" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2644" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2645" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2646" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2647" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2648" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2649" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2650" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2651" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.95em; vertical-align: -0.173em;"></span></span></nobr></span><script id="MathJax-Element-90" type="math/tex">Associations \subseteq Interactions</script>, such that
-  <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-91-Frame" class="MathJax"><nobr><span id="MathJax-Span-2652" class="math"><span style="display: inline-block; position: relative; width: 179px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.719em, 1000em, 2.951em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2653" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2654" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2655" class="mi">y</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2656" class="mi">p</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2657" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2658" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2659" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2660" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2661" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2662" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2663" class="mi">c</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2664" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2665" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2666" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2667" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2668" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2669" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2670" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2671" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2672" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2673" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2674" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2675" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2676" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.069em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-91" type="math/tex">type(assoc) = association</script> if and only if <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-92-Frame" class="MathJax"><nobr><span id="MathJax-Span-2677" class="math"><span style="display: inline-block; position: relative; width: 147px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.727em, 1000em, 2.772em, -0.509em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2678" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2679" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2680" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2681" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2682" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2683" class="mi">c</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2684" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2685" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2686" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2687" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2688" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2689" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2690" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2691" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2692" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2693" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2694" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2695" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2696" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.869em; vertical-align: -0.091em;"></span></span></nobr></span><script id="MathJax-Element-92" type="math/tex">assoc \in Associations</script>.
-</p>
-  </div>
-  
-<p>Associations are used below in the <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-93-Frame" class="MathJax"><nobr><span id="MathJax-Span-2697" class="math"><span style="display: inline-block; position: relative; width: 56px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.727em, 1000em, 2.758em, -0.577em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2698" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2699" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2700" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2701" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2702" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2703" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2704" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2705" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.854em; vertical-align: -0.076em;"></span></span></nobr></span><script id="MathJax-Element-93" type="math/tex">ActsFor</script> and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-94-Frame" class="MathJax"><nobr><span id="MathJax-Span-2706" class="math"><span style="display: inline-block; position: relative; width: 109px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.763em, -0.577em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2707" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2708" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2709" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2710" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2711" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2712" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2713" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2714" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2715" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2716" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2717" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2718" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2719" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2720" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2721" class="mi">h</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.875em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-94" type="math/tex">AssociatedWith</script> relations.
-
-
-</p></section>
-  <section id="communications">
-  <h5><span class="secno">3.2.4.3 </span>Communications</h5>
-  <div class="note"><div class="note-title"><span>Note</span></div><div class="">TODO</div></div>
-  
-</section>
-  <section id="delegations">
-  <h5><span class="secno">3.2.4.4 </span>Delegations</h5>
-  <div class="note"><div class="note-title"><span>Note</span></div><div class="">TODO</div></div>
-</section>
-  <section id="derivations-1">
-  
-  <h5><span class="secno">3.2.4.5 </span> Derivations </h5>
-
-<p>A <em>Derivation</em> is an interaction chaining one or more generation and use steps.  
-</p>
-<div data-title="Component 10 (derivations)" data-count="10" class="component" id="derivations"><div class="ruleTitle"><a href="#derivations" class="internalDFN">Component 10 (derivations)</a></div>
-<p>  A set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-95-Frame" class="MathJax"><nobr><span id="MathJax-Span-2722" class="math"><span style="display: inline-block; position: relative; width: 188px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.741em, 1000em, 2.848em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2723" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2724" class="mi">D</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2725" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2726" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2727" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2728" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2729" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2730" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2731" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2732" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2733" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2734" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2735" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2736" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2737" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2738" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2739" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2740" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2741" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2742" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2743" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2744" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2745" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2746" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2747" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.935em; vertical-align: -0.173em;"></span></span></nobr></span><script id="MathJax-Element-95" type="math/tex">Derivations \subseteq Interactions</script>, such that
-  <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-96-Frame" class="MathJax"><nobr><span id="MathJax-Span-2748" class="math"><span style="display: inline-block; position: relative; width: 427px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.954em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2749" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2750" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2751" class="mi">y</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2752" class="mi">p</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2753" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2754" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2755" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2756" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2757" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2758" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2759" class="mi">v</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2760" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2761" class="mo">∈</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2762" class="mo">{</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2763" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2764" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2765" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2766" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2767" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2768" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2769" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2770" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2771" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2772" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2773" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2774" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2775" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2776" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2777" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2778" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2779" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2780" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2781" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2782" class="mi">p</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2783" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2784" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2785" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2786" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2787" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2788" class="mi">y</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2789" class="mi">S<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.008em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2790" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2791" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2792" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2793" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2794" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2795" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2796" class="mi">q</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2797" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2798" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2799" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2800" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2801" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2802" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2803" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2804" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2805" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.079em; vertical-align: -0.286em;"></span></span></nobr></span><script id="MathJax-Element-96" type="math/tex">type(deriv) \in \{derivation, version, primarySource,quotation\}</script>
-  if and only if <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-97-Frame" class="MathJax"><nobr><span id="MathJax-Span-2806" class="math"><span style="display: inline-block; position: relative; width: 140px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.772em, -0.511em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2807" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2808" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2809" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2810" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2811" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2812" class="mi">v</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2813" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2814" class="mi">D</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2815" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2816" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2817" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2818" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2819" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2820" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2821" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2822" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2823" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2824" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.885em; vertical-align: -0.091em;"></span></span></nobr></span><script id="MathJax-Element-97" type="math/tex">deriv \in Derivations</script>.
-</p></div>
-<p>See below for the associated derivation path and DerivedFrom relation.
-</p>
-</section>
-</section>
-</section>
-<section id="relations">
-
-<h3><span class="secno">3.3 </span> Relations </h3>
-
-<h3 id="simple-relations"> Simple relations </h3>
-<p>The entities, interactions, and activities in a structure are related in the following ways:
-</p>
-<ol><li> A relation <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-98-Frame" class="MathJax"><nobr><span id="MathJax-Span-2825" class="math"><span style="display: inline-block; position: relative; width: 178px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.848em, -0.424em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2826" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2827" class="mi">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.043em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2828" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2829" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2830" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2831" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2832" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2833" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2834" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2835" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2836" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2837" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2838" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-2839" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2840" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2841" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2842" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2843" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2844" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2845" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2846" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.966em; vertical-align: -0.173em;"></span></span></nobr></span><script id="MathJax-Element-98" type="math/tex">Used \subseteq Events \times Entities</script> saying when an event used an entity.  An event can use at most one entity, and if <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-99-Frame" class="MathJax"><nobr><span id="MathJax-Span-2847" class="math"><span style="display: inline-block; position: relative; width: 106px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.922em, -0.478em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2848" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2849" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2850" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2851" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2852" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2853" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2854" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2855" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2856" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2857" class="mi">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.043em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2858" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2859" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2860" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.045em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-99" type="math/tex">(evt,e)\in Used</script> then <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-100-Frame" class="MathJax"><nobr><span id="MathJax-Span-2861" class="math"><span style="display: inline-block; position: relative; width: 157px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2862" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2863" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2864" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2865" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2866" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2867" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2868" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2869" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2870" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2871" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2872" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2873" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2874" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2875" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2876" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2877" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2878" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2879" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2880" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2881" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2882" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2883" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-100" type="math/tex">time(evt) \in lifetime(e)</script> and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-101-Frame" class="MathJax"><nobr><span id="MathJax-Span-2884" class="math"><span style="display: inline-block; position: relative; width: 94px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.719em, 1000em, 2.951em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2885" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2886" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2887" class="mi">y</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2888" class="mi">p</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2889" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2890" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2891" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2892" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2893" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2894" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2895" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2896" class="mi">e</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.069em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-101" type="math/tex">type(g) = use</script> must hold.
-</li><li> A relation <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-102-Frame" class="MathJax"><nobr><span id="MathJax-Span-2897" class="math"><span style="display: inline-block; position: relative; width: 215px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.848em, -0.474em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2898" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2899" class="mi">G</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2900" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2901" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2902" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2903" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2904" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2905" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2906" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2907" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2908" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2909" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2910" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2911" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2912" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2913" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2914" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2915" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-2916" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2917" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2918" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2919" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2920" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2921" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2922" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2923" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.966em; vertical-align: -0.173em;"></span></span></nobr></span><script id="MathJax-Element-102" type="math/tex">Generated \subseteq Events \times Entities</script> saying when an event generated an entity.  An event can generate at most one entity, and if <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-103-Frame" class="MathJax"><nobr><span id="MathJax-Span-2924" class="math"><span style="display: inline-block; position: relative; width: 143px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.922em, -0.478em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2925" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2926" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2927" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2928" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2929" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2930" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2931" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2932" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2933" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2934" class="mi">G</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2935" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2936" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2937" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2938" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2939" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2940" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2941" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2942" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.045em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-103" type="math/tex">(evt,e)\in Generated</script> then <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-104-Frame" class="MathJax"><nobr><span id="MathJax-Span-2943" class="math"><span style="display: inline-block; position: relative; width: 194px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.514em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2944" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2945" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2946" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2947" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2948" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2949" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2950" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2951" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2952" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2953" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2954" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2955" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2956" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2957" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2958" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2959" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2960" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2961" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2962" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2963" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2964" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2965" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2966" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2967" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2968" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2969" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2970" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-104" type="math/tex">min(lifetime(e)) = time(evt)</script> and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-105-Frame" class="MathJax"><nobr><span id="MathJax-Span-2971" class="math"><span style="display: inline-block; position: relative; width: 145px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.719em, 1000em, 2.951em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2972" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2973" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2974" class="mi">y</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2975" class="mi">p</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2976" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2977" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2978" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2979" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2980" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2981" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2982" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2983" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2984" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2985" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2986" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2987" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2988" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2989" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2990" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.069em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-105" type="math/tex">type(g) = generation</script> must hold.
-</li><li> A relation <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-106-Frame" class="MathJax"><nobr><span id="MathJax-Span-2991" class="math"><span style="display: inline-block; position: relative; width: 222px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.848em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2992" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2993" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2994" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2995" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2996" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2997" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2998" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2999" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3000" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3001" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3002" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3003" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3004" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3005" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3006" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3007" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3008" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3009" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3010" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3011" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3012" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3013" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3014" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3015" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3016" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3017" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3018" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3019" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.966em; vertical-align: -0.173em;"></span></span></nobr></span><script id="MathJax-Element-106" type="math/tex">Invalidated \subseteq Events \times Entities</script> saying when an event invalidated an entity.  An event can invalidate at most one entity, and if <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-107-Frame" class="MathJax"><nobr><span id="MathJax-Span-3020" class="math"><span style="display: inline-block; position: relative; width: 149px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.922em, -0.478em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3021" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3022" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3023" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3024" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3025" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3026" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3027" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3028" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3029" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3030" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3031" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3032" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3033" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3034" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3035" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3036" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3037" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3038" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3039" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3040" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.045em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-107" type="math/tex">(evt,e)\in Invalidated</script> then <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-108-Frame" class="MathJax"><nobr><span id="MathJax-Span-3041" class="math"><span style="display: inline-block; position: relative; width: 194px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.514em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3042" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3043" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3044" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3045" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3046" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3047" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3048" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3049" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3050" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3051" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3052" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3053" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3054" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3055" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3056" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3057" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3058" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3059" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3060" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3061" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3062" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3063" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3064" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3065" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3066" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3067" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3068" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-108" type="math/tex">min(lifetime(e)) = time(evt)</script> and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-109-Frame" class="MathJax"><nobr><span id="MathJax-Span-3069" class="math"><span style="display: inline-block; position: relative; width: 154px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3070" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3071" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3072" class="mi">y</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3073" class="mi">p</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3074" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3075" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3076" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3077" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3078" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3079" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3080" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3081" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3082" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3083" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3084" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3085" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3086" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3087" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3088" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3089" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3090" class="mi">n</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-109" type="math/tex">type(g) = invalidation</script> must hold.
-</li><li> A relation <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-110-Frame" class="MathJax"><nobr><span id="MathJax-Span-3091" class="math"><span style="display: inline-block; position: relative; width: 245px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.727em, 1000em, 2.951em, -0.527em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3092" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3093" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3094" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3095" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3096" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3097" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3098" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3099" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3100" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3101" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3102" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3103" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3104" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3105" class="mi">y</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3106" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3107" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3108" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3109" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3110" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3111" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3112" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3113" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3114" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3115" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3116" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3117" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3118" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3119" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3120" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3121" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3122" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3123" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.06em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-110" type="math/tex">EventActivity \subseteq Events \times Activities</script> associating activities with events, such that <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-111-Frame" class="MathJax"><nobr><span id="MathJax-Span-3124" class="math"><span style="display: inline-block; position: relative; width: 175px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.719em, 1000em, 2.951em, -0.478em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3125" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3126" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3127" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3128" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3129" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3130" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3131" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3132" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3133" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3134" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3135" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3136" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3137" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3138" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3139" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3140" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3141" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3142" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3143" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3144" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3145" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3146" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3147" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3148" class="mi">y</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.069em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-111" type="math/tex">(act,evt) \in EventActivity</script> implies <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-112-Frame" class="MathJax"><nobr><span id="MathJax-Span-3149" class="math"><span style="display: inline-block; position: relative; width: 170px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3150" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3151" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3152" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3153" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3154" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3155" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3156" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3157" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3158" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3159" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3160" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3161" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3162" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3163" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3164" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3165" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3166" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3167" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3168" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3169" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3170" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3171" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3172" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3173" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-112" type="math/tex">time(evt) \in lifetime(act)</script>.
-</li><li> A relation <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-113-Frame" class="MathJax"><nobr><span id="MathJax-Span-3174" class="math"><span style="display: inline-block; position: relative; width: 430px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.577em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3175" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3176" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3177" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3178" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3179" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3180" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3181" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3182" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3183" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3184" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3185" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3186" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3187" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3188" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3189" class="mi">h</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3190" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3191" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3192" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3193" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3194" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3195" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3196" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3197" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3198" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3199" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3200" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3201" class="mi">n</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3202" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3203" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3204" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3205" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3206" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3207" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3208" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3209" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3210" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3211" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3212" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3213" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3214" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3215" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3216" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3217" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3218" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3219" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3220" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3221" class="mi">P</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3222" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3223" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3224" class="mi">n</span><span id="MathJax-Span-3225" class="msubsup"><span style="display: inline-block; position: relative; width: 0.951em; height: 0px;"><span style="position: absolute; clip: rect(1.953em, 1000em, 2.758em, -0.51em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3226" class="mi">s</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.409em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3227" class="mi">⊥</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-113" type="math/tex">AssociatedWith \subseteq Association \times Agents \times Activities \times Plans_\bot</script> indicating when an agent is associated with an activity, and giving the identity of the association relationship, and an optional plan. 
-</li><li> A relation <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-114-Frame" class="MathJax"><nobr><span id="MathJax-Span-3228" class="math"><span style="display: inline-block; position: relative; width: 377px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.577em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3229" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3230" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3231" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3232" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3233" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3234" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3235" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3236" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3237" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3238" class="mi">D</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3239" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3240" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3241" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3242" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3243" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3244" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3245" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3246" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3247" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3248" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3249" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3250" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3251" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3252" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3253" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3254" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3255" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3256" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3257" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3258" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3259" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3260" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3261" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3262" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3263" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3264" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3265" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3266" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3267" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3268" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3269" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3270" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3271" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3272" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3273" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-114" type="math/tex">ActsFor \subseteq Delegations \times Agents \times Agents \times Activities</script> indicating when one agent acts on behalf of another with respect to a given activity.  
-</li></ol>
-
-<div class="note"><div class="note-title"><span>Note</span></div><div class="">
-  TODO: Communication, start, end relations
-  </div></div>
-
-  <div class="note"><div class="note-title"><span>Note</span></div><div class="">
-  TODO: Specialization relation
-  </div></div>
-
-  <div class="note"><div class="note-title"><span>Note</span></div><div class="">
-    TODO: Explicit axioms concerning the relations
-  </div></div>
-
-  
-<section id="derivation-paths-and-derivedfrom">
-<h4><span class="secno">3.3.1 </span> Derivation paths and DerivedFrom </h4>
-
-<p>Recall that above we introduced a subset of interactions called <em>Derivations</em>.  These identify <em>paths</em> of the form </p>
-
-<span class="MathJax_Preview"></span><div style="text-align: center;" aria-readonly="true" role="textbox" class="MathJax_Display"><span style="" id="MathJax-Element-115-Frame" class="MathJax"><nobr><span id="MathJax-Span-3274" class="math"><span style="display: inline-block; position: relative; width: 388px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.993em, -0.495em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3275" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3276" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3277" class="mi">n</span><span id="MathJax-Span-3278" class="msubsup"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3279" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.274em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3280" class="mi">n</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3281" class="mo">⋅</span><span style="padding-left: 0.25em;" id="MathJax-Span-3282" class="msubsup"><span style="display: inline-block; position: relative; width: 0.951em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.951em, -0.518em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3283" class="mi">g</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.861em; left: 0.526em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3284" class="mi">n</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3285" class="mo">⋅</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3286" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3287" class="mi">c</span><span id="MathJax-Span-3288" class="msubsup"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3289" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.274em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3290" class="mi">n</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3291" class="mo">⋅</span><span style="padding-left: 0.25em;" id="MathJax-Span-3292" class="msubsup"><span style="display: inline-block; position: relative; width: 0.951em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.484em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3293" class="mi">u</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.526em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3294" class="mi">n</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3295" class="mo">⋅</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3296" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3297" class="mi">n</span><span id="MathJax-Span-3298" class="msubsup"><span style="display: inline-block; position: relative; width: 1.517em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3299" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.274em;"><span id="MathJax-Span-3300" class="texatom"><span id="MathJax-Span-3301" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3302" class="mi">n</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3303" class="mo">−</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3304" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3305" class="mo">⋅</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3306" class="mo">.</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-3307" class="mo">.</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-3308" class="mo">.</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-3309" class="mo">⋅</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3310" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3311" class="mi">n</span><span id="MathJax-Span-3312" class="msubsup"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3313" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.274em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3314" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3315" class="mo">⋅</span><span style="padding-left: 0.25em;" id="MathJax-Span-3316" class="msubsup"><span style="display: inline-block; position: relative; width: 0.951em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.951em, -0.518em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3317" class="mi">g</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.861em; left: 0.526em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3318" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3319" class="mo">⋅</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3320" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3321" class="mi">c</span><span id="MathJax-Span-3322" class="msubsup"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3323" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.274em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3324" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3325" class="mo">⋅</span><span style="padding-left: 0.25em;" id="MathJax-Span-3326" class="msubsup"><span style="display: inline-block; position: relative; width: 0.951em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.484em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3327" class="mi">u</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.526em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3328" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3329" class="mo">⋅</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3330" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3331" class="mi">n</span><span id="MathJax-Span-3332" class="msubsup"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3333" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.274em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3334" class="mn">0</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.974em; vertical-align: -0.328em;"></span></span></nobr></span></div><script id="MathJax-Element-115" type="math/tex; mode=display">ent_n\cdot g_n\cdot  act_n\cdot  u_n\cdot  ent_{n-1}\cdot  ...\cdot
-ent_1\cdot  g_1\cdot  act_1\cdot  u_1\cdot  ent_0</script>
-
-<p>where the <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-116-Frame" class="MathJax"><nobr><span id="MathJax-Span-3335" class="math"><span style="display: inline-block; position: relative; width: 25px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.903em, -0.495em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3336" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3337" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3338" class="mi">n</span><span id="MathJax-Span-3339" class="msubsup"><span style="display: inline-block; position: relative; width: 0.524em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3340" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.274em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3341" class="mi">i</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.878em; vertical-align: -0.231em;"></span></span></nobr></span><script id="MathJax-Element-116" type="math/tex">ent_i</script> are entities, <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-117-Frame" class="MathJax"><nobr><span id="MathJax-Span-3342" class="math"><span style="display: inline-block; position: relative; width: 25px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.903em, -0.509em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3343" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3344" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3345" class="mi">c</span><span id="MathJax-Span-3346" class="msubsup"><span style="display: inline-block; position: relative; width: 0.524em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3347" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.274em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3348" class="mi">i</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.878em; vertical-align: -0.231em;"></span></span></nobr></span><script id="MathJax-Element-117" type="math/tex">act_i</script> are activities, <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-118-Frame" class="MathJax"><nobr><span id="MathJax-Span-3349" class="math"><span style="display: inline-block; position: relative; width: 13px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.428em, 1000em, 2.469em, -0.518em); top: -2.044em; left: 0em;"><span id="MathJax-Span-3350" class="mrow"><span id="MathJax-Span-3351" class="msubsup"><span style="display: inline-block; position: relative; width: 0.776em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.951em, -0.518em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3352" class="mi">g</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.861em; left: 0.526em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3353" class="mi">i</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.044em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.863em; vertical-align: -0.329em;"></span></span></nobr></span><script id="MathJax-Element-118" type="math/tex">g_i</script> are generations, and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-119-Frame" class="MathJax"><nobr><span id="MathJax-Span-3354" class="math"><span style="display: inline-block; position: relative; width: 13px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.428em, 1000em, 2.377em, -0.484em); top: -2.044em; left: 0em;"><span id="MathJax-Span-3355" class="mrow"><span id="MathJax-Span-3356" class="msubsup"><span style="display: inline-block; position: relative; width: 0.776em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.484em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3357" class="mi">u</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.526em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3358" class="mi">i</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.044em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.766em; vertical-align: -0.231em;"></span></span></nobr></span><script id="MathJax-Element-119" type="math/tex">u_i</script> are usages.
-</p>
-<p>Formally, we consider the (regular) language:
-</p>
-<span class="MathJax_Preview"></span><div style="text-align: center;" aria-readonly="true" role="textbox" class="MathJax_Display"><span style="" id="MathJax-Element-120-Frame" class="MathJax"><nobr><span id="MathJax-Span-3359" class="math"><span style="display: inline-block; position: relative; width: 473px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.595em, 1000em, 2.922em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3360" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3361" class="mi">D</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3362" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3363" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3364" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3365" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3366" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3367" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3368" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3369" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3370" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3371" class="mi">P</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3372" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3373" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3374" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3375" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3376" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3377" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3378" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3379" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3380" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3381" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3382" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3383" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3384" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3385" class="mo">⋅</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3386" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3387" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3388" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3389" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3390" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3391" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3392" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3393" class="mo">⋅</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3394" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3395" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3396" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3397" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3398" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3399" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3400" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3401" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3402" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3403" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3404" class="mo">⋅</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3405" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3406" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3407" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3408" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3409" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3410" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3411" class="mo">⋅</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3412" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3413" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3414" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3415" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3416" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3417" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3418" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3419" class="mi">s</span><span id="MathJax-Span-3420" class="msubsup"><span style="display: inline-block; position: relative; width: 0.893em; height: 0px;"><span style="position: absolute; clip: rect(1.719em, 1000em, 2.922em, -0.497em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-3421" class="mo">)</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -2.516em; left: 0.35em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3422" class="mo">+</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.17em; vertical-align: -0.252em;"></span></span></nobr></span></div><script id="MathJax-Element-120" type="math/tex; mode=display">DerivationPaths = Entities \cdot (Events \cdot Activities \cdot
-Events \cdot Entities)^+</script>
-<p>with the constraints that for each derivation path:
-</p>
-<ul>
-<li>for each substring <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-121-Frame" class="MathJax"><nobr><span id="MathJax-Span-3423" class="math"><span style="display: inline-block; position: relative; width: 78px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.951em, -0.495em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3424" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3425" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3426" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3427" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3428" class="mo">⋅</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3429" class="mi">g</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3430" class="mo">⋅</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3431" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3432" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3433" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.93em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-121" type="math/tex">ent\cdot g \cdot act</script> we have <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-122-Frame" class="MathJax"><nobr><span id="MathJax-Span-3434" class="math"><span style="display: inline-block; position: relative; width: 145px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.478em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3435" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3436" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3437" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3438" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3439" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3440" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3441" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3442" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3443" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3444" class="mi">G</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3445" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3446" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3447" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3448" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3449" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3450" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3451" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3452" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-122" type="math/tex">(g,ent) \in Generated</script>  and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-123-Frame" class="MathJax"><nobr><span id="MathJax-Span-3453" class="math"><span style="display: inline-block; position: relative; width: 175px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.719em, 1000em, 2.951em, -0.478em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3454" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3455" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3456" class="mi">g</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3457" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3458" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3459" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3460" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3461" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3462" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3463" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3464" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3465" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3466" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3467" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3468" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3469" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3470" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3471" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3472" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3473" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3474" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3475" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3476" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3477" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.069em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-123" type="math/tex">(g,act) \in EventActivities</script>, and
-</li>
-<li>for each substring <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-124-Frame" class="MathJax"><nobr><span id="MathJax-Span-3478" class="math"><span style="display: inline-block; position: relative; width: 78px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3479" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3480" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3481" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3482" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3483" class="mo">⋅</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3484" class="mi">u</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3485" class="mo">⋅</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3486" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3487" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3488" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.721em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-124" type="math/tex">act \cdot u \cdot ent</script> we have <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-125-Frame" class="MathJax"><nobr><span id="MathJax-Span-3489" class="math"><span style="display: inline-block; position: relative; width: 107px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.922em, -0.478em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3490" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3491" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3492" class="mi">u</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3493" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3494" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3495" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3496" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3497" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3498" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3499" class="mi">U<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.043em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3500" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3501" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3502" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.045em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-125" type="math/tex">(u,ent) \in Used</script> and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-126-Frame" class="MathJax"><nobr><span id="MathJax-Span-3503" class="math"><span style="display: inline-block; position: relative; width: 175px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.719em, 1000em, 2.922em, -0.478em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3504" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3505" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3506" class="mi">u</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3507" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3508" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3509" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3510" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3511" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3512" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3513" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3514" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3515" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3516" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3517" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3518" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3519" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3520" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3521" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3522" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3523" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3524" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3525" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3526" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3527" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.038em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-126" type="math/tex">(u,act) \in EventActivities</script>.
-</li>
-</ul>
-<p>We also consider a function <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-127-Frame" class="MathJax"><nobr><span id="MathJax-Span-3528" class="math"><span style="display: inline-block; position: relative; width: 329px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.763em, -0.511em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3529" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3530" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3531" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3532" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3533" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3534" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3535" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3536" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3537" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3538" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3539" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3540" class="mi">m</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3541" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3542" class="mi">D</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3543" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3544" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3545" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3546" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3547" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3548" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3549" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3550" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3551" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3552" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3553" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3554" class="mi">D</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3555" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3556" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3557" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3558" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3559" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3560" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3561" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3562" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3563" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3564" class="mi">P</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3565" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3566" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3567" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3568" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.875em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-127" type="math/tex">derivedFrom : Derivations \to
-DerivationPaths</script>  linking each derivation to its path.</p>
-
-
-<div class="remark">The reason why we need paths and not just individual derivation steps is that imprecise wasDerivedFrom formulas can represent multiple derivation steps.</div>
-</section>
-</section>
-
-<section id="putting-it-all-together">
-<h3><span class="secno">3.4 </span> Putting it all together </h3>
-
-<p>A <em>structure</em> <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-128-Frame" class="MathJax"><nobr><span id="MathJax-Span-3569" class="math"><span style="display: inline-block; position: relative; width: 16px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.763em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3570" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3571" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.843em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-128" type="math/tex">W</script> is a structure containing all of the above
-described data.  If we need to talk about the objects or relations of
-more than one structure then we may write <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-129-Frame" class="MathJax"><nobr><span id="MathJax-Span-3572" class="math"><span style="display: inline-block; position: relative; width: 82px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3573" class="mrow"><span id="MathJax-Span-3574" class="msubsup"><span style="display: inline-block; position: relative; width: 1.287em; height: 0px;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.763em, -0.455em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3575" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.862em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3576" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3577" class="mo">.</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3578" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3579" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3580" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3581" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3582" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3583" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3584" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-129" type="math/tex">W_1.Objects</script>, <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-130-Frame" class="MathJax"><nobr><span id="MathJax-Span-3585" class="math"><span style="display: inline-block; position: relative; width: 77px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3586" class="mrow"><span id="MathJax-Span-3587" class="msubsup"><span style="display: inline-block; position: relative; width: 1.287em; height: 0px;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.763em, -0.455em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3588" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.862em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3589" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3590" class="mo">.</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3591" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3592" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3593" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3594" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3595" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3596" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-130" type="math/tex">W_1.Things</script>,
-etc.; otherwise, to
-decrease notational clutter, when we consider a fixed structure then the names of the sets, relations and functions above refer to the components of that model.
-</p>
-
-</section>
-<section id="interpretations">
-<h3><span class="secno">3.5 </span> Interpretations </h3>
-
-<p>We need to link identifiers to the objects they denote.  We do this using a function which we shall call an <em>interpretation</em>.
- Thus, we consider interpretations as follows:
-An interpretation is a function  <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-131-Frame" class="MathJax"><nobr><span id="MathJax-Span-3597" class="math"><span style="display: inline-block; position: relative; width: 179px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.566em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3598" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3599" class="mi">ρ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3600" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3601" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3602" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3603" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3604" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3605" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3606" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3607" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3608" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3609" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3610" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3611" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3612" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3613" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3614" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3615" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3616" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3617" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3618" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3619" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-131" type="math/tex">\rho : Identifiers \to Objects</script> describing
-which object is the target of each identifier. The mapping from identifiers to objects may <b>not</b> change over time. 
-</p>
-</section>
-
-</section>
-<section id="semantics">
-<!--OddPage--><h2><span class="secno">4. </span> Semantics </h2>
-
-<p>In what follows, let <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-132-Frame" class="MathJax"><nobr><span id="MathJax-Span-3620" class="math"><span style="display: inline-block; position: relative; width: 16px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.763em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3621" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3622" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.843em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-132" type="math/tex">W</script> be a fixed structure with the associated sets and relations discussed in the previous section, and let <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-133-Frame" class="MathJax"><nobr><span id="MathJax-Span-3623" class="math"><span style="display: inline-block; position: relative; width: 7px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.745em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3624" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3625" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.824em; vertical-align: -0.063em;"></span></span></nobr></span><script id="MathJax-Element-133" type="math/tex">I</script> be an interpretation of identifiers as objects in <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-134-Frame" class="MathJax"><nobr><span id="MathJax-Span-3626" class="math"><span style="display: inline-block; position: relative; width: 16px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.763em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3627" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3628" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.843em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-134" type="math/tex">W</script>.
-The annotations [WF] refer to well-formedness constraints that correspond to typing constraints.
-</p>
-
-<section id="satisfaction">
-<h3><span class="secno">4.1 </span> Satisfaction </h3>
-
-<p>Consider a formula <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-135-Frame" class="MathJax"><nobr><span id="MathJax-Span-3629" class="math"><span style="display: inline-block; position: relative; width: 11px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3630" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3631" class="mi">ϕ</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.075em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-135" type="math/tex">\phi</script>, a structure <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-136-Frame" class="MathJax"><nobr><span id="MathJax-Span-3632" class="math"><span style="display: inline-block; position: relative; width: 16px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.763em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3633" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3634" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.843em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-136" type="math/tex">W</script> and an interpretation
- <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-137-Frame" class="MathJax"><nobr><span id="MathJax-Span-3635" class="math"><span style="display: inline-block; position: relative; width: 7px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.745em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3636" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3637" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.824em; vertical-align: -0.063em;"></span></span></nobr></span><script id="MathJax-Element-137" type="math/tex">I</script>.
-We define notation <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-138-Frame" class="MathJax"><nobr><span id="MathJax-Span-3638" class="math"><span style="display: inline-block; position: relative; width: 65px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3639" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3640" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3641" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3642" class="mi">ρ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3643" class="mo">⊨</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3644" class="mi">ϕ</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.075em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-138" type="math/tex">W,\rho \models \phi</script> which means that <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-139-Frame" class="MathJax"><nobr><span id="MathJax-Span-3645" class="math"><span style="display: inline-block; position: relative; width: 11px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3646" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3647" class="mi">ϕ</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.075em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-139" type="math/tex">\phi</script> is
- satisfied in <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-140-Frame" class="MathJax"><nobr><span id="MathJax-Span-3648" class="math"><span style="display: inline-block; position: relative; width: 32px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.95em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3649" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3650" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3651" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3652" class="mi">ρ</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.043em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-140" type="math/tex">W,\rho</script>. For atomic formulas, the definition of the
- satisfaction relation is given in the next few subsections.  We give
- the standard definition of the semantics of the other formulas:
-</p>
-
-<div data-title="Formalism 11 (first-order-logic)" data-count="11" class="formalism" id="first-order-logic"><div class="ruleTitle"><a href="#first-order-logic" class="internalDFN">Formalism 11 (first-order-logic)</a></div>
-<ol>
-  <li><span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-141-Frame" class="MathJax"><nobr><span id="MathJax-Span-3653" class="math"><span style="display: inline-block; position: relative; width: 88px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.733em, 1000em, 2.95em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3654" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3655" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3656" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3657" class="mi">ρ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3658" class="mo">⊨</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3659" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3660" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3661" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3662" class="mi">e</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.053em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-141" type="math/tex">W,\rho \models True</script> always holds.</li>
-  <li><span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-142-Frame" class="MathJax"><nobr><span id="MathJax-Span-3663" class="math"><span style="display: inline-block; position: relative; width: 93px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3664" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3665" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3666" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3667" class="mi">ρ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3668" class="mo">⊨</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3669" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3670" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3671" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3672" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3673" class="mi">e</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.075em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-142" type="math/tex">W,\rho \models False</script> never holds.</li>
-  <li><span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-143-Frame" class="MathJax"><nobr><span id="MathJax-Span-3674" class="math"><span style="display: inline-block; position: relative; width: 76px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3675" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3676" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3677" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3678" class="mi">ρ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3679" class="mo">⊨</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3680" class="mi">¬</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3681" class="mi">ϕ</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.075em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-143" type="math/tex">W,\rho \models \neg \phi</script> holds if and only if <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-144-Frame" class="MathJax"><nobr><span id="MathJax-Span-3682" class="math"><span style="display: inline-block; position: relative; width: 65px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3683" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3684" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3685" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3686" class="mi">ρ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3687" class="mo">⊨</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3688" class="mi">ϕ</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.075em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-144" type="math/tex">W,\rho \models
-  \phi</script> does not hold.</li>
-  <li><span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-145-Frame" class="MathJax"><nobr><span id="MathJax-Span-3689" class="math"><span style="display: inline-block; position: relative; width: 95px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3690" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3691" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3692" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3693" class="mi">ρ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3694" class="mo">⊨</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3695" class="mi">ϕ</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3696" class="mo">∧</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3697" class="mi">ψ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.075em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-145" type="math/tex">W,\rho \models \phi \wedge \psi</script> holds if and only if <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-146-Frame" class="MathJax"><nobr><span id="MathJax-Span-3698" class="math"><span style="display: inline-block; position: relative; width: 65px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3699" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3700" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3701" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3702" class="mi">ρ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3703" class="mo">⊨</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3704" class="mi">ϕ</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.075em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-146" type="math/tex">W,\rho \models
-  \phi</script> and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-147-Frame" class="MathJax"><nobr><span id="MathJax-Span-3705" class="math"><span style="display: inline-block; position: relative; width: 62px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.733em, 1000em, 2.95em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3706" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3707" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3708" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3709" class="mi">i</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3710" class="mo">⊨</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3711" class="mi">ψ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.053em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-147" type="math/tex">W,i\models \psi</script>.</li>
-  <li><span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-148-Frame" class="MathJax"><nobr><span id="MathJax-Span-3712" class="math"><span style="display: inline-block; position: relative; width: 95px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3713" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3714" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3715" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3716" class="mi">ρ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3717" class="mo">⊨</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3718" class="mi">ϕ</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3719" class="mo">∨</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-3720" class="mi">ψ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.075em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-148" type="math/tex">W,\rho \models \phi \vee \psi</script> holds if either <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-149-Frame" class="MathJax"><nobr><span id="MathJax-Span-3721" class="math"><span style="display: inline-block; position: relative; width: 65px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3722" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3723" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3724" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3725" class="mi">ρ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3726" class="mo">⊨</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3727" class="mi">ϕ</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.075em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-149" type="math/tex">W,\rho \models \phi</script>
-  or <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-150-Frame" class="MathJax"><nobr><span id="MathJax-Span-3728" class="math"><span style="display: inline-block; position: relative; width: 65px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.733em, 1000em, 2.95em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3729" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3730" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3731" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3732" class="mi">ρ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3733" class="mo">⊨</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3734" class="mi">ψ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.053em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-150" type="math/tex">W,\rho \models \psi</script> holds.</li>
-  <li><span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-151-Frame" class="MathJax"><nobr><span id="MathJax-Span-3735" class="math"><span style="display: inline-block; position: relative; width: 103px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3736" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3737" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3738" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3739" class="mi">ρ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3740" class="mo">⊨</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3741" class="mi">ϕ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3742" class="mo">⇒</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3743" class="mi">ψ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.075em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-151" type="math/tex">W,\rho \models \phi \Rightarrow \psi</script> holds if <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-152-Frame" class="MathJax"><nobr><span id="MathJax-Span-3744" class="math"><span style="display: inline-block; position: relative; width: 65px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3745" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3746" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3747" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3748" class="mi">ρ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3749" class="mo">⊨</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3750" class="mi">ϕ</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.075em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-152" type="math/tex">W,\rho \models \phi</script>
-  implies <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-153-Frame" class="MathJax"><nobr><span id="MathJax-Span-3751" class="math"><span style="display: inline-block; position: relative; width: 65px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.733em, 1000em, 2.95em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3752" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3753" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3754" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3755" class="mi">ρ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3756" class="mo">⊨</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3757" class="mi">ψ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.084em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.053em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-153" type="math/tex">W,\rho \models \psi</script>.</li>
-  <li><span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-154-Frame" class="MathJax"><nobr><span id="MathJax-Span-3758" class="math"><span style="display: inline-block; position: relative; width: 90px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3759" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3760" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3761" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3762" class="mi">ρ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3763" class="mo">⊨</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3764" class="mi">∃</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3765" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3766" class="mo">.</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3767" class="mi">ϕ</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.075em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-154" type="math/tex">W,\rho \models \exists x. \phi</script> holds if there exists some <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-155-Frame" class="MathJax"><nobr><span id="MathJax-Span-3768" class="math"><span style="display: inline-block; position: relative; width: 97px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.499em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3769" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3770" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3771" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3772" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3773" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3774" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3775" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3776" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3777" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3778" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3779" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3780" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-155" type="math/tex">obj \in
-  Objects</script> such that <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-156-Frame" class="MathJax"><nobr><span id="MathJax-Span-3781" class="math"><span style="display: inline-block; position: relative; width: 136px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3782" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3783" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3784" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3785" class="mi">ρ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3786" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3787" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3788" class="mo">:<span style="font-family: STIXGeneral; font-style: normal; font-weight: normal;">=</span></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3789" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3790" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3791" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3792" class="mo">]</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3793" class="mo">⊨</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3794" class="mi">ϕ</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-156" type="math/tex">W,\rho[x:=obj] \models \phi</script>.</li>
-  <li><span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-157-Frame" class="MathJax"><nobr><span id="MathJax-Span-3795" class="math"><span style="display: inline-block; position: relative; width: 90px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3796" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3797" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3798" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3799" class="mi">ρ</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3800" class="mo">⊨</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3801" class="mi">∀</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3802" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3803" class="mo">.</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3804" class="mi">ϕ</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.075em; vertical-align: -0.282em;"></span></span></nobr></span><script id="MathJax-Element-157" type="math/tex">W,\rho \models \forall x. \phi</script> holds if there for every <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-158-Frame" class="MathJax"><nobr><span id="MathJax-Span-3805" class="math"><span style="display: inline-block; position: relative; width: 97px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.499em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3806" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3807" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3808" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3809" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3810" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3811" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3812" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3813" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3814" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3815" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3816" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3817" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-158" type="math/tex">obj \in
-  Objects</script> we have <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-159-Frame" class="MathJax"><nobr><span id="MathJax-Span-3818" class="math"><span style="display: inline-block; position: relative; width: 136px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3819" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3820" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3821" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3822" class="mi">ρ</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3823" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3824" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3825" class="mo">:<span style="font-family: STIXGeneral; font-style: normal; font-weight: normal;">=</span></span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3826" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3827" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3828" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3829" class="mo">]</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3830" class="mo">⊨</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3831" class="mi">ϕ</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-159" type="math/tex">W,\rho[x:=obj] \models \phi</script>.</li>
-  </ol></div>
-
-<div class="remark">
-  <p>In the semantics above, note that the domain of quantification is
-  the set of <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-160-Frame" class="MathJax"><nobr><span id="MathJax-Span-3832" class="math"><span style="display: inline-block; position: relative; width: 53px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.466em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3833" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3834" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3835" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3836" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3837" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3838" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3839" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3840" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-160" type="math/tex">Objects</script>; that is, quantifiers range over entities,
-  activities, agents, or interactions (which are in turn further
-  subdivided into types of interactions).  <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-161-Frame" class="MathJax"><nobr><span id="MathJax-Span-3841" class="math"><span style="display: inline-block; position: relative; width: 48px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.467em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3842" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3843" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3844" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3845" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3846" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3847" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3848" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-161" type="math/tex">Things</script> and relations
-  cannot be referenced directly by identifiers.  
-</p>
-  </div>
-
-  <div class="remark">
-    <p>A PROV instance <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-162-Frame" class="MathJax"><nobr><span id="MathJax-Span-3849" class="math"><span style="display: inline-block; position: relative; width: 7px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.745em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3850" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3851" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.824em; vertical-align: -0.063em;"></span></span></nobr></span><script id="MathJax-Element-162" type="math/tex">I</script> consists of a set of statements, each of
-    which can be translated to an atomic formula following the
-    definitional rules in PROV-CONSTRAINTS, possibly by introducing
-    fresh existential variables.  Thus, we can view an
-    instance <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-163-Frame" class="MathJax"><nobr><span id="MathJax-Span-3852" class="math"><span style="display: inline-block; position: relative; width: 7px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.745em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3853" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3854" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.824em; vertical-align: -0.063em;"></span></span></nobr></span><script id="MathJax-Element-163" type="math/tex">I</script> as a set of atomic formulas <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-164-Frame" class="MathJax"><nobr><span id="MathJax-Span-3855" class="math"><span style="display: inline-block; position: relative; width: 84px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.992em, -0.426em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3856" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3857" class="mo">{</span><span id="MathJax-Span-3858" class="msubsup"><span style="display: inline-block; position: relative; width: 1.068em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3859" class="mi">ϕ</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.862em; left: 0.643em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3860" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3861" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-3862" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-3863" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-3864" class="msubsup"><span style="display: inline-block; position: relative; width: 1.068em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3865" class="mi">ϕ</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.862em; left: 0.643em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3866" class="mi">n</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3867" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.12em; vertical-align: -0.326em;"></span></span></nobr></span><script id="MathJax-Element-164" type="math/tex">\{\phi_1,\ldots,\phi_n\}</script>, or equivalently a
-    single formula <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-165-Frame" class="MathJax"><nobr><span id="MathJax-Span-3868" class="math"><span style="display: inline-block; position: relative; width: 176px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.992em, -0.453em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3869" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3870" class="mi">∃</span><span id="MathJax-Span-3871" class="msubsup"><span style="display: inline-block; position: relative; width: 0.89em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.553em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3872" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.464em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3873" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3874" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-3875" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-3876" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-3877" class="msubsup"><span style="display: inline-block; position: relative; width: 0.89em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.553em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3878" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.464em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3879" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3880" class="mo">.</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-3881" class="mtext">&nbsp;</span><span id="MathJax-Span-3882" class="msubsup"><span style="display: inline-block; position: relative; width: 1.068em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3883" class="mi">ϕ</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.862em; left: 0.643em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3884" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3885" class="mo">∧</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3886" class="mo">⋯</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-3887" class="mo">∧</span><span style="padding-left: 0.25em;" id="MathJax-Span-3888" class="msubsup"><span style="display: inline-block; position: relative; width: 1.068em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3889" class="mi">ϕ</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.862em; left: 0.643em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3890" class="mi">n</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.12em; vertical-align: -0.326em;"></span></span></nobr></span><script id="MathJax-Element-165" type="math/tex">\exists x_1,\ldots,x_k.~\phi_1 \wedge \cdots
-    \wedge \phi_n</script>, where <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-166-Frame" class="MathJax"><nobr><span id="MathJax-Span-3891" class="math"><span style="display: inline-block; position: relative; width: 64px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.903em, -0.553em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3892" class="mrow"><span id="MathJax-Span-3893" class="msubsup"><span style="display: inline-block; position: relative; width: 0.89em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.553em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3894" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.464em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3895" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3896" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-3897" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-3898" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-3899" class="msubsup"><span style="display: inline-block; position: relative; width: 0.89em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.553em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3900" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.464em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3901" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.766em; vertical-align: -0.231em;"></span></span></nobr></span><script id="MathJax-Element-166" type="math/tex">x_1,\ldots,x_k</script> are the existential
-    variables of <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-167-Frame" class="MathJax"><nobr><span id="MathJax-Span-3902" class="math"><span style="display: inline-block; position: relative; width: 7px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.745em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3903" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3904" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.824em; vertical-align: -0.063em;"></span></span></nobr></span><script id="MathJax-Element-167" type="math/tex">I</script>.
-    </p>
-    </div>
-
-</section>
-    <section id="attribute-matching">
-    
-<h3><span class="secno">4.2 </span> Attribute matching </h3>
-
-
-<p>We say that an object <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-168-Frame" class="MathJax"><nobr><span id="MathJax-Span-3905" class="math"><span style="display: inline-block; position: relative; width: 22px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.499em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3906" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3907" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3908" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3909" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-168" type="math/tex">obj</script> matches attributes <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-169-Frame" class="MathJax"><nobr><span id="MathJax-Span-3910" class="math"><span style="display: inline-block; position: relative; width: 126px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.901em, -0.438em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3911" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-3912" class="mo">[</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3913" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3914" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3915" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span id="MathJax-Span-3916" class="msubsup"><span style="display: inline-block; position: relative; width: 0.811em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.745em, -0.481em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3917" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.386em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3918" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3919" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3920" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3921" class="mi">a</span><span id="MathJax-Span-3922" class="msubsup"><span style="display: inline-block; position: relative; width: 0.717em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.756em, -0.485em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3923" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.291em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-3924" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3925" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-3926" class="mo">.</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-3927" class="mo">.</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-3928" class="mo">.</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-3929" class="mo">]</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.023em; vertical-align: -0.229em;"></span></span></nobr></span><script id="MathJax-Element-169" type="math/tex">[attr_1=val_1,...]</script> in structure <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-170-Frame" class="MathJax"><nobr><span id="MathJax-Span-3930" class="math"><span style="display: inline-block; position: relative; width: 16px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.763em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3931" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3932" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.843em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-170" type="math/tex">W</script> provided:
-for each attribute <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-171-Frame" class="MathJax"><nobr><span id="MathJax-Span-3933" class="math"><span style="display: inline-block; position: relative; width: 30px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.903em, -0.509em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3934" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3935" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3936" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3937" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span id="MathJax-Span-3938" class="msubsup"><span style="display: inline-block; position: relative; width: 0.636em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.745em, -0.481em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3939" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.386em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3940" class="mi">i</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.878em; vertical-align: -0.231em;"></span></span></nobr></span><script id="MathJax-Element-171" type="math/tex">attr_i</script>, we have <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-172-Frame" class="MathJax"><nobr><span id="MathJax-Span-3941" class="math"><span style="display: inline-block; position: relative; width: 178px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3942" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3943" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3944" class="mi">a</span><span id="MathJax-Span-3945" class="msubsup"><span style="display: inline-block; position: relative; width: 0.541em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.756em, -0.485em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3946" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.291em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3947" class="mi">i</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-3948" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-3949" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3950" class="mo">.</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3951" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3952" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3953" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3954" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3955" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3956" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3957" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3958" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3959" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3960" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3961" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3962" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3963" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span id="MathJax-Span-3964" class="msubsup"><span style="display: inline-block; position: relative; width: 0.636em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.745em, -0.481em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3965" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.386em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3966" class="mi">i</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3967" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-172" type="math/tex">val_i \in W.value(obj,attr_i)</script>.
-This is sometimes abbreviated as: <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-173-Frame" class="MathJax"><nobr><span id="MathJax-Span-3968" class="math"><span style="display: inline-block; position: relative; width: 138px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.514em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3969" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3970" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3971" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3972" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3973" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3974" class="mi">h</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3975" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3976" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3977" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3978" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3979" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3980" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3981" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-3982" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3983" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3984" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3985" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3986" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3987" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-173" type="math/tex">match(W,obj,attrs)</script>.
-</p>
-    
-</section>
-
-<section id="semantics-of-element-formulas">
-<h3><span class="secno">4.3 </span> Semantics of Element Formulas </h3>
-
-<section id="entity">
-
-<h4><span class="secno">4.3.1 </span> Entity </h4>
-
-<p>An entity formula is of the form <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-174-Frame" class="MathJax"><nobr><span id="MathJax-Span-3988" class="math"><span style="display: inline-block; position: relative; width: 104px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.495em); top: -2.57em; left: 0em;"><span id="MathJax-Span-3989" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3990" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3991" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3992" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3993" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3994" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3995" class="mi">y</span><span style="font-family: STIXGeneral;" id="MathJax-Span-3996" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3997" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3998" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-3999" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4000" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4001" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4002" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4003" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4004" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4005" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-174" type="math/tex">entity(id,attrs)</script> where <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-175-Frame" class="MathJax"><nobr><span id="MathJax-Span-4006" class="math"><span style="display: inline-block; position: relative; width: 14px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.758em, -0.477em); top: -2.57em; left: 0em;"><span id="MathJax-Span-4007" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4008" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4009" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.87em; vertical-align: -0.076em;"></span></span></nobr></span><script id="MathJax-Element-175" type="math/tex">id</script> denotes an entity.
-</p>
-<p>Entity formulas <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-176-Frame" class="MathJax"><nobr><span id="MathJax-Span-4010" class="math"><span style="display: inline-block; position: relative; width: 104px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.495em); top: -2.57em; left: 0em;"><span id="MathJax-Span-4011" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4012" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4013" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4014" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4015" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4016" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4017" class="mi">y</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4018" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4019" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4020" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4021" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-4022" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4023" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4024" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4025" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4026" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-4027" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-176" type="math/tex">entity(id,attrs)</script> can be interpreted as follows:
-</p>
-<div data-title="Formalism 12 (rule_12)" data-count="12" id="rule_12" class="formalism"><div class="ruleTitle"><a href="#rule_12" class="internalDFN">Formalism 12 (rule_12)</a></div>
- <p><span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-177-Frame" class="MathJax"><nobr><span id="MathJax-Span-4028" class="math"><span style="display: inline-block; position: relative; width: 158px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-4029" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4030" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-4031&qu