--- a/semantics/constraints.pl Wed Jan 09 17:24:26 2013 +0000
+++ b/semantics/constraints.pl Thu Jan 10 14:10:18 2013 +0000
@@ -13,6 +13,9 @@
%% A rule(Xs,Hyps,Ys,Body) is an implication
%% forall Xs. Hyps ==> exists Ys. Concls
%% A comment(Comment,Body) is a body with a surrounding comment
+%% - the body can be an atom or a list of atoms with math() fucntions
+%% the math() constructors are translated to appropriate
+%% delimiters of latex environments
%% A rules(L) is a list of rules, which will be formatted as a list.
%% Nested rules() are not guaranteed to work, but comments within list
%% elements are OK.
@@ -66,7 +69,10 @@
decl('Inference', 11, 'derivation-generation-use-inference',
- comment('In this inference, none of <em>a</em>, <em>gen<sub>2</sub></em> or <em>use_1</em> can be placeholders -.',
+ comment(['In this inference, none of ',
+ math(a), ',',
+ math(gen_2), ', or ',
+ math(use_1),' can be placeholders -.'],
rule( [id, e_2, e_1, a, gen_2, use_1, attrs],
[notNull(a),
notNull(gen_2),
@@ -78,7 +84,10 @@
decl('Inference', 12, 'revision-is-alternate-inference',
- comment('In this inference, any of <em>a</em>, <em>g</em> or <em>u</em> may be placeholders.',
+ comment(['In this inference, any of ',
+ math(a), ',',
+ math(gen_2), ', or ',
+ math(use_1),' can be placeholders -.'],
rule( [id, e_1, e_2, a, g,u],
[wasDerivedFrom(id, e_2,e_1,a,g,u,['prov:type = prov:Revision'])],
[alternateOf(e_2,e_1)]))).
@@ -119,11 +128,14 @@
rule( [id, e_2, e_1, a, g, u, attrs],
[wasDerivedFrom(id, e_2, e_1, a, g, u, attrs)],
[wasInfluencedBy(id, e_2, e_1, attrs)]),
- comment('In this rule, <em>a</em>, <em>g</em>, <em>u</em> may be placeholders -.',
+ comment(['In this rule, ',
+ math(a), ',',
+ math(g), ', or ',
+ math(u),' may be placeholders -.'],
rule( [id, e, ag, attrs],
[wasAttributedTo(id, e,ag,attrs)],
[wasInfluencedBy(id, e, ag, attrs)])),
- comment('In this rule, <em>pl</em> may be a placeholder -.',
+ comment(['In this rule, ',math(pl),' may be a placeholder -.'],
rule( [id, a, ag, pl, attrs],
[wasAssociatedWith(id, a,ag,pl,attrs)],
[wasInfluencedBy(id, a, ag, attrs)])),
@@ -321,7 +333,10 @@
[ preceq(inv_1, inv_2)])).
decl('Constraint', 41, 'derivation-usage-generation-ordering',
- comment('In this constraint, <math>a</math>, <math>gen_2</math>, <math>use_1</math> must not be placeholders.',
+ comment(['In this constraint',
+ math(a), ',',
+ math(gen_2), ', or ',
+ math(use_1),' must not be placeholders -.'],
rule( [ d,e_1,e_2,a,gen_2,use_1,attrs ],
[ notNull(a) ,
notNull(gen_2) ,
@@ -330,7 +345,10 @@
[ preceq(use_1, gen_2)]))).
decl('Constraint', 42, 'derivation-generation-generation-ordering',
- comment('In this constraint, any of <math>a</math>, <math>g</math>, <math>u</math> may be placeholders.',
+ comment(['In this constraint, any of ',
+ math(a), ',',
+ math(g), ', or ',
+ math(u),' may be placeholders -.'],
rule( [ d,gen_1,gen_2,e_1,e_2,a,a_1,a_2,g,u,t_1,t_2,attrs,attrs_1,attrs_2 ],
[ wasDerivedFrom(d, e_2,e_1,a,g,u,attrs) ,
wasGeneratedBy(gen_1, e_1,a_1,t_1,attrs_1) ,
@@ -372,7 +390,8 @@
[ preceq(inv_1, inv_2)])).
decl('Constraint', 47, 'wasAssociatedWith-ordering',
- comment('In the following inferences, pl may be a placeholder -.',
+ comment(['In the following inferences, ',
+ math(pl),' may be a placeholder -.'],
rules([rule([ assoc,start_1,inv_2,ag,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2 ],
[ wasAssociatedWith(assoc, a,ag,pl,attrs) ,
wasStartedBy(start_1, a,e_1,a_1,t_1,attrs_1) ,
@@ -440,6 +459,28 @@
[],
['False'])).
+decl('Constraint', 53,'impossible-property-overlap',
+ comment(['For each ',
+ math(r),' and ',
+ math('s \\in \\{ used, wasGeneratedBy, wasInvalidatedBy, wasStartedBy, wasEndedBy, wasInformedBy, wasAttributedTo, wasAssociatedWith, actedOnBehalfOf\\}'),
+ ' such that ',
+ math(r),' and ',
+ math(s),
+ ' are different relation names, the following constraint holds:'],
+ rule([id,a_1,'\\ldots',a_m,b_1,'\\ldots',b_n],
+ [r(id,a_1,'\\ldots',a_m),s(id,b_1,'\\ldots',b_n)],
+ [],[ 'False']))).
+
+decl('Constraint', 54,'impossible-object-property-overlap',
+ comment(['For each ',
+ math('p \\in \\{entity,activity,agent\\}'),' and each ',
+ math('r \\in \\{ used, wasGeneratedBy, wasInvalidatedBy, wasStartedBy, wasEndedBy, wasInformedBy, wasAttributedTo, wasAssociatedWith, actedOnBehalfOf\\}'),
+ ', the following constraint holds:'],
+ rule([id,a_1,'\\ldots',a_m,b_1,'\\ldots',b_n],
+ [p(id,a_1,'\\ldots',a_m),r(id,b_1,'\\ldots',b_n)],
+ [],[ 'False']))).
+
+
decl('Constraint', 55,'entity-activity-disjoint',
rule([id],
[typeOf(id,entity), typeOf(id,activity)],
@@ -459,6 +500,15 @@
emit(X,Y,Z) :- atom(X),atom_concat(Y,X,Z).
emit(X,Y,Z) :- number(X),number_codes(X,Xc),atom_codes(X2,Xc),atom_concat(Y,X2,Z).
+emitComment(A,_) --> {atom(A)}, emit(A).
+emitComment([X|Xs],T) --> emitComments([X|Xs],T).
+emitComment(math(A),latex) --> emit('$'),emit(A),emit('$').
+emitComment(math(A),html) --> emit('<span class="math">'),emit(A),emit('</span>').
+emitComment(math(A),wiki) --> emit('<math>'),emit(A),emit('</math>').
+
+emitComments([],_) --> [].
+emitComments([X|Xs],T) --> emitComment(X,T), emitComments(Xs,T).
+
emitListSep(_Sep,_Emit,[]) --> [].
emitListSep(_Sep,Emit,[P]) --> call(Emit,P).
emitListSep(Sep,Emit,[P,Q|Ps]) -->
@@ -492,15 +542,16 @@
latexRule(Xs,Hyps,Ys,Concls) -->
latexQuantify('\\forall ',Xs),
latexConj(Hyps),
- emit(' \\Longrightarrow '),
+ emit(' \\Rightarrow '),
latexQuantify('\\exists ',Ys),
latexConj(Concls).
latexInference(Xs,Hyps,Ys,Concls) -->
emit('\\begin{array}[t]{l}\n'),
latexQuantify('\\forall ',Xs),
+ emit('\n\\\\\n\\qquad '),
latexConj(Hyps),
- emit('\n\\\\\n\\Longrightarrow\n\\\\\n'),
+ emit('\n\\\\\n\\quad\\Rightarrow\n'),
latexQuantify('\\exists ',Ys),
latexConj(Concls),
emit('\n\\end{array}').
@@ -529,7 +580,7 @@
emit('\n</math>').
wikiRules(comment(Comment,Rules)) -->
- emit(Comment),
+ emitComment(Comment,wiki),
wikiRules(Rules).
wikiRules(rules(L)) -->
wikiRuleList(L).
@@ -623,7 +674,7 @@
emit('</span>').
htmlRules(comment(Comment,Rules)) -->
- emit(Comment),
+ emitComment(Comment,html),
htmlRules(Rules).
htmlRules(rules(L)) -->
emit('<ol>'),
@@ -652,6 +703,64 @@
emit('</div>\n').
+% html wrapping latex/MathJax
+
+htmlLatexRules(key(Id,Formula)) -->
+ emit('The identifier field '),
+ emit('$'),
+ emit(Id),
+ emit('$'),
+ emit(' is a <span class="conditional">KEY</span> for the '),
+ emit('$'),
+ latexPred(Formula),
+ emit('$'),
+ emit(' statement.').
+
+htmlLatexRules(rule(Xs,Hyps,Concls)) -->
+ emit('$'),
+ latexInference(Xs,Hyps,[],Concls),
+ emit('$').
+
+htmlLatexRules(rule(Xs,Hyps,Ys,Concls)) -->
+ emit('$'),
+ latexInference(Xs,Hyps,Ys,Concls),
+ emit('$').
+
+htmlLatexRules(comment(Comment,Rules)) -->
+ emitComment(Comment,latex),
+ emit('<br />'),
+ htmlLatexRules(Rules).
+htmlLatexRules(rules(L)) -->
+ emit('<ol>'),
+ htmlLatexRuleList(L),
+ emit('</ol>').
+
+% only handles one nesting level
+htmlLatexRuleList([]) --> [].
+htmlLatexRuleList([R|Rs]) -->
+ emit('<li>'),
+ htmlLatexRules(R),
+ emit('</li>'),
+ htmlLatexRuleList(Rs).
+
+htmlLatexDecl('Inference',Num,Name,Rules) -->
+ emit('<div class="inference" number="'),
+ emit(Num),
+ emit('" id="'),
+ emit(Name),
+ emit('">'),
+ htmlLatexRules(Rules),
+ emit('</div>\n').
+htmlLatexDecl('Constraint',Num,Name,Rules) -->
+ emit('<div class="constraint" number="'),
+ emit(Num),
+ emit('" id="'),
+ emit(Name),
+ emit('">'),
+ htmlLatexRules(Rules),
+ emit('</div>\n').
+
+
writeEach(_Target,_Out,[]).
writeEach(Target,Out,[Num|Nums]) :-
decl(Type,Num,Name,Rules),
@@ -667,5 +776,5 @@
close(Out).
main :- findall(X, decl(_,X,_,_),Xs),
- writeEach(wikiDecl,user,Xs).
+ writeEach(htmlLatexDecl,user,Xs).
--- a/semantics/prov-sem.html Wed Jan 09 17:24:26 2013 +0000
+++ b/semantics/prov-sem.html Thu Jan 10 14:10:18 2013 +0000
@@ -70,6 +70,15 @@
background: #fff;
}
+.formal {
+ padding: 1em;
+ margin: 1em 0em 0em;
+ border: 1px solid #0f0;
+ background: #fff;
+}
+
+
+
/* .inference[id]::before { */
/* content: "Inference: " attr(id); */
/* width: 380px; /\* How can we compute the length of "Constraint: " attr(id) *\/ */
@@ -562,11 +571,13 @@
function updateRules() {
var count=1;
- $('.constraint,.definition,.inference').each(function(index) {
+ $('.constraint,.definition,.inference,.formal').each(function(index) {
var myid=$(this).attr('id');
- var mycount=count++;
-
+ var mycount=$(this).attr('number');
+ if(mycount==undefined) {
+ mycount=count++;
+ }
if (myid==undefined) {
myid='rule_' + mycount;
$(this).attr('id',myid);
@@ -930,9 +941,9 @@
<ul>
<li> <a href="http://www.w3.org/TR/2012/WD-prov-overview-20121211/">PROV-OVERVIEW</a> (Note), an overview of the PROV family of documents [[PROV-OVERVIEW]];</li>
<li> <a href="http://www.w3.org/TR/2012/WD-prov-primer-20121211/">PROV-PRIMER</a> (Note), a primer for the PROV data model [[PROV-PRIMER]];</li>
-<li> <a href="http://www.w3.org/TR/2012/CR-prov-o-20121211/">PROV-O</a> (Recommendation), the PROV ontology, an OWL2 ontology allowing the mapping of PROV to RDF [[!PROV-O]];</li>
-<li> <a href="http://www.w3.org/TR/2012/CR-prov-dm-20121211/">PROV-DM</a> (Recommendation), the PROV data model for provenance [[!PROV-DM]];</li>
-<li> <a href="http://www.w3.org/TR/2012/CR-prov-n-20121211/">PROV-N</a> (Recommendation), a notation for provenance aimed at human consumption [[!PROV-N]];</li>
+<li> <a href="http://www.w3.org/TR/2012/CR-prov-o-20121211/">PROV-O</a> (Recommendation), the PROV ontology, an OWL2 ontology allowing the mapping of PROV to RDF [[PROV-O]];</li>
+<li> <a href="http://www.w3.org/TR/2012/CR-prov-dm-20121211/">PROV-DM</a> (Recommendation), the PROV data model for provenance [[PROV-DM]];</li>
+<li> <a href="http://www.w3.org/TR/2012/CR-prov-n-20121211/">PROV-N</a> (Recommendation), a notation for provenance aimed at human consumption [[PROV-N]];</li>
<li> <a
href="http://www.w3.org/TR/2012/CR-prov-constraints-20121211/">PROV-CONSTRAINTS</a>
(Recommendation), a set of constraints applying to the PROV data model;</li>
@@ -965,16 +976,11 @@
- <section id="conventions">
+<!--
+<section id="conventions">
<h3>Conventions</h3>
-
-<p>The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL
- NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and
- "OPTIONAL" in this document are to be interpreted as described in
- [[!RFC2119]].</p>
-
<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
@@ -983,7 +989,7 @@
provided merely as an aid to the reader. </p>
</section>
-
+-->
<section id="purpose">
@@ -1029,32 +1035,1156 @@
</section>
+<section id="wiki">
+<h2> Wiki stuff </h2>
+
+<h2> Overview </h2>
-<section id="inferences">
-<h2>Inferences</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>
-<div class="inference" id="communication-generation-use-inference"><span class="math">∀ id,a<sub>2</sub>,a<sub>1</sub>,attrs. wasInformedBy(id,a<sub>2</sub>,a<sub>1</sub>,attrs) ⟹ ∃ e,gen,t<sub>1</sub>,use,t<sub>2</sub>. wasGeneratedBy(gen,e,a<sub>1</sub>,t<sub>1</sub>,[]) ∧ used(use,a<sub>2</sub>,e,t<sub>2</sub>,[])</span></div>
-<div class="inference" id="generation-use-communication-inference"><span class="math">∀ gen,a<sub>1</sub>,t<sub>1</sub>,attrs<sub>1</sub>,id<sub>2</sub>,a<sub>2</sub>,t<sub>2</sub>,attrs<sub>2</sub>. wasGeneratedBy(gen,e,a<sub>1</sub>,t<sub>1</sub>,attrs<sub>1</sub>) ∧ used(id<sub>2</sub>,a<sub>2</sub>,e,t<sub>2</sub>,attrs<sub>2</sub>) ⟹ ∃ id. wasInformedBy(id,a<sub>2</sub>,a<sub>1</sub>,[])</span></div>
-<div class="inference" id="entity-generation-invalidation-inference"><span class="math">∀ gen,e,a<sub>1</sub>,t<sub>1</sub>,attrs<sub>1</sub>,id<sub>2</sub>,a<sub>2</sub>,t<sub>2</sub>,attrs<sub>2</sub>. wasGeneratedBy(gen,e,a<sub>1</sub>,t<sub>1</sub>,attrs<sub>1</sub>) ∧ used(id<sub>2</sub>,a<sub>2</sub>,e,t<sub>2</sub>,attrs<sub>2</sub>) ⟹ ∃ id. wasInformedBy(id,a<sub>2</sub>,a<sub>1</sub>,[])</span></div>
-<div class="inference" id="activity-start-end-inference"><span class="math">∀ a,t<sub>1</sub>,t<sub>2</sub>,attrs. activity(a,t<sub>1</sub>,t<sub>2</sub>,attrs) ⟹ ∃ start,e<sub>1</sub>,a<sub>1</sub>,end,a<sub>2</sub>,e<sub>2</sub>. wasStartedBy(start,a,e<sub>1</sub>,a<sub>1</sub>,t<sub>1</sub>,[]) ∧ wasEndedBy(end,a,e<sub>2</sub>,a<sub>2</sub>,t<sub>2</sub>,[])</span></div>
-<div class="inference" id="wasStartedBy-inference"><span class="math">∀ id,a,e<sub>1</sub>,a<sub>1</sub>,t,attrs. wasStartedBy(id,a,e<sub>1</sub>,a<sub>1</sub>,t,attrs) ⟹ ∃ gen,t<sub>1</sub>. wasGeneratedBy(gen,e<sub>1</sub>,a<sub>1</sub>,t<sub>1</sub>,[])</span></div>
-<div class="inference" id="wasEndedBy-inference"><span class="math">∀ id,a,e<sub>1</sub>,a<sub>1</sub>,t,attrs. wasEndedBy(id,a,e<sub>1</sub>,a<sub>1</sub>,t,attrs) ⟹ ∃ gen,t<sub>1</sub>. wasGeneratedBy(gen,e<sub>1</sub>,a<sub>1</sub>,t<sub>1</sub>,[])</span></div>
-<div class="inference" id="derivation-generation-use-inference">In this inference, none of <em>a</em>, <em>gen<sub>2</sub></em> or <em>use_1</em> can be placeholders -.<span class="math">∀ id,e<sub>2</sub>,e<sub>1</sub>,a,gen<sub>2</sub>,use<sub>1</sub>,attrs. wasDerivedFrom(id,e<sub>2</sub>,e<sub>1</sub>,a,gen<sub>2</sub>,use<sub>1</sub>,attrs) ⟹ ∃ s,t<sub>1</sub>,t<sub>2</sub>. used(use<sub>1</sub>,a,e<sub>1</sub>,t<sub>1</sub>,[]) ∧ wasGeneratedBy(gen<sub>2</sub>,e<sub>2</sub>,a,t<sub>2</sub>,[])</span></div>
-<div class="inference" id="revision-is-alternate-inference">In this inference, any of <em>a</em>, <em>g</em> or <em>u</em> may be placeholders.<span class="math">∀ id,e<sub>1</sub>,e<sub>2</sub>,a,g,u. wasDerivedFrom(id,e<sub>2</sub>,e<sub>1</sub>,a,g,u,.(prov:type = prov:Revision,[])) ⟹ alternateOf(e<sub>2</sub>,e<sub>1</sub>)</span></div>
-<div class="inference" id="attribution-inference"><span class="math">∀ att,e,ag,attrs. wasAttributedTo(att,e,ag,attrs) ⟹ ∃ a,t,gen,assoc,pl. wasGeneratedBy(gen,e,a,t,[]) ∧ wasAssociatedWith(assoc,a,ag,pl,[])</span></div>
-<div class="inference" id="delegation-inference"><span class="math">∀ id,ag<sub>1</sub>,ag<sub>2</sub>,a,attrs. actedOnBehalfOf(id,ag<sub>1</sub>,ag<sub>2</sub>,a,attrs) ⟹ ∃ id<sub>1</sub>,pl<sub>1</sub>,id<sub>2</sub>,pl<sub>2</sub>. wasAssociatedWith(id<sub>1</sub>,a,ag<sub>1</sub>,pl<sub>1</sub>,[]) ∧ wasAssociatedWith(id<sub>2</sub>,a,ag<sub>2</sub>,pl<sub>2</sub>,[])</span></div>
-<div class="inference" id="influence-inference"><ol><li><span class="math">∀ id,e,a,t,attrs. wasGeneratedBy(id,e,a,t,attrs) ⟹ wasInfluencedBy(id,e,a,attrs)</span></li><li><span class="math">∀ id,a,e,t,attrs. used(id,a,e,t,attrs) ⟹ wasInfluencedBy(id,a,e,attrs)</span></li><li><span class="math">∀ id,a<sub>2</sub>,a<sub>1</sub>,attrs. wasInformedBy(id,a<sub>2</sub>,a<sub>1</sub>,attrs) ⟹ wasInfluencedBy(id,a<sub>2</sub>,a<sub>1</sub>,attrs)</span></li><li><span class="math">∀ id,a<sub>2</sub>,e,a<sub>1</sub>,t,attrs. wasStartedBy(id,a<sub>2</sub>,e,a<sub>1</sub>,t,attrs) ⟹ wasInfluencedBy(id,a<sub>2</sub>,e,attrs)</span></li><li><span class="math">∀ id,a<sub>2</sub>,e,a<sub>1</sub>,t,attrs. wasEndedBy(id,a<sub>2</sub>,e,a<sub>1</sub>,t,attrs) ⟹ wasInfluencedBy(id,a<sub>2</sub>,e,attrs)</span></li><li><span class="math">∀ id,e,a,t,attrs. wasInvalidatedBy(id,e,a,t,attrs) ⟹ wasInfluencedBy(id,e,a,attrs)</span></li><li><span class="math">∀ id,e<sub>2</sub>,e<sub>1</sub>,a,g,u,attrs. wasDerivedFrom(id,e<sub>2</sub>,e<sub>1</sub>,a,g,u,attrs) ⟹ wasInfluencedBy(id,e<sub>2</sub>,e<sub>1</sub>,attrs)</span></li><li>In this rule, <em>a</em>, <em>g</em>, <em>u</em> may be placeholders -.<span class="math">∀ id,e,ag,attrs. wasAttributedTo(id,e,ag,attrs) ⟹ wasInfluencedBy(id,e,ag,attrs)</span></li><li>In this rule, <em>pl</em> may be a placeholder -.<span class="math">∀ id,a,ag,pl,attrs. wasAssociatedWith(id,a,ag,pl,attrs) ⟹ wasInfluencedBy(id,a,ag,attrs)</span></li><li><span class="math">∀ id,ag<sub>2</sub>,ag<sub>1</sub>,a,attrs. actedOnBehalfOf(id,ag<sub>2</sub>,ag<sub>1</sub>,a,attrs) ⟹ wasInfluencedBy(id,ag<sub>2</sub>,ag<sub>1</sub>,attrs)</span></li></ol></div>
-<div class="inference" id="alternate-reflexive"><span class="math">∀ e. entity(e) ⟹ alternateOf(e,e)</span></div>
-<div class="inference" id="alternate-transitive"><span class="math">∀ e<sub>1</sub>,e<sub>2</sub>,e<sub>3</sub>. alternateOf(e<sub>1</sub>,e<sub>2</sub>) ∧ alternateOf(e<sub>2</sub>,e<sub>3</sub>) ⟹ alternateOf(e<sub>1</sub>,e<sub>3</sub>)</span></div>
-<div class="inference" id="alternate-symmetric"><span class="math">∀ e<sub>1</sub>,e<sub>2</sub>. alternateOf(e<sub>1</sub>,e<sub>2</sub>) ⟹ alternateOf(e<sub>2</sub>,e<sub>1</sub>)</span></div>
-<div class="inference" id="specialization-transitive"><span class="math">∀ e<sub>1</sub>,e<sub>2</sub>,e<sub>3</sub>. specializationOf(e<sub>1</sub>,e<sub>2</sub>) ∧ specializationOf(e<sub>2</sub>,e<sub>3</sub>) ⟹ specializationOf(e<sub>1</sub>,e<sub>3</sub>)</span></div>
-<div class="inference" id="specialization-alternate-inference"><span class="math">∀ e<sub>1</sub>,e<sub>2</sub>. specializationOf(e<sub>1</sub>,e<sub>2</sub>) ⟹ alternateOf(e<sub>1</sub>,e<sub>2</sub>)</span></div>
-<div class="inference" id="specialization-attributes-inference"><span class="math">∀ e<sub>1</sub>,attrs,e<sub>2</sub>. entity(e<sub>1</sub>,attrs) ∧ specializationOf(e<sub>2</sub>,e<sub>1</sub>) ⟹ entity(e<sub>2</sub>,attrs)</span></div>
-<div class="constraint" id="membership-empty-collection"><span class="math">∀ c,e. hasMember(c,e) ∧ prov:EmptyCollection \in typeOf(c) ⟹ False</span></div>
+<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 assertions 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 assertions 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 "world model" to describe the mathematical structure that corresponds to actual state of affairs, and will try to avoid ambiguous, unqualified uses of the word "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>
+<h2> Basics </h2>
+
+
+
+
+<p>I will use syntax for PROV-DM records (which I will usually call
+formulas) as described in the Candidate Recommendation of PROV-DM (<a href="http://www.w3.org/TR/2012/CR-prov-dm-20121211"> PROV-DM CR</a>).
+</p>
+<p>A PROV-DM instance, or set of atomic formulas $\phi_1$...$\phi_n$, is interpreted as a conjunction, that is, the overall instance is considered to hold in a given structure if each atomic formula in it holds.</p>
+
+<p>The rest of the document will discuss the structures and define when an atomic assertion holds in a given world.
+</p>
+<h3> Identifiers </h3>
+
+<p>A lowercase symbol $x,y,...$ on its own denotes an identifier. Identifiers may or may not be URIs. I view identifiers as being like variables in logic (or blank nodes in RDF): just because we have two different identifiers $x$ and $y$ doesn't tell us that they denote different things, since we could discover that they are actually the same later. We write $Identifiers$ for the set of identifiers of interest in a given situation (typically, the set of identifiers present in the PROV instance of interest).
+</p>
+<h3> Times and Intervals </h3>
+
+<p>We assume a linearly ordered set $(Times,\leq)$ 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 $Intervals$ of closed intervals of the form $\{t \mid t_1 \leq t \leq t_2\}$.
+</p>
+
+<h3> Attributes and Values </h3>
+
+<p>We assume a set $Attributes$ of attribute labels and a set $Values$ of possible values of attributes.
+</p>
+<h2>Formulas </h2>
+
+<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 $r(id,a_1,\ldots,a_n)$ instead of the semicolon notation $r(id;a_1,\ldots,a_n)$.
+</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 $wasAssociatedWith$ and $wasDerivedFrom$, as shown in the grammar below.
+</p>
+$$
+\begin{array}{rcl}
+ formula &::=& element\_formula\\
+ & | & relation\_formula\\
+ element\_formula
+ &::= &entity(id,attrs) \\
+ & |& activity(id,st,et,attrs)\\
+ & |& agent(id,attrs)\\
+ relation\_formula
+ &::=& wasGeneratedBy(id,e,a,t,attrs)\\
+ & |& used(id,e,a,t,attrs)\\
+ & |& wasInvalidatedBy(id,e,a,t,attrs)\\
+ & |& wasAssociatedWith(id,ag,act,pl,attrs)\\
+ & |& wasAssociatedWith(id,ag,act,-,attrs)\\
+ & |& wasStartedBy(id,a_2,e,a_1,attrs)\\
+ & |& wasEndedBy(id,a_2,e,a_1,attrs)\\
+ & |& wasAttributedTo(id,e,ag,attrs)\\
+ & |& actedOnBehalfOf(if,ag_2,ag_1,act,attrs)\\
+ & |& wasDerivedFrom(id,e_2,e_1,act,g,u,attrs)\\
+ & |& wasDerivedFrom(id,e_2,e_1,-,-,-,attrs)\\
+ & |& alternateOf(e_1,e_2)\\
+ & |& specializationOf(e_1,e_2)
+\end{array}
+$$
+
+<h2> World Models </h2>
+
+<h3> Things </h3>
+
+<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 world model $W$ includes
+</p>
+<ul>
+ <li> a set $Things$ of things</li>
+ <li> a function $lifetime : Things \to Intervals$ from objects to time intervals</li>
+ <li>a function $value : Things \times Attributes \times Times \to Values_\bot$
+</li>
+</ul>
+
+<p>Note that this description does not say what the structure of an object 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 $\pi$. All that matters from our point of view is that we know how to map the object to its time interval and attribute mapping.
+</p>
+
+<p>The range of the $value$ function us $Values_\bot$, that is, $Values \uplus \{\bot\}$, the set of values with an additional element $\bot \notin Values$. When $value(x,a,t) = \bot$, we say that attribute $a$ is undefined for $x$ at time $t$.
+</p>
+
+<p>It is possible for two Things to be indistinguishable by their attribute values and lifetime, but have different identity.
+</p>
+
+<h3> Objects </h3>
+
+A <em>Object</em> is described by a time interval and attributes with unchanging values. Objects encompass entities, interactions, and activities.
+
+To model this, a world includes
+
+<ul><li> a set $Objects$
+</li><li> a function $lifetime : Objects \to Intervals$ from objects to time intervals
+</li><li> a function $value : Objects \times Attributes \to Values_\bot$
+</li></ul>
+
+Intuitively, $lifetime(e)$ is the time interval during which object $e$ exists. The value $value(e,a)$ is the value of attribute $a$ during the object's lifetime.
+
+As with <em>Things</em>, the range of $value$ includes the special undefined value $\bot$, making $value$ effectively a partial 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 $Objects$ and $Things$ are disjoint; however, certain objects, namely entities, are linked to things.
+
+<h4> 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:
+
+<ul><li> a set $Entities \subseteq Objects$ of entities, disjoint from $Activities$ and $Events$ below.
+</li><li> a function $thingOf : Entities \to Things$ that associates each Entity with a Thing, such that for each $t \in lifetime(obj)$, and for each attribute $a$ such that $value(obj,a) \neq \bot$, we have $value(obj,a) = value(thingOf(obj),a,t)$.
+</li><li> $lifetime(e) \subseteq lifetime(t)$.
+</li></ul>
+
+<div class="remark"><p> Although both entities and things can have undefined attribute values, their meaning is slightly different: for a thing, $value(x,a,t) = \bot$ means that the attribute $a$ has no value at time $t$, whereas for an entity, $value(x,a) = \bot$ only means that the entity does not record a fixed value for $a$. This does not imply that $value(thingOf(e),a,t) = \bot$ when $t \in lifetime(e)$. In particular, if the $value(x,a,t)$ has multiple values during the lifetime of $e$, then $value(e,a)$ <b>must</b> be $\bot$, since assigning a value to $value(e,a)$ would violate condition (3) above.
+</p>
+ </div>
+
+<h5> Plans </h5>
+<p>We identify a specific subset of the entities called <em>plans</em>, $Plans \subseteq Entities$.
+</p>
+<h4> Agents </h4>
+
+<p>An agent is an entity that can act, by controlling, starting, ending, or participating in activities. Agents can act on behalf of other agents. We introduce:
+</p>
+* a set $Agents \subseteq Objects$ of agents.
+
+<h4> Actvities </h4>
+
+
+<p>An <em>activity</em> is an object that encompasses a set of events. We introduce
+</p>
+* a set $Activities \subseteq Objects$ of activities, disjoint from $Entities$ and $Events$
+
+
+<h4> Interactions </h4>
+
+<p>We consider a $Interactions \subseteq Objects$ which are split into <em>Events</em> between entities and activities, <em>Associations</em> between agents and activities, 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>
+<ul><li> $Interactions = (Events \cup Associations) \cup Derivations \subseteq Objects$
+</li><li> $(Events \cup Associations) \cap Derivations = \emptyset$
+</li><li> $Interactions \cap (Entities \cup Activities \cup Agents) = \emptyset$
+</li></ul>
+
+<h5> 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, starting and ending (possibly more may be added such as destruction/invalidation of an entity). Events are instantaneous. We introduce:
+</p>
+<div class="formal">
+<ul><li> A set $Events \subseteq Interactions$ of events.
+</li><li> A function $time : Events \to Times$ giving the time of each event; i.e. $lifetime(evt) = \{time(t)\}$.
+</li><li> The derived ordering on events given by $evt_1 \leq evt_2 \iff time(evt_1) \leq time(evt_2)$
+</li><li> A function $type: Events \to \{start,end,use,generate\}$ such that Events have types in $\{start,end,use,generate\}$.
+</li></ul>
+</div>
+<h5> Associations </h5>
+
+<p>An <em>Association</em> is an interaction relating an agent to an activity. Associations can overlap with events; for example, a start event is also an association. To model associations, we introduce:
+</p>
+<div class="formal">
+ <p>A set $Associations \subseteq Interactions$, such that every event $evt \in Events$ that is a start or end event is also an association. That is, $type(evt) \in \{start,end\}$ implies $evt \in Associations$
+</p>
+ </div>
+
+<p>Associations are used below in the $ActsFor$ and $AssociatedWith$ relations.
+<b>TODO: Add types for association or delegation?</b>
+</p>
+<h5> Derivations </h5>
+
+<p>A <em>Derivation</em> is an interaction chaining one or more generation and use steps. Derivations can also carry attributes, so we introduce an explicit kind of interaction for them that can carry attributes.
+</p>
+<div class="formal">
+<p> * A set $Derivations \subseteq Interactions$, disjoint from $Events \cup Associations$.
+</p></div>
+<p>See below for the associated derivation path and DerivedFrom relation.
+</p>
+<h3> Relations </h3>
+
+<h4> Simple relations </h4>
+<p>The entities, interactions, and activities in a world model are related in the following ways:
+</p>
+<ul><li> A relation $Used \subseteq Events \times Entities$ saying when an event used an entity. An event can use at most one entity, and if $(evt,e)\in Used$ then $time(evt) \in lifetime(e)$ and $type(g) = use$ must hold.
+</li><li> A relation $Generated \subseteq Events \times Entities$ saying when an event generated an entity. An event can generate at most one entity, and if $(evt,e)\in Generated$ then $min(lifetime(e)) = time(evt)$ and $type(g) = generation$ must hold.
+</li><li> A relation $Invalidated \subseteq Events \times Entities$ saying when an event invalidated an entity. An event can invalidate at most one entity, and if $(evt,e)\in Invalidated$ then $min(lifetime(e)) = time(evt)$ and $type(g) = invalidation$ must hold.
+</li><li> A relation $EventActivity \subseteq Events \times Activities$ associating activities with events, such that $(act,evt) \in EventActivity$ implies $time(evt) \in lifetime(act)$.
+</li><li> A relation $AssociatedWith \subseteq Association \times Agents \times Activities \times Plans^?$ 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 $ActsFor \subseteq Agents \times Agents \times Activities$ indicating when one agent acts on behalf of another with respect to a given activity.
+</li></ul>
+
+<h4> 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>
+
+$$ent_n\cdot g_n\cdot act_n\cdot u_n\cdot ent_{n-1}\cdot ...\cdot
+ent_1\cdot g_1\cdot act_1\cdot u_1\cdot ent_0$$
+
+<p>where the $ent_i$ are entities, $act_i$ are activities, $g_i$ are generations, and $u_i$ are usages.
+</p>
+<p>Formally, we consider the (regular) language:
+</p>
+$$DerivationPaths = Entities \cdot (Events \cdot Activities \cdot
+Events \cdot Entities)^+$$
+<p>with the constraints that for each derivation path:
+</p>
+<ol>
+<li>for each substring $ent\cdot g \cdot act$ we have $(g,ent) \in Generated$ and $(g,act) \in EventActivities$, and
+</li>
+<li>for each substring $act \cdot u \cdot ent$ we have $(u,ent) \in Used$ and $(u,act) \in EventActivities$.
+and we use this language to give meaning to derivations:
+</li>
+</ol>
+<p>We also consider a function $derivedFrom : Derivations \to
+DerivationPaths$ 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>
+
+<h3> Putting it all together </h3>
+
+<p>A <em>world model</em> W is a structure containing all of the above described data. If we need to talk about the objects or relations of more than one world model then we may write $W_1.Objects$; otherwise, to decrease notational clutter, when we consider a fixed world model then the names of the sets, relations and functions above refer to the components of that model.
+</p>
+<p><b>TODO: List the components.</b>
+</p>
+<h2> Semantics </h2>
+
+<p>In what follows, let $W$ be a fixed world model with the associated sets and relations discussed in the previous section, and let $I$ be an interpretation of identifiers as objects in $W$.
+</p>
+<p>The annotations [WF] refer to well-formedness constraints that correspond to typing constraints.
+</p>
+<h3> 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>.
+</p>
+<p>The mapping from identifiers to objects may <b>not</b> change over time. Thus, we consider interpretations as follows:
+</p>
+<div class="formal"> An interpretation function $I : Identifiers \to Objects$ describing which object is the target of each identifier.
+</div>
+
+<h3> Satisfaction </h3>
+
+<p>Consider an atomic formula $\phi$, a world $W$ and an interpretation $I$. We define notation $W,I \models \phi$ which means that $\phi$ is satisfied in $W,I$. For basic assertions, the definition of the satisfaction relation is given in the next few subsections. For a conjunction of assertions $\phi_1,\ldots,\phi_n$ we write $W,I \models \phi_1,\ldots,\phi_n$ to indicate that $W,I \models \phi_1$ and ... and $W,I \models \phi_n$ hold.
+</p>
+<p><b>TODO: Satisfiability of additional formulas to explain constraints/inferences</b>
+</p>
+<h3> Attribute matching </h3>
+
+
+<p>We say that an object $obj$ matches attributes $[attr_1=val_1,...]$ in world $W$ provided:
+</p>
+<ul><li> for each attribute $attr_i$, we have $W.value(obj,attr_i) = val_i$.
+This is sometimes abbreviated as: $match(W,obj,attrs)$
+</li></ul>
+
+<h3> Semantics of Element Records </h3>
+
+<h4> Entity Records </h4>
+
+<p>An entity formula is of the form $entity(id,attrs)$ where $id$ denotes an entity.
+</p>
+<p>Entity assertions $entity(id,attrs)$ can be interpreted as follows:
+</p>
+<div class="formal">
+ $W,I \models entity(id,attrs)$ if and only if:
+<ol>
+<li>[WF] $id$ denotes an entity $ent = I(id) \in Entities$
+</li>
+<li>the attributes match: $match(W,ent, attrs)$.
+</li>
+</ol>
+</div>
+
+<p>For example, the following formulas both hold if $x$ denotes an entity $e$ such that $value(e,a) = 4, value(e,b) = 5, value(e,c) = 6$ hold:
+</p><pre>
+ entity(x,[a=4,b=5])
+ entity(x,[a=4,b=5,c=6])
+</pre>
+
+<h4> Activity Records </h4>
+
+<p>An activity record is of the form $activity(id,st,et,attrs)$ where $id$ is a identifier referring to the activity, $st$ is a start time and $et$ is an end time.
+</p>
+<div class="formal">
+ <p>We say that $W,I \models activity(id,st,et,attrs)$ if and only if:</p>
+<ol>
+<li>[WF] The identifier $id$ maps to an activity $act = I(id) \in Activities$
+</li>
+<li>If $st$ is specified then it is equal to the start time of the activity, that is: $min(lifetime(id)) = st$
+</li>
+<li>If $et$ is specified then it is equal to the end time of the activity, that is: $max(lifetime(id)) = et$
+</li>
+<li>The attributes match: $match(W,act,attrs)$.
+</li>
+</ol>
+</div>
+
+<h4> Agent Records </h4>
+
+<p>An agent formula is of the form $agent(id,attrs)$ where $id$ denotes the agent and $attrs$ describes additional attributes.
+</p>
+<div class="formal">
+ <p>Agent assertions $agent(id,attrs)$ can be interpreted as follows:
+ $W,I \models agent(id,attrs)$ if and only if:
+ </p>
+ <ol>
+ <li>[WF] $id$ denotes an agent $ag = I(id) \in Agents$
+ </li>
+ <li>The attributes match: $match(W,ag,attrs)$.
+ </li>
+ </ol>
+</div>
+
+<h3> Semantics of Relations </h3>
+
+<h4> Entity-Activity </h4>
+
+<h5> Generation </h5>
+
+<p>The generation assertion is of the form $wasGeneratedBy(id,e,a,t,attrs)$ where $id$ is an event identifier, $e$ is an entity identifier, $a$ is an activity identifier, $attrs$ is a set of attribute-value pairs, and $t$ is an optional time.
+</p>
+<div class="formal">
+ $W,I \models wasGeneratedBy(id,e,a,t,attrs)$ if and only if:
+<ol>
+<li>[WF] The identifier $id$ denotes an event $evt = I(id) \in Events$
+</li>
+<li>[WF] The identifier $e$ denotes an entity $ent = I(e) \in Entities$
+</li>
+<li>[WF] The identifier $a$ denotes an activity $act = I(a) \in Activities$.
+</li>
+<li>The event $evt$ is involved in $act$, that is, such that $(evt,act) \in EventActivities$.
+</li>
+<li>The type of $evt$ is $generation$, i.e. $type(evt) = generation$
+</li>
+<li>The event $evt$ occurred at time $t$, i.e. $time(evt) = t$
+</li>
+<li>The event $evt$ generated $ent$, i.e. $(evt,ent) \in Generated$.
+</li>
+<li>The attribute values match: $match(W,evt,attrs)$
+</li>
+</ol>
+</div>
+
+<h5> Use </h5>
+
+The use assertion is of the form $used(id,a,e,t,attrs)$ where $id$ denotes an event, $a$ is an activity identifier, $e$ is an object identifier, $attrs$ is a set of attribute-value pairs, and $t$ is an optional time.
+
+<div class="formal"> $W,I \models used(id,a,e,t,attrs)$ if and only if:
+</li>
+<li>[WF] The identifier $id$ denotes an event $evt = I(id) \in Events$
+</li>
+<li>[WF] The identifier $a$ denotes an activity $act = I(id) \in Activities$
+</li>
+<li>[WF] The identifier $e$ denotes an entity $ent = I(e) \in Entities$
+</li>
+<li>The event $evt$ is part of $act$, i.e. $(evt,act) \in EventActivities$.
+</li>
+<li>The type of $evt$ is $use$, i.e., $type(evt) = use$.
+</li>
+<li>The event $evt$ occurred at time $t$, i.e. $time(evt) = t$
+</li>
+<li>The event $evt$ used $obj$, i.e. $(evt,ent) \in Used$.
+</li>
+<li>The attribute values match: $match(W,evt,attrs)$
+</li></ol></div>
+
+<h5> Invalidation </h5>
+
+The invalidation assertion is of the form $wasInvalidatedBy(id,e,a,t,attrs)$ where $id$ is an event identifier, $e$ is an entity identifier, $a$ is an activity identifier, $attrs$ is a set of attribute-value pairs, and $t$ is an optional time.
+
+<div class="formal"> $W,I \models wasInvalidatedBy(id,e,a,t,attrs)$ if and only if:
+<ol>
+<li>[WF] The identifier $id$ denotes an event $evt = I(id) \in Events$
+</li>
+<li>[WF] The identifier $e$ denotes an entity $ent = I(e) \in Entities$
+</li>
+<li>[WF] The identifier $a$ denotes an activity $act = I(a) \in Activities$.
+</li>
+<li>The event $evt$ is involved in $act$, that is, such that $(evt,act) \in EventActivities$.
+</li>
+<li>The type of $evt$ is $invalidation$, i.e. $type(evt) = invalidation$
+</li>
+<li>The event $evt$ occurred at time $t$, i.e. $time(evt) = t$
+</li>
+<li>The event $evt$ invalidated $ent$, i.e. $(evt,ent) \in Invalidated$.
+</li>
+<li>The attribute values match: $match(W,evt,attrs)$
+</li></ol></div>
+<h4> Agent-Activity </h4>
+
+<h5> Association Records </h5>
+
+An association record has the form $wasAssociatedWith(id,a,ag,pl,attrs)$.
+
+<div class="formal"> $W,I \models wasAssociatedWith(id,a,ag,pl,attrs)$ holds if and only if:
+<ol>
+<li>[WF] $assoc$ denotes an association $assoc = I(id) \in Associations$.
+</li>
+<li>[WF] $a$ denotes an activity $act = I(a) \in Activities$.
+</li>
+<li>[WF] $ag$ denotes an agent $agent = I(ag) \in Agents$.
+</li>
+<li>[WF] $pl$ is either the placeholder $-$ or denotes a plan $plan=I(pl) \in Plans$.
+</li>
+<li>The association associates the agent with the activity and plan, i.e. $(assoc,agent,act,plan) \in AssociatedWith$.
+</li>
+<li>The attributes match: $match(W,assoc,attrs)$.
+</li></ol></div>
+
+<h5> Start Records </h5>
+
+A start record $wasStartedBy(id,a2,e,a1,attrs)$ is interpreted as follows:
+
+<div class="formal"> $W,I \models wasStartedBy(id,a2,e,a1,attrs)$ holds if and only if:
+<ol>
+<li>[WF] $id$ denotes an event $evt = I(id) \in Events$
+</li>
+<li>[WF] $a2$ denotes an activity $act2 = I(a2)$
+</li>
+<li>[WF] $e$ denotes an entity $ent = I(e)$
+</li>
+<li>[WF] $a1$ denotes an activity $act1 = I(a1)$
+</li>
+<li>The event $evt$ has type $start$, i.e. $type(evt) = start$.
+</li>
+<li>The event happened at the start of $act2$, that is, $(act2,evt) \in ActivitiesEvents$, and $min(lifetime(act2)) = time(evt)$.
+</li>
+<li>The event happened during $act1$, that is, $(act1,evt) \in ActivitiesEvents$.
+</li>
+<li>The attributes match: $match(W,evt,attrs)$.
+</li></ol></div>
+
+<h5> End Records </h5>
+
+An activity end record $wasEndedBy(id,a2,e,a1,attrs)$ is interpreted as follows:
+
+<div class="formal">
+ $W,I \models wasEndedBy(id,a2,e,a1,attrs)$ holds if and only if:
+<ol>
+ <li>[WF] $id$ denotes an event $evt = I(id) \in Events$</li>
+<li> [WF] $a2$ denotes an activity $act2 = I(a2)$</li>
+<li> [WF] $e$ denotes an entity $ent = I(e)$</li>
+<li>[WF] $a1$ denotes an activity $act1 = I(a1)$</li>
+<li>The event $evt$ has type $end$, i.e. $type(evt) = end$.</li>
+<li>The event happened at the end of $act2$, that is, $(act2,evt) \in ActivitiesEvents$, and $max(lifetime(act2)) = time(evt)$.</li>
+<li>The event happened during $act1$, that is, $(act1,evt) \in ActivitiesEvents$.</li>
+<li>The attributes match: $match(W,evt,attrs)$.</li>
+</ol></div>
+
+<h4> Agent-Entity </h4>
+
+<h5> Attribution Records </h5>
+
+An attribution record $wasAttributedTo(id,e,ag,attrs)$ is interpreted as follows:
+
+<div class="formal"> $W,I \models wasAttributedTo(id,e,ag,attrs)$ holds if and only if:
+<ol>
+<li>[WF] $id$ denotes an event $evt = I(id)$ that is also an association $evt \in Associations$
+</li>
+<li>[WF] $e$ denotes an entity $ent = I(e)$
+</li>
+<li>[WF] $ag$ denotes an agent $agent = I(ag)$
+</li>
+<li>The event $evt$ has type $attribution$, i.e. $type(evt) = attribution$.
+</li>
+<li>The entity was attributed to the agent, i.e. $(id,ent,ag) \in AttributedTo$
+</li>
+<li>The attributes match: $match(W,evt,attrs)$.
+</li></ol></div>
+
+<h4> Agent-Agent </h4>
+
+<h5> Responsibility </h5>
+
+The $actedOnBehalfOf(id,ag2,ag1,act,attrs)$ relation is interpreted using the $ActsFor$ relation as follows:
+
+<div class="formal"> $W,I \models actedOnBehalfOf(id,ag2,ag1,act,attrs)$ holds if and only if:
+<ol>
+<li>[WF] $id$ denotes an association $assoc=I(id) \in Associations$ that is an association interaction, and $type(id) = responsibility$.
+</li>
+<li>[WF] $a$ denotes an activity $act=I(a) \in Activities$ is an activity.
+</li>
+<li>[WF] $ag1,ag2$ denote agents $agent1=I(ag1), agent2=I(ag2) \in Agents$ are agents.
+</li>
+<li>The agent $agent2$ acts for the agent $agent1$ with respect to the activity $act$, i.e. $(agent2,agent1,act) \in ActsFor$.
+</li>
+<li>[Redundant?] The association $id$ associates both agents with the activity, i.e. $(assoc,agent1,act),(assoc,agent2,act) \in AssociatedWith$.
+</li>
+<li>The attributes match: $match(W,assoc,attrs)$.
+</li></ol></div>
+
+<h4> Entity-Entity </h4>
+
+<h5> Derivation </h5>
+
+<h6> Precise </h6>
+
+<p>A precise derivation record has the form $wasDerivedFrom(id,e2,e1,a,g,u,attrs)$.
+</p>
+<div class="formal"> $W,I \models wasDerivedFrom(id,e2,e1,act,g,u,attrs)$ if and only if:
+<ol>
+<li>[WF] $id$ denotes a derivation $deriv = I(id) \in Derivations$
+</li>
+<li>[WF] $e1,e2$ denote entities $ent1 = I(e1), ent2=I(e2) \in Entities$
+</li>
+<li>[WF] $a$ denotes an activity $act = I(a) \in Activities$
+</li>
+<li>[WF] $g$ denotes a generation event $gen = I(g) \in Events$ and $type(I(g)) = generation$
+</li>
+<li>[WF] $u$ denotes a use event $I(u) \in Events$ and $type(I(u)) = use$
+</li>
+<li>The derivation denotes a valid one-step derivation $derivedFrom(deriv) = I(e2) \cdot I(g) \cdot I(act) \cdot I(u) \cdot I(e1)$
+</li>
+<li>The attribute values match: $match(W,deriv,attrs)$.</li>
+</ol>
+</div>
+
+<h6> Imprecise </h6>
+<p>
+An imprecise derivation record has the form $wasDerivedFrom(id,e2,e1,-,-,-,attrs)$.</p>
+
+<div class="formal"> $W,I \models wasDerivedFrom(id,e2,e1,-,-,-,attrs)$ if and only if there exists $path \in DerivationPaths$ such that:
+<ol>
+<li>[WF] $id$ denotes a derivation $deriv = I(id) \in Derivations$
+</li>
+<li>[WF] $e1,e2$ denote entities $ent1 = I(e1), ent2=I(e2) \in Entities$
+</li>
+<li>$derivedFrom(deriv)= ent2 \cdot w \cdot ent1$ for some $w$
+</li>
+<li>The attribute values match: $match(W,deriv,attrs)$.
+</li></ol></div>
+
+
+<h5> Specialization </h5>
+
+<p>The $specializationOf(e1,e2)$ relation indicates when one entity record presents more specific aspects of another.
+</p>
+<div class="formal"> $W,I \models specializationOf(e1,e2)$ if and only if:
+<ol>
+<li>[WF] Both $e1$ and $e2$ are entity identifiers, denoting $ent_1 = I(e1) \in Entities$ and $ent_2 = I(e2) \in Entities$.
+</li>
+<li>The two Entities refer to the same Thing, that is, $thingOf(ent_1) = thingOf(ent_2)$.
+</li>
+<li>The lifetime of $obj_1$ is contained in that of $ent_2$,i.e. $lifetime(ent_1) \subseteq lifetime(ent_2)$.
+</li>
+<li>For each attribute such that $value(obj_2,a) \neq \bot$ we have $value(obj_1,attr) = value(obj_2,attr)$.
+</li></ol></div>
+
+<p>The second criterion says that the two Entities present aspects of the same Thing. Note that the third criterion allows $obj_1$ and $obj_2$ to have the same lifetime (or that of $obj_2$ can be larger). The last criterion allows $obj_1$ to have more defined attributes than $obj_2$, but they must agree on the attributes defined by $obj_2$.
+</p>
+<div class="remark">
+ <p>There has been discussion whether $specializationOf$ is <em>transitive</em> and/or<em>anti-symmetric</em>:</p>
+<ul><li>
+ Transitivity: If $specializationOf(a,b)$ and $specializationOf(b,c)$ hold then $specializationOf(a,c)$ hold. This holds for the above definition.
+</li>
+<li>Antisymmetry: If $specializationOf(a,b)$ and
+$specializationOf(b,a)$ hold then $a=b$. This doesn't follow from the
+current definition (but it would if we stipulated that two entities
+that have the same interval, attribute and thing are equal).
+</li>
+</ul>
+</div>
+
+<h5> Alternate </h5>
+
+<p>The $alternateOf$ relation indicates when two entity records present (possibly different) aspects of the same thing. The two entities may or may not overlap in time.
+</p>
+<div class="formal"> $W,I \models alternateOf(e1,e2)$ if and only if:
+<ol>
+<li>[WF] Both $e1$ and $e2$ are entity identifiers, denoting $ent_1 = I(e1)$ and $ent_2 = I(e2)$.
+</li>
+<li>The two objects refer to the same underlying Thing: $thingOf(ent_1) = thingOf(ent_2)$
+</li></ol>
+</div>
+
+<div class="remark">
+ <p>There has been discussion whether $alternateOf$ is <em>symmetric</em> and <em>transitive</em>:</p>
+<ul><li> <b>Symmetry</b>: If $alternateOf(a,b)$ holds then $alternateOf(b,a)$ holds.
+</li><li> <b>Transitivity</b>: If $alternateOf(a,b)$ and $alternateOf(b,c)$ hold then $alternateOf(a,c)$ hold. This holds of the above definition.
+</li></ul>
+
+<p>We also consider the following properties which have been suggested:</p>
+<ul>
+ <li> $specializationOf(e_1,e_2)$ implies $alternateOf(e_1,e_2)$? (This holds at the moment.)</li>
+<li> $alternateOf(a, b)$ if and only if there exists c such that $specializationOf(a,c)$ and $specializationOf(b,c)$? This does <b>not</b> necessarily hold without further assumptions about the Entities.</li>
+</ul>
</section>
+<section id="infconstr">
+<h2> Inferences and Constraints </h2>
+
+<div class="inference" number="5" id="communication-generation-use-inference">$\begin{array}[t]{l}
+\forall id,a_2,a_1,attrs.~
+\\
+\qquad wasInformedBy(id,a_2,a_1,attrs)
+\\
+\quad\Rightarrow
+\exists e,gen,t_1,use,t_2.~wasGeneratedBy(gen,e,a_1,t_1,[]) \wedge used(use,a_2,e,t_2,[])
+\end{array}$</div>
+<div class="inference" number="6" id="generation-use-communication-inference">$\begin{array}[t]{l}
+\forall gen,e,a_1,t_1,attrs_1,id_2,a_2,t_2,attrs_2.~
+\\
+\qquad wasGeneratedBy(gen,e,a_1,t_1,attrs_1) \wedge used(id_2,a_2,e,t_2,attrs_2)
+\\
+\quad\Rightarrow
+\exists id.~wasInformedBy(id,a_2,a_1,[])
+\end{array}$</div>
+<div class="inference" number="7" id="entity-generation-invalidation-inference">$\begin{array}[t]{l}
+\forall e,attrs.~
+\\
+\qquad entity(e,attrs)
+\\
+\quad\Rightarrow
+\exists gen,a_1,t_1,inv,a_2,t_2.~wasGeneratedBy(gen,e,a_1,t_1,[]) \wedge wasInvalidatedBy(inv,e,a_2,t_2,[])
+\end{array}$</div>
+<div class="inference" number="8" id="activity-start-end-inference">$\begin{array}[t]{l}
+\forall a,t_1,t_2,attrs.~
+\\
+\qquad activity(a,t_1,t_2,attrs)
+\\
+\quad\Rightarrow
+\exists start,e_1,a_1,end,a_2,e_2.~wasStartedBy(start,a,e_1,a_1,t_1,[]) \wedge wasEndedBy(end,a,e_2,a_2,t_2,[])
+\end{array}$</div>
+<div class="inference" number="9" id="wasStartedBy-inference">$\begin{array}[t]{l}
+\forall id,a,e_1,a_1,t,attrs.~
+\\
+\qquad wasStartedBy(id,a,e_1,a_1,t,attrs)
+\\
+\quad\Rightarrow
+\exists gen,t_1.~wasGeneratedBy(gen,e_1,a_1,t_1,[])
+\end{array}$</div>
+<div class="inference" number="10" id="wasEndedBy-inference">$\begin{array}[t]{l}
+\forall id,a,e_1,a_1,t,attrs.~
+\\
+\qquad wasEndedBy(id,a,e_1,a_1,t,attrs)
+\\
+\quad\Rightarrow
+\exists gen,t_1.~wasGeneratedBy(gen,e_1,a_1,t_1,[])
+\end{array}$</div>
+<div class="inference" number="11" id="derivation-generation-use-inference">In this inference, none of $a$,$gen_2$, or $use_1$ can be placeholders -.<br />$\begin{array}[t]{l}
+\forall id,e_2,e_1,a,gen_2,use_1,attrs.~
+\\
+\qquad notNull(a) \wedge notNull(gen_2) \wedge notNull(use_1) \wedge wasDerivedFrom(id,e_2,e_1,a,gen_2,use_1,attrs)
+\\
+\quad\Rightarrow
+\exists t_1,t_2.~used(use_1,a,e_1,t_1,[]) \wedge wasGeneratedBy(gen_2,e_2,a,t_2,[])
+\end{array}$</div>
+<div class="inference" number="12" id="revision-is-alternate-inference">In this inference, any of $a$,$gen_2$, or $use_1$ can be placeholders -.<br />$\begin{array}[t]{l}
+\forall id,e_1,e_2,a,g,u.~
+\\
+\qquad wasDerivedFrom(id,e_2,e_1,a,g,u,.(prov:type = prov:Revision,[]))
+\\
+\quad\Rightarrow
+alternateOf(e_2,e_1)
+\end{array}$</div>
+<div class="inference" number="13" id="attribution-inference">$\begin{array}[t]{l}
+\forall att,e,ag,attrs.~
+\\
+\qquad wasAttributedTo(att,e,ag,attrs)
+\\
+\quad\Rightarrow
+\exists a,t,gen,assoc,pl.~wasGeneratedBy(gen,e,a,t,[]) \wedge wasAssociatedWith(assoc,a,ag,pl,[])
+\end{array}$</div>
+<div class="inference" number="14" id="delegation-inference">$\begin{array}[t]{l}
+\forall id,ag_1,ag_2,a,attrs.~
+\\
+\qquad actedOnBehalfOf(id,ag_1,ag_2,a,attrs)
+\\
+\quad\Rightarrow
+\exists id_1,pl_1,id_2,pl_2.~wasAssociatedWith(id_1,a,ag_1,pl_1,[]) \wedge wasAssociatedWith(id_2,a,ag_2,pl_2,[])
+\end{array}$</div>
+<div class="inference" number="15" id="influence-inference"><ol><li>$\begin{array}[t]{l}
+\forall id,e,a,t,attrs.~
+\\
+\qquad wasGeneratedBy(id,e,a,t,attrs)
+\\
+\quad\Rightarrow
+wasInfluencedBy(id,e,a,attrs)
+\end{array}$</li><li>$\begin{array}[t]{l}
+\forall id,a,e,t,attrs.~
+\\
+\qquad used(id,a,e,t,attrs)
+\\
+\quad\Rightarrow
+wasInfluencedBy(id,a,e,attrs)
+\end{array}$</li><li>$\begin{array}[t]{l}
+\forall id,a_2,a_1,attrs.~
+\\
+\qquad wasInformedBy(id,a_2,a_1,attrs)
+\\
+\quad\Rightarrow
+wasInfluencedBy(id,a_2,a_1,attrs)
+\end{array}$</li><li>$\begin{array}[t]{l}
+\forall id,a_2,e,a_1,t,attrs.~
+\\
+\qquad wasStartedBy(id,a_2,e,a_1,t,attrs)
+\\
+\quad\Rightarrow
+wasInfluencedBy(id,a_2,e,attrs)
+\end{array}$</li><li>$\begin{array}[t]{l}
+\forall id,a_2,e,a_1,t,attrs.~
+\\
+\qquad wasEndedBy(id,a_2,e,a_1,t,attrs)
+\\
+\quad\Rightarrow
+wasInfluencedBy(id,a_2,e,attrs)
+\end{array}$</li><li>$\begin{array}[t]{l}
+\forall id,e,a,t,attrs.~
+\\
+\qquad wasInvalidatedBy(id,e,a,t,attrs)
+\\
+\quad\Rightarrow
+wasInfluencedBy(id,e,a,attrs)
+\end{array}$</li><li>$\begin{array}[t]{l}
+\forall id,e_2,e_1,a,g,u,attrs.~
+\\
+\qquad wasDerivedFrom(id,e_2,e_1,a,g,u,attrs)
+\\
+\quad\Rightarrow
+wasInfluencedBy(id,e_2,e_1,attrs)
+\end{array}$</li><li>In this rule, $a$,$g$, or $u$ may be placeholders -.<br />$\begin{array}[t]{l}
+\forall id,e,ag,attrs.~
+\\
+\qquad wasAttributedTo(id,e,ag,attrs)
+\\
+\quad\Rightarrow
+wasInfluencedBy(id,e,ag,attrs)
+\end{array}$</li><li>In this rule, $pl$ may be a placeholder -.<br />$\begin{array}[t]{l}
+\forall id,a,ag,pl,attrs.~
+\\
+\qquad wasAssociatedWith(id,a,ag,pl,attrs)
+\\
+\quad\Rightarrow
+wasInfluencedBy(id,a,ag,attrs)
+\end{array}$</li><li>$\begin{array}[t]{l}
+\forall id,ag_2,ag_1,a,attrs.~
+\\
+\qquad actedOnBehalfOf(id,ag_2,ag_1,a,attrs)
+\\
+\quad\Rightarrow
+wasInfluencedBy(id,ag_2,ag_1,attrs)
+\end{array}$</li></ol></div>
+<div class="inference" number="16" id="alternate-reflexive">$\begin{array}[t]{l}
+\forall e.~
+\\
+\qquad entity(e)
+\\
+\quad\Rightarrow
+alternateOf(e,e)
+\end{array}$</div>
+<div class="inference" number="17" id="alternate-transitive">$\begin{array}[t]{l}
+\forall e_1,e_2,e_3.~
+\\
+\qquad alternateOf(e_1,e_2) \wedge alternateOf(e_2,e_3)
+\\
+\quad\Rightarrow
+alternateOf(e_1,e_3)
+\end{array}$</div>
+<div class="inference" number="18" id="alternate-symmetric">$\begin{array}[t]{l}
+\forall e_1,e_2.~
+\\
+\qquad alternateOf(e_1,e_2)
+\\
+\quad\Rightarrow
+alternateOf(e_2,e_1)
+\end{array}$</div>
+<div class="inference" number="19" id="specialization-transitive">$\begin{array}[t]{l}
+\forall e_1,e_2,e_3.~
+\\
+\qquad specializationOf(e_1,e_2) \wedge specializationOf(e_2,e_3)
+\\
+\quad\Rightarrow
+specializationOf(e_1,e_3)
+\end{array}$</div>
+<div class="inference" number="20" id="specialization-alternate-inference">$\begin{array}[t]{l}
+\forall e_1,e_2.~
+\\
+\qquad specializationOf(e_1,e_2)
+\\
+\quad\Rightarrow
+alternateOf(e_1,e_2)
+\end{array}$</div>
+<div class="inference" number="21" id="specialization-attributes-inference">$\begin{array}[t]{l}
+\forall e_1,attrs,e_2.~
+\\
+\qquad entity(e_1,attrs) \wedge specializationOf(e_2,e_1)
+\\
+\quad\Rightarrow
+entity(e_2,attrs)
+\end{array}$</div>
+<div class="constraint" number="22" id="key-object"><ol><li>The identifier field $id$ is a <span class="conditional">KEY</span> for the $entity(id,attrs)$ statement.</li><li>The identifier field $id$ is a <span class="conditional">KEY</span> for the $activity(id,t1,t2,attrs)$ statement.</li><li>The identifier field $id$ is a <span class="conditional">KEY</span> for the $agent(id,attrs)$ statement.</li></ol></div>
+<div class="constraint" number="23" id="key-properties"><ol><li>The identifier field $id$ is a <span class="conditional">KEY</span> for the $wasGeneratedBy(id,e,a,t,attrs)$ statement.</li><li>The identifier field $id$ is a <span class="conditional">KEY</span> for the $used(id,a,e,t,attrs)$ statement.</li><li>The identifier field $id$ is a <span class="conditional">KEY</span> for the $wasInformedBy(id,a_2,a_1,attrs)$ statement.</li><li>The identifier field $id$ is a <span class="conditional">KEY</span> for the $wasStartedBy(id,a_2,e,a_1,t,attrs)$ statement.</li><li>The identifier field $id$ is a <span class="conditional">KEY</span> for the $wasEndedBy(id,a_2,e,a_1,t,attrs)$ statement.</li><li>The identifier field $id$ is a <span class="conditional">KEY</span> for the $wasInvalidatedBy(id,e,a,t,attrs)$ statement.</li><li>The identifier field $id$ is a <span class="conditional">KEY</span> for the $wasDerivedFrom(id,e_2,e_1,a,g2,u1,attrs)$ statement.</li><li>The identifier field $id$ is a <span class="conditional">KEY</span> for the $wasAttributedTo(id,e,ag,attr)$ statement.</li><li>The identifier field $id$ is a <span class="conditional">KEY</span> for the $wasAssociatedWith(id,a,ag,pl,attrs)$ statement.</li><li>The identifier field $id$ is a <span class="conditional">KEY</span> for the $actedOnBehalfOf(id,ag_2,ag_1,a,attrs)$ statement.</li><li>The identifier field $id$ is a <span class="conditional">KEY</span> for the $wasInfluencedBy(id,o2,o1,attrs)$ statement.</li></ol></div>
+<div class="constraint" number="24" id="unique-generation">$\begin{array}[t]{l}
+\forall gen_1,gen_2,e,a,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasGeneratedBy(gen_1,e,a,t_1,attrs_1) \wedge wasGeneratedBy(gen_2,e,a,t_2,attrs_2)
+\\
+\quad\Rightarrow
+gen_1 = gen_2
+\end{array}$</div>
+<div class="constraint" number="25" id="unique-invalidation">$\begin{array}[t]{l}
+\forall inv_1,inv_2,e,a,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasInvalidatedBy(inv_1,e,a,t_1,attrs_1) \wedge wasInvalidatedBy(inv_2,e,a,t_2,attrs_2)
+\\
+\quad\Rightarrow
+inv_1 = inv_2
+\end{array}$</div>
+<div class="constraint" number="26" id="unique-wasStartedBy">$\begin{array}[t]{l}
+\forall start_1,start_2,a,e_1,e_2,a_0,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasStartedBy(start_1,a,e_1,a_0,t_1,attrs_1) \wedge wasStartedBy(start_2,a,e_2,a_0,t_2,attrs_2)
+\\
+\quad\Rightarrow
+start_1 = start_2
+\end{array}$</div>
+<div class="constraint" number="27" id="unique-wasEndedBy">$\begin{array}[t]{l}
+\forall end_1,end_2,a,e_1,e_2,a_0,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasEndedBy(end_1,a,e_1,a_0,t_1,attrs_1) \wedge wasEndedBy(end_2,a,e_2,a_0,t_2,attrs_2)
+\\
+\quad\Rightarrow
+end_1 = end_2
+\end{array}$</div>
+<div class="constraint" number="28" id="unique-startTime">$\begin{array}[t]{l}
+\forall start,a_1,a_2,t,t_1,t_2,e,attrs,attrs_1.~
+\\
+\qquad activity(a_2,t_1,t_2,attrs) \wedge wasStartedBy(start,a_2,e,a_1,t,attrs_1)
+\\
+\quad\Rightarrow
+t_1 = t
+\end{array}$</div>
+<div class="constraint" number="29" id="unique-endTime">$\begin{array}[t]{l}
+\forall end,a_1,a_2,t,t_1,t_2,e,attrs,attrs_1.~
+\\
+\qquad activity(a_2,t_1,t_2,attrs) \wedge wasEndedBy(end,a_2,e,a_1,t,attrs_1)
+\\
+\quad\Rightarrow
+t_2 = t
+\end{array}$</div>
+<div class="constraint" number="30" id="start-precedes-end">$\begin{array}[t]{l}
+\forall start,end,a,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasStartedBy(start,a,e_1,a_1,t_1,attrs_1) \wedge wasEndedBy(end,a,e_2,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+start \preceq end
+\end{array}$</div>
+<div class="constraint" number="31" id="start-start-ordering">$\begin{array}[t]{l}
+\forall start_1,start_2,a,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasStartedBy(start_1,a,e_1,a_1,t_1,attrs_1) \wedge wasStartedBy(start_2,a,e_2,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+start_1 \preceq start_2
+\end{array}$</div>
+<div class="constraint" number="32" id="end-end-ordering">$\begin{array}[t]{l}
+\forall end_1,end_2,a,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasEndedBy(end_1,a,e_1,a_1,t_1,attrs_1) \wedge wasEndedBy(end_2,a,e_2,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+end_1 \preceq end2
+\end{array}$</div>
+<div class="constraint" number="33" id="usage-within-activity"><ol><li>$\begin{array}[t]{l}
+\forall start,use,a,e_1,e_2,a_1,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasStartedBy(start,a,e_1,a_1,t_1,attrs_1) \wedge used(use,a,e_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+start \preceq use
+\end{array}$</li><li>$\begin{array}[t]{l}
+\forall use,end,a,e_1,e_2,a_2,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad used(use,a,e_1,t_1,attrs_1) \wedge wasEndedBy(end,a,e_2,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+use \preceq end
+\end{array}$</li></ol></div>
+<div class="constraint" number="34" id="generation-within-activity"><ol><li>$\begin{array}[t]{l}
+\forall start,gen,e_1,e_2,a,a_1,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasStartedBy(start,a,e_1,a_1,t_1,attrs_1) \wedge wasGeneratedBy(gen,e_2,a,t_2,attrs_2)
+\\
+\quad\Rightarrow
+start \preceq gen
+\end{array}$</li><li>$\begin{array}[t]{l}
+\forall gen,end,e,e_1,a,a_1,t,t_1,attrs,attrs_1.~
+\\
+\qquad wasGeneratedBy(gen,e,a,t,attrs) \wedge wasEndedBy(end,a,e_1,a_1,t_1,attrs_1)
+\\
+\quad\Rightarrow
+gen \preceq end
+\end{array}$</li></ol></div>
+<div class="constraint" number="35" id="wasInformedBy-ordering">$\begin{array}[t]{l}
+\forall id,start,end,a_1,a_1',a_2,a_2',e_1,e_2,t_1,t_2,attrs,attrs_1,attrs_2.~
+\\
+\qquad wasInformedBy(id,a_2,a_1,attrs) \wedge wasStartedBy(start,a_1,e_1,a_1',t_1,attrs_1) \wedge wasEndedBy(end,a_2,e_2,a_2',t_2,attrs_2)
+\\
+\quad\Rightarrow
+start \preceq end
+\end{array}$</div>
+<div class="constraint" number="36" id="generation-precedes-invalidation">$\begin{array}[t]{l}
+\forall gen,inv,e,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasGeneratedBy(gen,e,a_1,t_1,attrs_1) \wedge wasInvalidatedBy(inv,e,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+gen \preceq inv
+\end{array}$</div>
+<div class="constraint" number="37" id="generation-precedes-usage">$\begin{array}[t]{l}
+\forall gen,use,e,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasGeneratedBy(gen,e,a_1,t_1,attrs_1) \wedge used(use,a_2,e,t_2,attrs_2)
+\\
+\quad\Rightarrow
+gen \preceq use
+\end{array}$</div>
+<div class="constraint" number="38" id="usage-precedes-invalidation">$\begin{array}[t]{l}
+\forall use,inv,a_1,a_2,e,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad used(use,a_1,e,t_1,attrs_1) \wedge wasInvalidatedBy(inv,e,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+use \preceq inv
+\end{array}$</div>
+<div class="constraint" number="39" id="generation-generation-ordering">$\begin{array}[t]{l}
+\forall gen_1,gen_2,e,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasGeneratedBy(gen_1,e,a_1,t_1,attrs_1) \wedge wasGeneratedBy(gen_2,e,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+gen_1 \preceq gen_2
+\end{array}$</div>
+<div class="constraint" number="40" id="invalidation-invalidation-ordering">$\begin{array}[t]{l}
+\forall inv_1,inv_2,e,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasInvalidatedBy(inv_1,e,a_1,t_1,attrs_1) \wedge wasInvalidatedBy(inv_2,e,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+inv_1 \preceq inv_2
+\end{array}$</div>
+<div class="constraint" number="41" id="derivation-usage-generation-ordering">In this constraint$a$,$gen_2$, or $use_1$ must not be placeholders -.<br />$\begin{array}[t]{l}
+\forall d,e_1,e_2,a,gen_2,use_1,attrs.~
+\\
+\qquad notNull(a) \wedge notNull(gen_2) \wedge notNull(use_1) \wedge wasDerivedFrom(d,e_2,e_1,a,gen_2,use_1,attrs)
+\\
+\quad\Rightarrow
+use_1 \preceq gen_2
+\end{array}$</div>
+<div class="constraint" number="42" id="derivation-generation-generation-ordering">In this constraint, any of $a$,$g$, or $u$ may be placeholders -.<br />$\begin{array}[t]{l}
+\forall d,gen_1,gen_2,e_1,e_2,a,a_1,a_2,g,u,t_1,t_2,attrs,attrs_1,attrs_2.~
+\\
+\qquad wasDerivedFrom(d,e_2,e_1,a,g,u,attrs) \wedge wasGeneratedBy(gen_1,e_1,a_1,t_1,attrs_1) \wedge wasGeneratedBy(gen_2,e_2,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+gen_1 \prec gen_2
+\end{array}$</div>
+<div class="constraint" number="43" id="wasStartedBy-ordering"><ol><li>$\begin{array}[t]{l}
+\forall gen,start,e,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasGeneratedBy(gen,e,a_1,t_1,attrs_1) \wedge wasStartedBy(start,a,e,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+gen \preceq start
+\end{array}$</li><li>$\begin{array}[t]{l}
+\forall start,inv,e,a,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasStartedBy(start,a,e,a_1,t_1,attrs_1) \wedge wasInvalidatedBy(inv,e,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+start \preceq inv
+\end{array}$</li></ol></div>
+<div class="constraint" number="44" id="wasEndedBy-ordering"><ol><li>$\begin{array}[t]{l}
+\forall gen,end,e,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasGeneratedBy(gen,e,a_1,t_1,attrs_1) \wedge wasEndedBy(end,a,e,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+gen \preceq end
+\end{array}$</li><li>$\begin{array}[t]{l}
+\forall end,inv,e,a,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasEndedBy(end,a,e,a_1,t_1,attrs_1) \wedge wasInvalidatedBy(inv,e,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+end \preceq inv
+\end{array}$</li></ol></div>
+<div class="constraint" number="45" id="specialization-generation-ordering">$\begin{array}[t]{l}
+\forall gen_1,gen_2,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad specializationOf(e_2,e_1) \wedge wasGeneratedBy(gen_1,e_1,a_1,t_1,attrs_1) \wedge wasGeneratedBy(gen_2,e_2,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+gen_1 \preceq gen_2
+\end{array}$</div>
+<div class="constraint" number="46" id="specialization-invalidation-ordering">$\begin{array}[t]{l}
+\forall inv_1,inv_2,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad specializationOf(e_1,e_2) \wedge wasInvalidatedBy(inv_1,e_1,a_1,t_1,attrs_1) \wedge wasInvalidatedBy(inv_2,e_2,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+inv_1 \preceq inv_2
+\end{array}$</div>
+<div class="constraint" number="47" id="wasAssociatedWith-ordering">In the following inferences, $pl$ may be a placeholder -.<br /><ol><li>$\begin{array}[t]{l}
+\forall assoc,start_1,inv_2,ag,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasAssociatedWith(assoc,a,ag,pl,attrs) \wedge wasStartedBy(start_1,a,e_1,a_1,t_1,attrs_1) \wedge wasInvalidatedBy(inv_2,ag,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+start_1 \preceq inv_2
+\end{array}$</li><li>$\begin{array}[t]{l}
+\forall assoc,gen_1,end_2,ag,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasAssociatedWith(assoc,a,ag,pl,attrs) \wedge wasGeneratedBy(gen_1,ag,a_1,t_1,attrs_1) \wedge wasEndedBy(end_2,a,e_2,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+gen_1 \preceq end_2
+\end{array}$</li><li>$\begin{array}[t]{l}
+\forall assoc,start_1,end_2,ag,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasAssociatedWith(assoc,a,ag,pl,attrs) \wedge wasStartedBy(start_1,a,e_1,a_1,t_1,attrs_1) \wedge wasEndedBy(end_2,ag,e_2,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+start_1 \preceq end_2
+\end{array}$</li><li>$\begin{array}[t]{l}
+\forall assoc,start_1,end_2,ag,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
+\\
+\qquad wasAssociatedWith(assoc,a,ag,pl,attrs) \wedge wasStartedBy(start_1,ag,e_1,a_1,t_1,attrs_1) \wedge wasEndedBy(end_2,a,e_2,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+start_1 \preceq end_2
+\end{array}$</li></ol></div>
+<div class="constraint" number="48" id="wasAttributedTo-ordering"><ol><li>$\begin{array}[t]{l}
+\forall att,gen_1,gen_2,e,a_1,a_2,t_1,t_2,ag,attrs,attrs_1,attrs_2.~
+\\
+\qquad wasAttributedTo(att,e,ag,attrs) \wedge wasGeneratedBy(gen_1,ag,a_1,t_1,attrs_1) \wedge wasGeneratedBy(gen_2,e,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+gen_1 \preceq gen_2
+\end{array}$</li><li>$\begin{array}[t]{l}
+\forall att,start_1,gen_2,e,e_1,a,a_2,ag,t_1,t_2,attrs,attrs_1,attrs_2.~
+\\
+\qquad wasAttributedTo(att,e,ag,attrs) \wedge wasStartedBy(start_1,ag,e_1,a_1,t_1,attrs_1) \wedge wasGeneratedBy(gen_2,e,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+start_1 \preceq gen_2
+\end{array}$</li></ol></div>
+<div class="constraint" number="49" id="actedOnBehalfOf-ordering"><ol><li>$\begin{array}[t]{l}
+\forall del,gen_1,inv_2,ag_1,ag_2,a,a_1,a_2,t_1,t_2,attrs,attrs_1,attrs_2.~
+\\
+\qquad actedOnBehalfOf(del,ag_2,ag_1,a,attrs) \wedge wasGeneratedBy(gen_1,ag_1,a_1,t_1,attrs_1) \wedge wasInvalidatedBy(inv_2,ag_2,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+gen_1 \preceq inv_2
+\end{array}$</li><li>$\begin{array}[t]{l}
+\forall del,start_1,end_2,ag_1,ag_2,a,a_1,a_2,e_1,e_2,t_1,t_2,attrs,attrs_1,attrs_2.~
+\\
+\qquad actedOnBehalfOf(del,ag_2,ag_1,a,attrs) \wedge wasStartedBy(start_1,ag_1,e_1,a_1,t_1,attrs_1) \wedge wasEndedBy(end2,ag_2,e_2,a_2,t_2,attrs_2)
+\\
+\quad\Rightarrow
+start_1 \preceq end2
+\end{array}$</li></ol></div>
+<div class="constraint" number="51" id="impossible-unspecified-derivation-generation-use"><ol><li>$\begin{array}[t]{l}
+\forall id,e_1,e_2,g,attrs.~
+\\
+\qquad notNull(g) \wedge wasDerivedFrom(id,e_2,e_1,-,g,-,attrs)
+\\
+\quad\Rightarrow
+False
+\end{array}$</li><li>$\begin{array}[t]{l}
+\forall id,e_1,e_2,u,attrs.~
+\\
+\qquad notNull(u) \wedge wasDerivedFrom(id,e_2,e_1,-,-,u,attrs)
+\\
+\quad\Rightarrow
+False
+\end{array}$</li><li>$\begin{array}[t]{l}
+\forall id,e_1,e_2,g,u,attrs.~
+\\
+\qquad notNull(g) \wedge notNull(u) \wedge wasDerivedFrom(id,e_2,e_1,-,g,u,attrs)
+\\
+\quad\Rightarrow
+False
+\end{array}$</li></ol></div>
+<div class="constraint" number="52" id="impossible-specialization-reflexive">$\begin{array}[t]{l}
+\forall e.~
+\\
+\qquad specializationOf(e,e)
+\\
+\quad\Rightarrow
+False
+\end{array}$</div>
+<div class="constraint" number="53" id="impossible-property-overlap">For each $r$ and $s \in \{ used, wasGeneratedBy, wasInvalidatedBy, wasStartedBy, wasEndedBy, wasInformedBy, wasAttributedTo, wasAssociatedWith, actedOnBehalfOf\}$ such that $r$ and $s$ are different relation names, the following constraint holds:<br />$\begin{array}[t]{l}
+\forall id,a_1,\ldots,a_m,b_1,\ldots,b_n.~
+\\
+\qquad r(id,a_1,\ldots,a_m) \wedge s(id,b_1,\ldots,b_n)
+\\
+\quad\Rightarrow
+False
+\end{array}$</div>
+<div class="constraint" number="54" id="impossible-object-property-overlap">For each $p \in \{entity,activity,agent\}$ and each $r \in \{ used, wasGeneratedBy, wasInvalidatedBy, wasStartedBy, wasEndedBy, wasInformedBy, wasAttributedTo, wasAssociatedWith, actedOnBehalfOf\}$, the following constraint holds:<br />$\begin{array}[t]{l}
+\forall id,a_1,\ldots,a_m,b_1,\ldots,b_n.~
+\\
+\qquad p(id,a_1,\ldots,a_m) \wedge r(id,b_1,\ldots,b_n)
+\\
+\quad\Rightarrow
+False
+\end{array}$</div>
+<div class="constraint" number="55" id="entity-activity-disjoint">$\begin{array}[t]{l}
+\forall id.~
+\\
+\qquad entity \in typeOf(id) \wedge activity \in typeOf(id)
+\\
+\quad\Rightarrow
+False
+\end{array}$</div>
+<div class="constraint" number="56" id="membership-empty-collection">$\begin{array}[t]{l}
+\forall c,e.~
+\\
+\qquad hasMember(c,e) \wedge prov:EmptyCollection \in typeOf(c)
+\\
+\quad\Rightarrow
+False
+\end{array}$</div>
+
+</section>
+
+
<section class="appendix">
<h2>Acknowledgements</h2>
<p>