author | Peter F. Patel-Schneider <pfpschneider@gmail.com> |

Sun, 09 Jun 2013 09:28:04 -0700 | |

changeset 832 | 994c80a52937 |

parent 831 | 026eab0d94aa |

child 833 | cabfe9583550 |

Editorial changes in response to Ivan's comments; fix up wording in Appendix B a bit

rdf-mt/index.html |

--- a/rdf-mt/index.html Sun Jun 09 08:45:14 2013 -0700 +++ b/rdf-mt/index.html Sun Jun 09 09:28:04 2013 -0700 @@ -294,8 +294,8 @@ - <p>The denotation of a ground RDF graph in I is then given by the following - rules, where an interpretation is treated as a function from expressions (names, triples and graphs) to semantic values:</p> + <p>The denotation of a ground RDF graph in an interpretation I is then given by the following + rules, where the interpretation is also treated as a function from expressions (names, triples and graphs) to semantic values:</p> <div class="tabletitle">Semantic conditions for ground graphs.</div> @@ -432,7 +432,7 @@ If S entails a finite graph E, then some finite subset S' of S entails E. <!-- <a href="#compactlemmaprf" class="termref"> [Proof]</a> --> </p> -<p>The property described here is called <em>compactness</em>. RDF is compact. As RDF graphs can be infinite, this is sometimes important.</p> +<p>The property just above is called <em>compactness</em> - RDF is compact. As RDF graphs can be infinite, this is sometimes important.</p> <p class="fact"> If E contains an IRI which does not occur anywhere in S, then S does not entail E.</p> @@ -1151,9 +1151,12 @@ <section class="appendix"><h2 id="proofs">Finite interpretations (Informative)</h2> <p>To keep the exposition simple, the RDF semantics has been phrased in a way which requires interpretations to be larger than absolutely necessary. For example, all interpretations are required to interpret the whole IRI vocabulary, and the universes of all D-interpretations must contain all possible strings and therefore be infinite. This appendix sketches, without proof, how to re-state the semantics using smaller semantic structures, without changing any entailments. </p> -<p>Basically, it is only necessary for an interpretation structure to interpret the <a>name</a>s actually used in the graphs whose entailment is being considered, and to have a universe which contains referents for all the <a>name</a>s and blank nodes, plus a random member of each nonempty class or type, in order to provide for the truth of existential assertions made using blank nodes. Formally, we can define a <dfn>pre-interpretation</dfn> over a <a>vocabulary</a> V to be a structure I similar to a <a>simple interpretation</a> but with a mapping only from V to its universe IR, and when considering whether G entails E, consider only pre-interpretations over the finite vocabulary of <a>name</a>s actually used in G union E. The universe of such a pre-interpretation can be restricted to the cardinality N+B, where N is the size of the vocabulary and B is the number of blank nodes in the graphs. Any such pre-interpretation may be extended to <a>simple interpretation</a>s all of which which will give the same truth values for any triples in G or E. Satisfiability, entailment and so on can then be defined with respect to these finite pre-interpretations, and shown to be identical to the ideas defined in the body of the specification.</p> -<p>When considering D-entailment, pre-interpretations may be kept finite by weakening the semantic conditions for datatyped literals so that IR need contain literal values only for literals which actually occur in G or E, and the size of the universe restricted to (N+B).(D+1), where D is the number of recognized datatypes. (A tighter bound is possible.) For RDF entailment, only the finite part of the RDF vocabulary which include those container membership properties which actually occur in the graphs need to be interpreted, and the second RDF semantic condition is weakened to apply only to values which are values of literals which actually occur in the vocabulary. For RDFS interpretations, again only that finite part of the infinite container membership property vocabulary which actually occurs in the graphs under consideration needs to be interpreted. In all these cases, a pre-interpretation of the vocabulary of a set of graphs may be extended to a full interpretation of the appropriate type without changing the truth-values of any triples in the graphs.</p> -<p>The whole semantics could be stated in terms of pre-interpretations, yielding the same entailments, and allowing finite RDF graphs to be interpreted in finite structures, if this <em>finite model property</em> is considered important. + +<p>Basically, it is only necessary for an interpretation structure to interpret the <a>name</a>s actually used in the graphs whose entailment is being considered, and to consider interpretations whose universes are at most as big as the number of names and blank nodes in the graphs. More formally, we can define a <dfn>pre-interpretation</dfn> over a <a>vocabulary</a> V to be a structure I similar to a <a>simple interpretation</a> but with a mapping only from V to its universe IR. Then when determining whether G entails E, consider only pre-interpretations over the finite vocabulary of <a>name</a>s actually used in G union E. The universe of such a pre-interpretation can be restricted to the cardinality N+B, where N is the size of the vocabulary and B is the number of blank nodes in the graphs. Any such pre-interpretation may be extended to <a>simple interpretation</a>s, all of which which will give the same truth values for any triples in G or E. Satisfiability, entailment and so on can then be defined with respect to these finite pre-interpretations, and shown to be identical to the ideas defined in the body of the specification.</p> + +<p>When considering D-entailment, pre-interpretations may be kept finite by weakening the semantic conditions for datatyped literals so that IR need contain literal values only for literals which actually occur in G or E, and the size of the universe restricted to (N+B).(D+1), where D is the number of recognized datatypes. (A tighter bound is possible.) For RDF entailment, only the finite part of the RDF vocabulary which includes those container membership properties which actually occur in the graphs need to be interpreted, and the second RDF semantic condition is weakened to apply only to values which are values of literals which actually occur in the vocabulary. For RDFS interpretations, again only that finite part of the infinite container membership property vocabulary which actually occurs in the graphs under consideration needs to be interpreted. In all these cases, a pre-interpretation of the vocabulary of a set of graphs may be extended to a full interpretation of the appropriate type without changing the truth-values of any triples in the graphs.</p> + +<p>The whole semantics could be stated in terms of pre-interpretations, yielding the same entailments, and allowing finite RDF graphs to be interpreted in finite structures, if the <em>finite model property</em> is considered important. </section>