author | Pat Hayes <phayes@ihmc.us> |

Sun, 09 Jun 2013 19:43:25 -0500 | |

changeset 833 | cabfe9583550 |

parent 832 | 994c80a52937 |

child 834 | 6860c0785715 |

inserted multiplication character.

rdf-mt/index.html |

--- a/rdf-mt/index.html Sun Jun 09 09:28:04 2013 -0700 +++ b/rdf-mt/index.html Sun Jun 09 19:43:25 2013 -0500 @@ -938,7 +938,7 @@ <p> aaa <code>rdf:type rdfs:Resource .</code></p> <p>where aaa is an IRI, are true in all RDFS interpretations.</p> -<section> <h4>Patterns of RDFS entailment.</h4> +<section> <h4 id="rdfs_patterns">Patterns of RDFS entailment.</h4> <P>RDFS entailment holds for all the following patterns, which correspond closely to the RDFS semantic conditions:</p> @@ -1050,7 +1050,7 @@ <h2>Appendices</h2> -<section class="appendix"><h3>Entailment rules (Informative)</h3> +<section class="appendix"><h3 id="entailment_rules">Entailment rules (Informative)</h3> <p>This section is based on work described more fully in [[HORST04]], [[HORST05]], which should be consulted for technical details and proofs. </p> <p> The RDF and RDFS entailment patterns listed in the above tables can be viewed as left-to-right rules which add the entailed conclusion to a graph. These rule sets can be used to check RDF (or RDFS) entailment between graphs S and E, by the following sequence of operations:</p> @@ -1149,19 +1149,19 @@ </section> -<section class="appendix"><h2 id="proofs">Finite interpretations (Informative)</h2> +<section class="appendix"><h2 id="finite_interpretations">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 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>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> -<section class="appendix" class="informative"><h2>Proofs of some results (Informative)</h2> +<section class="appendix" class="informative"><h2 id="proofs">Proofs of some results (Informative)</h2> <p class="fact"> The <a>empty graph</a> is entailed by any graph, and does not entail any graph except itself.