Simple interpretations are required to interpret all names, and are therefore infinite. - This simplifies the exposition. - However, RDF can be interpreted using finite structures, - supporting decidable algorithms. - Details are given in .
+ It was shown in + Appendix B + of RDF 1.1 Semantics spec. + that RDF 1.1 could be interpreted using finite structures. +IEXT(x), called the extension of x, is a set of pairs which identify the arguments for which the property is true, @@ -1882,45 +1883,6 @@
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 where D contains
- xsd:string 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.
Basically, it is only necessary for an interpretation structure to interpret the names - 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 pre-interpretation over a vocabulary V to be a structure I - similar to a simple interpretation 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 names actually used in G union E. The universe of such a pre-interpretation can be restricted to the cardinality N+B+1, 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 simple interpretations, all of 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.
- -When considering D-entailment, pre-interpretations may be kept finite - by weakening the semantic conditions for literals so that IR needs to 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 graph may be extended to a full interpretation - of the appropriate type without changing the truth-values of any triples in the graphs.
- -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 finite model property is considered important.
- -