Formal Semantics of SGs

Abstract

The first section of this chapter presents a model semantic (or setsemantic) for SGs. First, a model of a vocabulary is defined. Itconsists of a set, the set of entities also called a universe, uponwhich the concept types, the relation types and the individuals areinterpreted. A concept type is interpreted as a subset of the universe,a relation type is interpreted as a set of tuples of elements of theuniverse and an individual is interpreted as an element of the universe.Secondly, a model of an SG over a vocabulary V is defined. It is a modelof V enriched by an interpretation of the concept nodes as elements ofthe universe. Then, an entailment relation is defined, i.e., what itmeans that an SG H entails an SG G, or equivalently, what means that Gis a consequence of H. The canonical model of a SG G, which plays aspecific role in the characterization of the SGs consequences of G, isdefined. This first section ends by the fundamental theorem stating thatif there is a homomorphism from G to H then H entails G and if H/core fentails G then there is a homomorphism from H/core f to G (i.e., asoundness and completeness theorem of SG homomorphism with respect toentailment).The second section concerns the presentation of a first order logicalsemantic for SGs. This semantic is defined through a mapping from SGs toFOL formulas called Phi. First, a FOL language corresponding to theelements of a vocabulary nu is defined and a set of FOL formulas,denoted (Phi)(nu), corresponding to the order relations over nu isdefined. Secondly, for any SG G a FOL formula Phi(G), built on thelanguage associated with nu, is defined. Thirdly, it is shown that theclassical model theory of FOL is equivalent to the model semanticpresented in the first section.In the third section relationships between SGs and the positive,conjunctive and existential fragment of FOL are studied. First, weintroduce the L-substitution notion and we use it to give another proofof the soundness and completeness theorem. Secondly, we prove the``equivalence{''} between the SGs and the positive, conjunctive andexistential fragment of FOL. Finally, we present a second FOL semantic,denoted Psi, which is less intuitive than Phi but has interestingtechnical properties.FOL is used to give a semantic to SGs, but not to reason with them.According to our claim that SGs have good computational properties, wewill build graph-based deduction algorithms that are not translation oflogical procedures. This is developed in Chap. 6 and Chap. 7.The final section is a note on the relationships between descriptionlogics and conceptual graphs.