Answer sets and nonmonotonic S4

Abstract

The semantics of answer sets ([10]) provides a flexible tool for interpreting various forms of extended logic programs and deductive databases. In particular, it is applicable to extended disjunctive databases whose rules may contain two kinds of negation as well as disjunctive conclusions. Extending our earlier work, [19, 20], we provide here a complete logical characterisation of answer set inference as a subsystem of nonmonotonic S4. The method consists in interpreting the rules of disjunctive databases as formulas of Nelson's constructive logic with strong negation, N. In particular, the two types of negation present in database rules are interpreted as Nelson's strong negation and Heyting's intuitionistic negation, respectively. We then make use of the well-known Gödel embedding of N to modal S4, and show that the inference relation associated with the answer set semantics is equivalent to that of nonmonotonic S4. As corollaries we obtain ini N a monotonic lower-bound for answer set inference as well as the related modal embeddings recently established in [12] and [13].

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