Synthesizing Strongly Equivalent Logic Programs: Beth Definability for Answer Set Programs via Craig Interpolation in First-Order Logic

We show a projective Beth definability theorem for logic programs under the stable model semantics: For given programs P and Q and vocabulary V (set of predicates) the existence of a program R in V such that \(P \cup R\) and \(P \cup Q\) are strongly equivalent can be expressed as a first-order entailment. Moreover, our result is effective: A program R can be constructed from a Craig interpolant for this entailment, using a known first-order encoding for testing strong equivalence, which we apply in reverse to extract programs from formulas. As a further perspective, this allows transforming logic programs via transforming their first-order encodings. In a prototypical implementation, the Craig interpolation is performed by first-order provers based on clausal tableaux or resolution calculi. Our work shows how definability and interpolation, which underlie modern logic-based approaches to advanced tasks in knowledge representation, transfer to answer set programming.