Definitions and (Uniform) Interpolants in First-Order Modal Logic

Kurucz, A. and Wolter, F. and Zakharyaschev, Michael (2023) Definitions and (Uniform) Interpolants in First-Order Modal Logic. In: The 20th International Conference on Principles of Knowledge Representation and Reasoning (KR2023), 02-08 Sep 2023, Rhodes, Greece.

Abstract

We first consider two decidable fragments of quantified modal logic S5: the one-variable fragment Q1S5 and its extension S5ALCu that combines S5 and the description logic ALC with the universal role. As neither of them enjoys Craig interpolation or projective Beth definability, the exis- tence of interpolants and explicit definitions of predicates— which is crucial in many knowledge engineering tasks—does not directly reduce to entailment. Our concern therefore is the computational complexity of deciding whether (uni- form) interpolants and definitions exist for given input for- mulas, signatures and ontologies. We prove that interpolant and definition existence in Q1S5 and S5ALCu is decidable in CON2EXPTIME, being 2EXPTIME-hard, while uniform in- terpolant existence is undecidable. Then we show that interpolant and definition existence in the one-variable fragment Q1K of quantified modal logic K is nonelementary decidable, while uniform interpolant existence is undecidable.

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