Definitions and (Uniform) Interpolants in First-Order Modal Logic
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We first consider two decidable fragments of quantified modal logic 𝖲5: the one-variable fragment 𝖰^1𝖲5 and its extension 𝖲5_𝒜ℒ𝒞^u that combines 𝖲5 and the description logic 𝒜ℒ𝒞 with the universal role. As neither of them enjoys Craig interpolation or projective Beth definability, the existence 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 (uniform) interpolants and definitions exist for given input formulas, signatures and ontologies. We prove that interpolant and definition existence in 𝖰^1𝖲5 and 𝖲5_𝒜ℒ𝒞^u is decidable in coN2ExpTime, being 2ExpTime-hard, while uniform interpolant existence is undecidable. Then we show that interpolant and definition existence in the one-variable fragment 𝖰^1𝖪 of quantified modal logic 𝖪 is nonelementary decidable, while uniform interpolant existence is undecidable.
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