Deciding the Existence of Interpolants and Definitions in First-Order Modal Logic

Abstract

None of the first-order modal logics between $\mathsf{K}$ and $S5$ under the constant domain semantics enjoys Craig interpolation or projective Beth definability, even in the language restricted to a single individual variable. It follows that the existence of a Craig interpolant for a given implication or of an explicit definition for a given predicate cannot be directly reduced to validity as in classical first-order and many other logics. Our concern here is the decidability and computational complexity of the interpolant and definition existence problems. We first consider two decidable fragments of first-order modal logic $S5$: the one-variable fragment ${\mathsf{Q}}^{1}S5$ and its extension ${S5}_{{\mathcal{A}\mathcal{LC}}^u}$ that combines $S5$ and the description logic$\mathcal{A}\mathcal{LC}$ with the universal role. We prove that interpolant and definition existence in ${\mathsf{Q}}^{1}S5$ and ${S5}_{{\mathcal{A}\mathcal{LC}}^u}$ is decidable in coN2ExpTime, being 2ExpTime-hard, while uniform interpolant existence is undecidable. These results transfer to the two-variable fragment $\mathsf{F}{\mathsf{O}}^2$ of classical first-order logic without equality. We also show that interpolant and definition existence in the one-variable fragment ${\mathsf{Q}}^1\mathsf{K}$ of first-order modal logic $\mathsf{K}$ is non-elementary decidable, while uniform interpolant existence is again undecidable.