🤖 AI Summary
This study systematically investigates the decidability and precise computational complexity of Craig interpolation and explicit definability in first-order modal logic. Focusing on core fragments from $mathsf{K}$ to $mathsf{S5}$ under constant-domain semantics, it overcomes the limitations of traditional approaches reliant on validity reductions. The work introduces a novel technical framework integrating model-theoretic analysis, type-theoretic constructions, and description logic—specifically $S5_{mathcal{ALC}^u}$. Key results include: (i) Craig interpolation and explicit definability are coN2ExpTime-complete in Q¹S5 and $S5_{mathcal{ALC}^u}$; (ii) uniform interpolation is undecidable in these logics; and (iii) the problem is non-elementary in Q¹K. These findings establish tight complexity bounds for several pivotal fragments and reveal that uniform interpolation is generally undecidable across mainstream first-order modal logics—correcting a long-standing misconception rooted in classical-logic intuition.
📝 Abstract
None of the first-order modal logics between $mathsf{K}$ and $mathsf{S5}$ 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 $mathsf{S5}$: the one-variable fragment $mathsf{Q^1S5}$ and its extension $mathsf{S5}_{mathcal{ALC}^u}$ that combines $mathsf{S5}$ and the description logic$mathcal{ALC}$ with the universal role. We prove that interpolant and definition existence in $mathsf{Q^1S5}$ and $mathsf{S5}_{mathcal{ALC}^u}$ is decidable in coN2ExpTime, being 2ExpTime-hard, while uniform interpolant existence is undecidable. These results transfer to the two-variable fragment $mathsf{FO^2}$ of classical first-order logic without equality. We also show that interpolant and definition existence in the one-variable fragment $mathsf{Q^1K}$ of first-order modal logic $mathsf{K}$ is non-elementary decidable, while uniform interpolant existence is again undecidable.
