Uniform Interpolation in Coalgebraic Modal Logic

TL;DR

This paper investigates uniform interpolation in coalgebraic modal logics, establishing conditions under which these logics possess the property, and applying the results to various examples like modal and coalition logics.

Contribution

It introduces the concept of one-step interpolation for coalgebraic modal logics and links it to uniform interpolation, providing criteria based on preservation of finite surjective weak pullbacks.

Findings

Most coalgebraic modal logics have uniform interpolation under certain conditions.

One-step interpolation implies uniform interpolation in these logics.

Preservation of finite surjective weak pullbacks is key for one-step interpolation.

Abstract

A logic has uniform interpolation if its formulas can be projected down to given subsignatures, preserving all logical consequences that do not mention the removed symbols; the weaker property of (Craig) interpolation allows the projected formula -- the interpolant -- to be different for each logical consequence of the original formula. These properties are of importance, e.g., in the modularization of logical theories. We study interpolation in the context of coalgebraic modal logics, i.e. modal logics axiomatized in rank 1, restricting for clarity to the case with finitely many modalities. Examples of such logics include the modal logics K and KD, neighbourhood logic and its monotone variant, finite-monoid-weighted logics, and coalition logic. We introduce a notion of one-step (uniform) interpolation, which refers only to a restricted logic without nesting of modalities, and show that a coalgebraic modal logic has uniform interpolation if it has one-step interpolation. Moreover, we identify preservation of finite surjective weak pullbacks as a sufficient, and in the monotone case necessary, condition for one-step interpolation. We thus prove or reprove uniform interpolation for most of the examples listed above.