Deciding the existence of uniform interpolants over transitive models

Abstract

We consider the problem of the existence of uniform interpolants in the modal logic K4. We first prove that all $${\square}$$ -free formulas have uniform interpolants in this logic. In the general case, we shall prove that given a modal formula $${\phi}$$ and a sublanguage L of the language of the formula, we can decide whether $${\phi}$$ has a uniform interpolant with respect to L in K4. The $${\square}$$ -free case is proved using a reduction to the Godel Lob Logic GL, while in the general case we prove that the question of whether a modal formula has uniform interpolants over transitive frames can be reduced to a decidable expressivity problem on the μ-calculus.