Products of ‘transitive’ modal logics

Abstract

We solve a major open problem concerning algorithmic properties of products of ‘transitive’ modal logics by showing that products and commutators of such standard logics as K4, S4, S4.1, K4.3, GL, or Grz are undecidable and do not have the finite model property. More generally, we prove that no Kripke complete extension of the commutator [K4,K4] with product frames of arbitrary finite or infinite depth (with respect to both accessibility relations) can be decidable. In particular, if 𝒞₁ and 𝒞₂ are classes of transitive frames such that their depth cannot be bounded by any fixed n <ω, then the logic of the class { 𝔉₁ × 𝔉₂ | 𝔉₁ ∈ 𝒞₁, 𝔉₂∈ 𝒞₂ } is undecidable. (On the contrary, the product of, say, K4 and the logic of all transitive Kripke frames of depth ≤ n, for some fixed n <ω, is decidable.) The complexity of these undecidable logics ranges from r.e. to co-r.e. and Π₁¹-complete. As a consequence, we give the first known examples of Kripke incomplete commutators of Kripke complete logics.