Local tabularity is decidable for bi-intermediate logics of trees and of co-trees

Abstract

A bi-Heyting algebra validates the Gödel-Dummett axiom (p → q) ∨ (q → p) iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called bi-Gödel algebras and form a variety that algebraizes the extension bi-GD of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we establish the decidability of the problem of determining if a finitely axiomatizable extension of bi-GD is locally tabular. Notably, if L is an axiomatic extension of bi-GD, then L is locally tabular iff L is not contained in Log(F C), the logic of a particular family of finite co-trees, called the finite combs. We prove that Log(F C) is finitely axiomatizable. Since this logic also has the finite model property, it is therefore decidable. Thus, the above characterization of local tabularity ensures the decidability of the aforementioned problem.