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

A bi-Heyting algebra validates the Gödel-Dummett axiom $(p\to q)\vee (q\to p)$ 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 $\mathrm{bi-GD}$ 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 $\mathrm{bi-GD}$ is locally tabular.

Notably, if L is an axiomatic extension of $\mathrm{bi-GD}$, then L is locally tabular iff L is not contained in $Log\left(FC\right)$, the logic of a particular family of finite co-trees, called the finite combs. We prove that $Log\left(FC\right)$ 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.