Universal proof theory: Feasible admissibility in intuitionistic modal logics

Abstract

We introduce a general and syntactically defined family of sequent-style calculi over the propositional language with the modalities $\{\square ,◇\}$ and its fragments as a formalization for constructively acceptable systems. Calling these calculi constructive, we show that any strong enough constructive sequent calculus, satisfying a mild technical condition, feasibly admits all Visser's rules. This means that there exists a polynomial-time algorithm that, given a proof of the premise of a Visser's rule, provides a proof for its conclusion. As a positive application, we establish the feasible admissibility of Visser's rules in sequent calculi for several intuitionistic modal logics, including $\mathrm{CK}$, $\mathrm{IK}$, their extensions by the modal axioms T, B, 4, 5, and the axioms for bounded width and depth and their fragments ${\mathrm{CK}}_{\square }$, propositional lax logic and $\mathrm{IPC}$. On the negative side, we show that if a strong enough intuitionistic modal logic (satisfying a mild technical condition) does not admit at least one of Visser's rules, it cannot have a constructive sequent calculus. Consequently, no intermediate logic other than $\mathrm{IPC}$ has a constructive sequent calculus.