Universal Proof Theory: Constructive Rules and Feasible Admissibility

Visser's rules have an essential role in intermediate logics. They form a basis for the admissible rules of intuitionistic logic and any intermediate logic in which they are admissible. In this paper, we follow the universal proof theory program introduced and developed in [1, 2, 24, 25] to establish a connection between the form of the rules in a sequent calculus for an intuitionistic modal logic and the admissibility of Visser's rules in that logic. More precisely, by investigating the form of the constructively acceptable rules, we first introduce a very general family of rules called the constructive rules. Then, defining a constructive sequent calculus as a calculus consisting of constructive rules and some basic modal rules, we prove that any constructive sequent calculus stronger than $\mathbf{CK}$ satisfying a mild technical condition, feasibly admits all Visser's rules, i.e., there is a polynomial time algorithm that reads a proof of the premise of a Visser's rule and provides a proof for its conclusion. This connection has two types of applications. On the positive side, it proves the feasible admissibility of Visser's rules in the sequent system for several intuitionistic modal logics, including $\mathsf{CK}$, $\mathsf{IK}$, their extensions by the usual modal axioms of $T$, $B$, $4$, $5$, the modal axioms of bounded width and depth and the propositional lax logic. On the negative side, though, it shows that if an intuitionistic modal logic satisfying a mild technical condition does not admit Visser's rules, then it cannot have a constructive sequent calculus.