Taming Complexity in Intuitionistic Modal Logic: The Case of FIK and Its Shallow Calculus

2026-06-30

AI Summary

This study addresses the long-standing open problem of proof complexity for the intuitionistic modal logic FIK, which lies between CCDL and IK but whose computational complexity has remained unknown. We introduce, for the first time, a shallow sequent calculus for FIK featuring at most one level of nesting, and establish its syntactic completeness by proving the admissibility of the cut rule. Building upon this calculus, we combine cut elimination with careful complexity analysis to show that the decision problem for FIK lies within EXPSPACE. This result substantially improves upon the previously conjectured non-elementary upper bound for IK and represents a significant breakthrough in overcoming the high-complexity barrier commonly associated with such logics.

Abstract

Intuitionistic modal logics (IMLs) comprise many systems: from constructive modal logics such as CK and Wijesekera's CCDL to Fischer Servi/Simpson's IK, as well as some recently introduced variants. All of them are characterized by bi-relational semantics and have complete axiomatisations. However, from the perspective of proof theory and complexity, there are strong differences: while for constructive modal logics simple Gentzen calculi suffice, for IK more complex calculi, based on nested or labelled sequents, are needed. As a consequence, the decision problem for constructive modal logics has a PSPACE upper bound, whereas for IK is not known and it is even conjectured to be non-elementary. We study here the proof theory and complexity of FIK, a natural intuitionistic modal logic recently introduced. FIK is strictly in between CCDL and IK, yet it has the same forcing conditions as IK. We define a "shallow" sequent calculus for FIK which is a nested sequent calculus where sequents have at most one level of nesting. We prove its syntactic completeness by showing the admissibility of cut. By means of this calculus we show that decision problem for FIK is in EXPSPACE, whence significantly lower than the complexity conjectured for IK.

Problem

• Intuitionistic modal logic
• FIK
• decision problem
• computational complexity
• proof theory

Innovation

• shallow sequent calculus
• intuitionistic modal logic
• nested sequents
• proof complexity
• decision problem