Decidability and Complexity in Weakening and Contraction Hypersequent Substructural Logics

TL;DR

This paper proves decidability for a broad class of hypersequent calculi in substructural logics and establishes complexity bounds, including the first known complexity result for the fuzzy logic MTL, addressing longstanding open problems.

Contribution

It introduces complexity bounds for weakening and contraction hypersequent extensions, including the first complexity estimate for MTL fuzzy logic, advancing understanding of their computational properties.

Findings

Decidability established for all analytic structural rule extensions of FLew.

Complexity upper bounds derived for these extensions, including MTL.

First complexity bound provided for the fuzzy logic MTL.

Abstract

We establish decidability for the infinitely many axiomatic extensions of the commutative Full Lambek logic with weakening FLew (i.e. IMALLW) that have a cut-free hypersequent proof calculus (specifically: every analytic structural rule extension). Decidability for the corresponding extensions of its contraction counterpart FLec was established recently but their computational complexity was left unanswered. In the second part of this paper, we introduce just enough on length functions for well-quasi-orderings and the fast-growing complexity classes to obtain complexity upper bounds for both the weakening and contraction extensions. A specific instance of this result yields the first complexity bound for the prominent fuzzy logic MTL (monoidal t-norm based logic) providing an answer to a long-standing open problem.