The complexity of admissible rules of {\L}ukasiewicz logic

TL;DR

This paper proves that determining whether inference rules are admissible in infinite-valued { extL}ukasiewicz logic is PSPACE-complete, establishing the computational complexity as optimal and contrasting it with the coNP-completeness of derivable rules.

Contribution

The paper establishes the PSPACE-completeness of admissibility in { extL}ukasiewicz logic, confirming the computational complexity as optimal and providing a complexity classification.

Findings

Admissibility in { extL}ukasiewicz logic is PSPACE-complete.

Derivability in { extL}ukasiewicz logic is coNP-complete.

The PSPACE complexity result is optimal.

Abstract

We investigate the computational complexity of admissibility of inference rules in infinite-valued {\L}ukasiewicz propositional logic (\L). It was shown in [13] that admissibility in {\L} is checkable in PSPACE. We establish that this result is optimal, i.e., admissible rules of {\L} are PSPACE-complete. In contrast, derivable rules of {\L} are known to be coNP-complete.