# Rules with parameters in modal logic II

## TL;DR

This paper classifies the computational complexity of admissibility and unifiability problems with parameters in transitive modal logics, providing precise complexity bounds for a broad class of such logics.

## Contribution

It offers a complete complexity classification for unifiability and inadmissibility in cluster-extensible (clx) logics, including parameterized and parameter-free cases, with general lower bounds.

## Key findings

- Complexity classified as $ ext{Sigma}^{ ext{exp}}_2$, NEXP, coNEXP, PSPACE, or $ ext{Pi}^p_2$.
- Upper bounds specific to clx logics and related classes.
- Lower bounds applicable to broad classes of transitive logics.

## Abstract

We analyze the computational complexity of admissibility and unifiability with parameters in transitive modal logics. The class of cluster-extensible (clx) logics was introduced in the first part of this series of papers. We completely classify the complexity of unifiability or inadmissibility in any clx logic as being complete for one of $\Sigma^{\exp}_2$, NEXP, coNEXP, PSPACE, or $\Pi^p_2$. In addition to the main case where arbitrary parameters are allowed, we consider restricted problems with the number of parameters bounded by a constant, and the parameter-free case. Our upper bounds are specific to clx logics, but we also include similar results for logics of bounded depth and width. In contrast, our lower bounds are very general: they apply each to a class of all transitive logics whose frames allow occurrence of certain finite subframes. We also discuss the baseline problem of complexity of derivability: it is coNP-complete or PSPACE-complete for each clx logic. In particular, we prove PSPACE-hardness of derivability for a broad class of transitive logics that includes all logics with the disjunction property.