# EXPSPACE-Completeness of the Logics K4xS5 and S4xS5 and the Logic of Subset Spaces, Part 2: EXPSPACE-Hardness

## TL;DR

This paper establishes that the satisfiability problems for the product logics K4xS5, S4xS5, and the logic of subset spaces are all EXPSPACE-complete, refining previous lower bounds to match known upper bounds.

## Contribution

The paper proves that these three logic satisfiability problems are EXPSPACE-hard, completing their complexity classification as EXPSPACE-complete.

## Key findings

- All three problems are EXPSPACE-hard under logspace reduction.
- These problems are in EXPSPACE, thus are EXPSPACE-complete.
- The results improve previous lower bounds from NEXPTIME and PSPACE.

## Abstract

It is known that the satisfiability problems of the product logics K4xS5 and S4xS5 are NEXPTIME-hard and that the satisfiability problem of the logic SSL of subset spaces is PSPACE-hard. We improve these lower bounds for the complexity of these problems by showing that all three problems are EXPSPACE-hard under logspace reduction. In another paper we show that these problems are in ESPACE. This shows that all three problems are EXPSPACE-complete.