The Intersection Problem for Finite Monoids

TL;DR

This paper explores the computational complexity of the intersection problem for finite monoids, establishing its PSPACE-completeness and NP-completeness in different cases, and introduces novel algorithms and reductions related to automata and transformation monoids.

Contribution

It provides a comprehensive complexity classification for the intersection problem in finite monoids and introduces new algorithms and reductions, extending classical results.

Findings

The problem is PSPACE-complete for varieties contained in DS.

The problem is NP-complete for varieties contained in DO.

The NP-algorithm employs novel compression and combinatorial techniques.

Abstract

We investigate the intersection problem for finite monoids, which asks for a given set of regular languages, represented by recognizing morphisms to finite monoids from a variety V, whether there exists a word contained in their intersection. Our main result is that the problem is PSPACE-complete if V is contained in DS and NP-complete if V is non-trivial and contained in DO. Our NP-algorithm for the case that V is contained in DO uses novel methods, based on compression techniques and combinatorial properties of DO. We also show that the problem is log-space reducible to the intersection problem for deterministic finite automata (DFA) and that a variant of the problem is log-space reducible to the membership problem for transformation monoids. In light of these reductions, our hardness results can be seen as a generalization of both a classical result by Kozen and a theorem by Beaudry, McKenzie and Therien.