The Intersection Problem for Finite Semigroups

TL;DR

This paper studies the computational complexity of the intersection problem for finite semigroups, introducing new tools and classifications to determine when the problem is tractable or hard.

Contribution

It introduces compressibility measures to classify the intersection problem's complexity for various classes of finite semigroups.

Findings

NP containment for groups and commutative semigroups

NP-hardness for semigroups with unbounded order and nilpotent semigroups of bounded order

Bounded order and commutativity imply qAC^k containment and quasi-polynomial decidability

Abstract

We investigate the intersection problem for finite semigroups, which asks for a given set of regular languages, represented by recognizing morphisms to finite semigroups, whether there exists a word contained in their intersection. We introduce compressibility measures as a useful tool to classify the intersection problem for certain classes of finite semigroups into circuit complexity classes and Turing machine complexity classes. Using this framework, we obtain a new and simple proof that for groups and commutative semigroups, the problem is contained in NP. We uncover certain structural and non-structural properties determining the complexity of the intersection problem for varieties of semigroups containing only trivial submonoids. More specifically, we prove NP-hardness for classes of semigroups having a property called unbounded order and for the class of all nilpotent semigroups of bounded order. On the contrary, we show that bounded order and commutativity imply containment in the circuit complexity class qAC^k (for some k) and decidability in quasi-polynomial time. We also establish connections to the monoid variant of the problem.