On semigroups with PSPACE-complete subpower membership problem

TL;DR

This paper classifies the computational complexity of the subpower membership problem for various finite semigroups, revealing cases where it is polynomial-time solvable, NP-complete, or PSPACE-complete, including some well-known semigroups.

Contribution

It provides a dichotomy and trichotomy classification for the SMP in combinatorial Rees matrix semigroups, identifying conditions leading to PSPACE-completeness.

Findings

SMP is in P for certain matrix forms

SMP is NP-complete for other matrix forms

SMP is PSPACE-complete for some semigroups like the Brandt monoid and matrix semigroups

Abstract

Fix a finite semigroup $S$ and let $a_1, \ldots, a_k, b$ be tuples in a direct power $S^n$. The subpower membership problem (SMP) for $S$ asks whether $b$ can be generated by $a_1, \ldots, a_k$. For combinatorial Rees matrix semigroups we establish a dichotomy result: if the corresponding matrix is of a certain form, then the SMP is in P; otherwise it is NP-complete. For combinatorial Rees matrix semigroups with adjoined identity, we obtain a trichotomy: the SMP is either in P, NP-complete, or PSPACE-complete. This result yields various semigroups with PSPACE-complete SMP including the $6$-element Brandt monoid, the full transformation semigroup on $3$ or more letters, and semigroups of all $n$ by $n$ matrices over a field for $n\ge 2$.