On the Complexity of Properties of Partial Bijection Semigroups

TL;DR

This paper investigates the computational complexity of properties of partial bijection semigroups, providing complexity classifications and algorithms for various problems, and establishing a connection to the Rectangle Tiling Problem.

Contribution

It establishes complexity bounds for checking properties of partial bijection semigroups and introduces algorithms, including a new proof linking membership problems to PSPACE-complete tiling problems.

Findings

Enumerating identities is in AC^0.

Checking if the semigroup is completely regular is in AC^0.

Membership of an idempotent is PSPACE-complete.

Abstract

We examine the computational complexity of problems in which we are given generators for a partial bijection semigroup and asked to check properties of the generated semigroup. We prove that the following problems are in AC$^0$: (1) enumerating left and right identities and (2) checking if the semigroup is completely regular. We also describe a nondeterministic logspace algorithm for checking if an inverse semigroup given by generators satisfies a fixed semigroup identity that may involve a unary inverse operation. We conclude with an alternative proof that checking membership of a given idempotent in a partial bijection semigroup is a PSPACE-complete problem. The proof reduces from the well-known PSPACE-complete Rectangle Tiling Problem, thereby illustrating a connection between Wang tilings and partial bijection semigroups.