Space functions and complexity of the word problem in semigroups

TL;DR

This paper introduces a space function for finitely presented semigroups to analyze the complexity of their word problem, establishing a criterion linking polynomial space decidability to subsemigroup embeddings.

Contribution

It defines the space function for semigroups and provides a criterion connecting polynomial space complexity of the word problem to subsemigroup embeddings.

Findings

The space function bounds the memory needed for word transformations.

A semigroup has a polynomial space word problem iff it embeds into one with polynomial space function.

The paper establishes a new complexity criterion for semigroup word problems.

Abstract

We introduce the space function $s(n)$ of a finitely presented semigroup $S =<A\mid R>.$ To define $s(n)$ we consider pairs of words $w,w'$ over $A$ of length at most $n$ equal in $S$ and use relations from $R$ for the transformations $w=w_0\to...\to w_t= w'$; $s(n)$ bounds from above the tape space (or computer memory) sufficient to implement all such transitions $w\to...\to w'.$ One of the results obtained is the following criterion: A finitely generated semigroup $S$ has decidable word problem of polynomial space complexity if and only if $S$ is a subsemigroup of a finitely presented semigroup $H$ with polynomial space function.