An explicit algorithm for normal forms in small overlap monoids

TL;DR

This paper introduces a practical quadratic-time algorithm for computing normal forms in small overlap monoids, improving efficiency and applicability for solving the word problem in these algebraic structures.

Contribution

The paper presents a new quadratic-time algorithm for normal forms in $C(4)$ monoids with smaller coefficients, enhancing practical computation over previous methods.

Findings

Normal form computation in $C(4)$ monoids can be done in $O(|u|^2)$ time.

The uniform word problem for small overlap monoids is solvable in $O(N imes ext{min}iglrace |u|, |v|igrrace)$ time.

The new algorithm has smaller coefficients, making it more practical for implementation.

Abstract

We describe a practical algorithm for computing normal forms for semigroups and monoids with finite presentations satisfying so-called small overlap conditions. Small overlap conditions are natural conditions on the relations in a presentation, which were introduced by J. H. Remmers and subsequently studied extensively by M. Kambites. Presentations satisfying these conditions are ubiquitous; Kambites showed that a randomly chosen finite presentation satisfies the $C(4)$ condition with probability tending to 1 as the sum of the lengths of relation words tends to infinity. Kambites also showed that several key problems for finitely presented semigroups and monoids are tractable in $C(4)$ monoids: the word problem is solvable in $O(\min\{|u|, |v|\})$ time in the size of the input words $u$ and $v$; the uniform word problem for $\langle A|R\rangle$ is solvable in $O(N ^ 2 \min\{|u|, |v|\})$ where $N$ is the sum of the lengths of the words in $R$; and a normal form for any given word $u$ can be found in $O(|u|)$ time. Although Kambites' algorithm for solving the word problem in $C(4)$ monoids is highly practical, it appears that the coefficients in the linear time algorithm for computing normal forms are too large in practice. In this paper, we present an algorithm for computing normal forms in $C(4)$ monoids that has time complexity $O(|u| ^ 2)$ for input word $u$, but where the coefficients are sufficiently small to allow for practical computation. Additionally, we show that the uniform word problem for small overlap monoids can be solved in $O(N \min\{|u|, |v|\})$ time.