A polynomial time algorithm for local testability and its level

TL;DR

This paper presents a quadratic time algorithm to determine whether a finite semigroup is locally testable and to find its level, advancing the computational methods in semigroup theory.

Contribution

It introduces a polynomial time algorithm for testing local testability and determining its level in finite semigroups, based on identities and structure analysis.

Findings

Algorithm solves local testability problem efficiently

Algorithm finds the level of local testability in quadratic time

Advances computational methods in semigroup theory

Abstract

A locally testable semigroup S is a semigroup with the property that for some nonnegative integer k, called the order or level of local testability, two words u and v in some set of generators for semigroup S are equal in the semigroup if (1) the prefix and suffix of the words of length k coincide, and (2) the set of intermediate substrings of length k of the words coincide. The local testability problem for semigroups is, given a finite semigroup, to decide, if the semigroup is locally testable or not. Recently, we introduced a polynomial time algorithm for the local testability problem and to find the level of local testability for semigroups based on our previous description of identities of $k$-testable semigroups and the structure of locally testable semigroups. The first part of the algorithm we introduce solves the local testability problem. The second part of the algorithm finds the order of local testability of a semigroup. The algorithm is of quadratic order where n is the order of the semigroup.