On the piecewise complexity of words

TL;DR

This paper introduces and studies the piecewise complexity and minimality index of words, providing algorithms to compute these measures, which help characterize words efficiently in formal language theory.

Contribution

It defines the measures of piecewise complexity and minimality index and presents efficient algorithms for their computation, advancing understanding of word characterization.

Findings

Algorithms for computing $h(u)$ and $ ho(u)$ are efficient.

The measures help characterize words precisely.

The study opens new avenues in formal language analysis.

Abstract

The piecewise complexity $h(u)$ of a word is the minimal length of subwords needed to exactly characterise $u$. Its piecewise minimality index $\rho(u)$ is the smallest length $k$ such that $u$ is minimal among its order-$k$ class $[u]_k$ in Simon's congruence. We initiate a study of these two descriptive complexity measures. Among other results we provide efficient algorithms for computing $h(u)$ and $\rho(u)$ for a given word $u$.