On the piecewise complexity of words
Philippe Schnoebelen, Isa Vialard
TL;DR
The paper investigates the descriptive complexity of individual words through two measures, $h(u)$ and $\\rho(u)$, grounded in Simon's order-$k$ congruence and subword order. It develops characterizations of these measures via side distances $r$ and $\\ell$, establishes monotonicity and convexity properties, and provides efficient algorithms to compute them with $h(u)$ in $O(|A|\\cdot|u|)$ and $\\rho(u)$ in $O(|A|+|u|)$. In the binary setting, the authors prove the exact relation $h(u)=\\rho(u)+1$ and present linear-time algorithms on run-length encodings, along with tight bounds on maximum word length for a given piecewise complexity. These results connect piecewise complexity to Simon's PT-languages and subword distances, enabling practical computation and offering new directions for the descriptive complexity of words and languages.
Abstract
The piecewise complexity $h(u)$ of a word is the minimal length of subwords needed to exactly characterise $u$. Its piecewise minimality index $ρ(u)$ 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 $ρ(u)$ for a given word $u$.