On state complexity for subword-closed languages

TL;DR

This paper explores the state complexity of subword-closed and superword-closed languages, providing bounds for the square root and substitution operators, revealing exponential and quadratic complexities in various cases.

Contribution

It offers new bounds and conditions for the state complexity of subword-closed and superword-closed languages, especially regarding the square root and substitution operators.

Findings

Exponential lower bound for superword-closed languages with the k-th root.

Quadratic complexity for a specific subword-closed language instance.

Exponential lower bound for the general substitution operator.

Abstract

This paper investigates the state complexities of subword-closed and superword-closed languages, comparing them to regular languages. We focus on the square root operator and the substitution operator. We establish an exponential lower bound for superword-closed languages for the k-th root. For subword-closed languages we analyze in detail a specific instance of the square root problem for which a quadratic complexity is proven. For the substitution operator, we show an exponential lower bound for the general substitution. We then find some conditions for which we prove a quadratic upper bound.