Syntactic Complexity of Prefix-, Suffix-, Bifix-, and Factor-Free Regular Languages

TL;DR

This paper investigates the maximum syntactic complexity of various subclasses of regular languages, providing tight bounds for prefix-free languages and conjectures for suffix-, bifix-, and factor-free languages, supported by example languages.

Contribution

It establishes tight bounds for prefix-free languages and proposes conjectured bounds for other subclasses, advancing understanding of their syntactic semigroup complexities.

Findings

$n^{n-2}$ is a tight bound for prefix-free languages

Conjectured bounds for suffix-, bifix-, and factor-free languages

Examples of languages achieving these complexities

Abstract

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of the class of regular languages is the maximal syntactic complexity of languages in that class, taken as a function of the state complexity $n$ of these languages. We study the syntactic complexity of prefix-, suffix-, bifix-, and factor-free regular languages. We prove that $n^{n-2}$ is a tight upper bound for prefix-free regular languages. We present properties of the syntactic semigroups of suffix-, bifix-, and factor-free regular languages, conjecture tight upper bounds on their size to be $(n-1)^{n-2}+(n-2)$, $(n-1)^{n-3} + (n-2)^{n-3} + (n-3)2^{n-3}$, and $(n-1)^{n-3} + (n-3)2^{n-3} + 1$, respectively, and exhibit languages with these syntactic complexities.