Syntactic Complexity of Ideal and Closed Languages

TL;DR

This paper investigates the maximum syntactic complexity of various subclasses of regular languages, establishing tight bounds for ideal and closed languages, which are important in automata theory and formal language analysis.

Contribution

It provides exact upper bounds on the syntactic complexity of ideal and closed languages, expanding understanding of their automata-theoretic properties.

Findings

n^{n-1} is the tight upper bound for right ideals and prefix-closed languages.

Existence of languages with complexity n^{n-1}+n-1 for left ideals and suffix-closed languages.

Complexity n^{n-2}+(n-2)2^{n-2}+1 for two-sided ideals and factor-closed languages.

Abstract

The state complexity of a regular language is the number of states in the minimal deterministic automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity $n$ of languages in that class. We study the syntactic complexity of the class of regular ideal languages and their complements, the closed languages. We prove that $n^{n-1}$ is a tight upper bound on the complexity of right ideals and prefix-closed languages, and that there exist left ideals and suffix-closed languages of syntactic complexity $n^{n-1}+n-1$, and two-sided ideals and factor-closed languages of syntactic complexity $n^{n-2}+(n-2)2^{n-2}+1$.