Syntactic Complexity of R- and J-Trivial Regular Languages

TL;DR

This paper investigates the maximum syntactic complexity of R- and J-trivial regular languages, establishing tight bounds and analyzing their properties related to state complexity and reversal operations.

Contribution

It provides the first tight bounds for the syntactic complexity of R- and J-trivial regular languages, including bounds for reversal operations.

Findings

Syntactic complexity of R-trivial languages is bounded by n!

Syntactic complexity of J-trivial languages is bounded by floor(e(n-1)!)

Reversal of J-trivial languages has a state complexity bound of 2^{n-1}

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 R- and J-trivial regular languages, and prove that n! and floor of [e(n-1)!] are tight upper bounds for these languages, respectively. We also prove that 2^{n-1} is the tight upper bound on the state complexity of reversal of J-trivial regular languages.