Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages

TL;DR

This paper investigates the complexity measures of three specific classes of suffix-convex regular languages—left-ideal, suffix-closed, and suffix-free—including their state complexity for various operations and their algebraic properties.

Contribution

It provides a detailed analysis of the complexity bounds and algebraic structures of these three classes, which were not comprehensively studied before.

Findings

Determined the quotient complexity bounds for boolean operations, product, star, and reversal.

Analyzed the size of syntactic semigroups for these language classes.

Computed the quotient complexity of their atoms.

Abstract

A language $L$ over an alphabet $\Sigma$ is suffix-convex if, for any words $x,y,z\in\Sigma^*$, whenever $z$ and $xyz$ are in $L$, then so is $yz$. Suffix-convex languages include three special cases: left-ideal, suffix-closed, and suffix-free languages. We examine complexity properties of these three special classes of suffix-convex regular languages. In particular, we study the quotient/state complexity of boolean operations, product (concatenation), star, and reversal on these languages, as well as the size of their syntactic semigroups, and the quotient complexity of their atoms.