Complexity of Prefix-Convex Regular Languages

TL;DR

This paper investigates the complexity measures of prefix-convex regular languages, including boolean operations, concatenation, star, reversal, and atoms, providing tight bounds and identifying most complex languages.

Contribution

It introduces new tight upper bounds for various complexity measures of prefix-convex languages and characterizes the most complex languages meeting these bounds.

Findings

Derived tight bounds for boolean operations, product, star, and reversal complexities.

Identified the size of the syntactic semigroup for prefix-convex languages.

Constructed most complex prefix-convex languages that achieve these bounds.

Abstract

A language $L$ over an alphabet $\Sigma$ is prefix-convex if, for any words $x,y,z\in\Sigma^*$, whenever $x$ and $xyz$ are in $L$, then so is $xy$. Prefix-convex languages include right-ideal, prefix-closed, and prefix-free languages. We study complexity properties of prefix-convex regular languages. In particular, we find the quotient/state complexity of boolean operations, product (concatenation), star, and reversal, the size of the syntactic semigroup, and the quotient complexity of atoms. For binary operations we use arguments with different alphabets when appropriate; this leads to higher tight upper bounds than those obtained with equal alphabets. We exhibit most complex prefix-convex languages that meet the complexity bounds for all the measures listed above.