Complexity of Suffix-Free Regular Languages

TL;DR

This paper investigates the complexity properties of suffix-free regular languages, demonstrating the absence of a most complex stream in this class and providing specific streams that meet various complexity bounds.

Contribution

It proves that no single most complex stream exists for suffix-free languages and constructs specific streams that achieve multiple known complexity bounds.

Findings

No most complex stream exists in suffix-free languages.

Constructed ternary stream meets bounds for product, star, and boolean operations.

Constructed quinary stream meets bounds for boolean operations, reversal, and syntactic semigroup size.

Abstract

We study various complexity properties of suffix-free regular languages. The quotient complexity of a regular language $L$ is the number of left quotients of $L$; this is the same as the state complexity of $L$. A regular language $L'$ is a dialect of a regular language $L$ if it differs only slightly from $L$. The quotient complexity of an operation on regular languages is the maximal quotient complexity of the result of the operation expressed as a function of the quotient complexities of the operands. A sequence $(L_k,L_{k+1},\dots)$ of regular languages in some class ${\mathcal C}$, where $n$ is the quotient complexity of $L_n$, is called a stream. A stream is most complex in class ${\mathcal C}$ if its languages $L_n$ meet the complexity upper bounds for all basic measures. It is known that there exist such most complex streams in the class of regular languages, in the class of prefix-free languages, and also in the classes of right, left, and two-sided ideals. In contrast to this, we prove that there does not exist a most complex stream in the class of suffix-free regular languages. However, we do exhibit one ternary suffix-free stream that meets the bound for product and whose restrictions to binary alphabets meet the bounds for star and boolean operations. We also exhibit a quinary stream that meets the bounds for boolean operations, reversal, size of syntactic semigroup, and atom complexities. Moreover, we solve an open problem about the bound for the product of two languages of quotient complexities $m$ and $n$ in the binary case by showing that it can be met for infinitely many $m$ and $n$.