Quotient Complexities of Atoms of Regular Languages

TL;DR

This paper investigates the maximum possible quotient complexity of atoms of regular languages, providing tight bounds and examples that meet these bounds for any number of quotients.

Contribution

It establishes exact upper bounds on the quotient complexity of atoms of regular languages and demonstrates their attainability with specific language examples.

Findings

Upper bounds on atom quotient complexity are proven.

Bounds depend on the number of complemented quotients in an atom.

Existence of languages meeting these bounds is shown.

Abstract

An atom of a regular language L with n (left) quotients is a non-empty intersection of uncomplemented or complemented quotients of L, where each of the n quotients appears in a term of the intersection. The quotient complexity of L, which is the same as the state complexity of L, is the number of quotients of L. We prove that, for any language L with quotient complexity n, the quotient complexity of any atom of L with r complemented quotients has an upper bound of 2^n-1 if r=0 or r=n, and 1+\sum_{k=1}^{r} \sum_{h=k+1}^{k+n-r} C_{h}^{n} \cdot C_{k}^{h} otherwise, where C_j^i is the binomial coefficient. For each n\ge 1, we exhibit a language whose atoms meet these bounds.