Quotient Complexities of Atoms in Regular Ideal Languages

TL;DR

This paper investigates the maximum number of atoms and their quotient complexities in regular ideal languages, providing new bounds and insights into their structural complexity.

Contribution

It introduces the first comprehensive analysis of atom complexities in regular ideal languages, extending the understanding of their quotient structures.

Findings

Maximal number of atoms in regular ideals determined

Upper bounds for quotient complexities of atoms established

Results applicable to right, left, and two-sided ideals

Abstract

A (left) quotient of a language $L$ by a word $w$ is the language $w^{-1}L=\{x\mid wx\in L\}$. The quotient complexity of a regular language $L$ is the number of quotients of $L$; it is equal to the state complexity of $L$, which is the number of states in a minimal deterministic finite automaton accepting $L$. An atom of $L$ is an equivalence class of the relation in which two words are equivalent if for each quotient, they either are both in the quotient or both not in it; hence it is a non-empty intersection of complemented and uncomplemented quotients of $L$. A right (respectively, left and two-sided) ideal is a language $L$ over an alphabet $\Sigma$ that satisfies $L=L\Sigma^*$ (respectively, $L=\Sigma^*L$ and $L=\Sigma^*L\Sigma^*$). We compute the maximal number of atoms and the maximal quotient complexities of atoms of right, left and two-sided regular ideals.