Maximally Atomic Languages

TL;DR

This paper introduces maximally atomic languages, a class of regular languages that meet known bounds on the number of atoms and their complexities, characterized by specific permutation and transformation properties.

Contribution

It defines maximally atomic languages and characterizes them using permutation group transitivity and transformation rank conditions.

Findings

Maximally atomic languages meet known bounds on atom counts.

Characterization involves transitive permutation groups on subsets.

Contains conditions on transformations of rank n-1.

Abstract

The atoms of a regular language are non-empty intersections of complemented and uncomplemented quotients of the language. Tight upper bounds on the number of atoms of a language and on the quotient complexities of atoms are known. We introduce a new class of regular languages, called the maximally atomic languages, consisting of all languages meeting these bounds. We prove the following result: If L is a regular language of quotient complexity n and G is the subgroup of permutations in the transition semigroup T of the minimal DFA of L, then L is maximally atomic if and only if G is transitive on k-subsets of 1,...,n for 0 <= k <= n and T contains a transformation of rank n-1.