# Primitivity, Uniform Minimality and State Complexity of Boolean Operations

## TL;DR

This paper explores the relationship between primitive groups and automata minimality, establishing conditions for uniform minimality and maximal state complexity in boolean operations on group languages.

## Contribution

It characterizes when permutation DFAs are uniformly minimal and introduces the concept of uniform boolean minimality for direct products, linking group theory with automata complexity.

## Key findings

- Permutation DFA is uniformly minimal iff its transition monoid is a primitive group.
- Direct products of permutation DFAs are generally not uniformly minimal, except trivial cases.
- Conditions for maximal boolean complexity in group languages are established.

## Abstract

A minimal deterministic finite automaton (DFA) is uniformly minimal if it always remains minimal when the final state set is replaced by a non-empty proper subset of the state set. We prove that a permutation DFA is uniformly minimal if and only if its transition monoid is a primitive group. We use this to study boolean operations on group languages, which are recognized by direct products of permutation DFAs. A direct product cannot be uniformly minimal, except in the trivial case where one of the DFAs in the product is a one-state DFA. However, non-trivial direct products can satisfy a weaker condition we call uniform boolean minimality, where only final state sets used to recognize boolean operations are considered. We give sufficient conditions for a direct product of two DFAs to be uniformly boolean minimal, which in turn gives sufficient conditions for pairs of group languages to have maximal state complexity under all binary boolean operations ("maximal boolean complexity"). In the case of permutation DFAs with one final state, we give necessary and sufficient conditions for pairs of group languages to have maximal boolean complexity. Our results demonstrate a connection between primitive groups and automata with strong minimality properties.