Regular languages defined by first-order formulas without quantifier alternation

TL;DR

This paper presents a simplified proof that regular languages definable by first-order formulas without quantifier alternation can be characterized using only regular atomic formulas, avoiding complex algebraic or circuit complexity arguments.

Contribution

The authors provide an elementary proof that simplifies understanding of the logical characterization of certain regular languages, bypassing advanced algebraic and circuit complexity techniques.

Findings

Regular languages with no quantifier alternation can be defined using only regular atomic formulas.

The proof relies solely on basic properties of finite automata.

This approach simplifies previous complex proofs.

Abstract

We give a simple new proof that regular languages defined by first-order sentences with no quantifier alteration can be defined by such sentences in which only regular atomic formulas appear. Earlier proofs of this fact relied on arguments from circuit complexity or algebra. Our proof is much more elementary, and uses only the most basic facts about finite automata.