The Regular Languages of First-Order Logic with One Alternation

TL;DR

This paper characterizes regular languages definable in first-order logic with one quantifier alternation and a neutral letter, proving a longstanding conjecture for a specific logical fragment and advancing understanding of logical definability.

Contribution

It proves the Central Conjecture of Straubing for  over languages with a neutral letter, showing these languages are definable with only the order predicate.

Findings

Characterization of -definable regular languages with a neutral letter

Proof that such languages can be defined using only the order predicate

Development of lower bounds against polynomial-size depth-3 Boolean circuits

Abstract

The regular languages with a neutral letter expressible in first-order logic with one alternation are characterized. Specifically, it is shown that if an arbitrary $\Sigma_2$ formula defines a regular language with a neutral letter, then there is an equivalent $\Sigma_2$ formula that only uses the order predicate. This shows that the so-called Central Conjecture of Straubing holds for $\Sigma_2$ over languages with a neutral letter, the first progress on the Conjecture in more than 20 years. To show the characterization, lower bounds against polynomial-size depth-3 Boolean circuits with constant top fan-in are developed. The heart of the combinatorial argument resides in studying how positions within a language are determined from one another, a technique of independent interest.