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Conference Papers Year : 2022

The Regular Languages of First-Order Logic with One Alternation

Abstract

The regular languages with a neutral letter expressible in firstorder logic with one alternation are characterized. Specifically, it is shown that if an arbitrary Σ 2 formula defines a regular language with a neutral letter, then there is an equivalent Σ 2 formula that only uses the order predicate. This shows that the so-called Central Conjecture of Straubing holds for Σ 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.
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Dates and versions

hal-03934389 , version 1 (11-01-2023)

Identifiers

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Corentin Barloy, Michael Cadilhac, Charles Paperman, Thomas Zeume. The Regular Languages of First-Order Logic with One Alternation. LICS 2022 - 37th Annual ACM/IEEE Symposium on Logic in Computer Science, Aug 2022, Haïfa, Israel. pp.1-11, ⟨10.1145/3531130.3533371⟩. ⟨hal-03934389⟩
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