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The regular languages of wire linear AC^0

July 2022
Acta Informatica 59(02)

Authors:

In this paper, the regular languages of wire linear AC0

\hbox {AC}^0

are characterized as the languages expressible in the two-variable fragment of first-order logic with regular predicates, FO2[reg]

\mathrm{FO}^2[\mathrm{reg}]

. Additionally, they are characterized as the languages recognized by the algebraic class QLDA

\mathbf {QLDA}

. The class is shown to be decidable and examples of languages in and outside of it are presented.

A circuit checking that each third nonneutral letter is a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a$$\end{document}

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Acta Informatica (2022) 59:321–336

https://doi.org/10.1007/s00236-022-00432-2

ORIGINAL ARTICLE

The regular languages of wire linear AC0

Michaël Cadilhac1·Charles Paperman2

Received: 15 September 2021 / Accepted: 17 June 2022 / Published online: 25 July 2022

Abstract

In this paper, the regular languages of wire linear AC0are characterized as the languages

expressible in the two-variable fragment of ﬁrst-order logic with regular predicates, FO2[reg].

Additionally, they are characterized as the languages recognized by the algebraic class

QLDA. The class is shown to be decidable and examples of languages in and outside of

it are presented.

Contents

1 Introduction .............................................. 321

2 Preliminaries .............................................. 323

3 Algebra, logic, and circuits ...................................... 326

3.1 From algebra to logic ....................................... 326

3.2 From logic to circuits ....................................... 327

3.3 Back to algebra .......................................... 329

3.4 Closing the circle: from circuits to algebra ............................ 332

4 Applications .............................................. 332

4.1 Decidability ............................................ 332

4.2 LAC0∩Reg, Straubing and Crane Beach properties ....................... 333

4.3 Bounded-depth Dyck languages ................................. 333

5 Conclusion ............................................... 334

References ................................................. 335

1 Introduction

A recurring theme in the work of Klaus–Jörn Lange is the interplay of logic, algebra, and

circuit complexity. In this paper dedicated to his 70th birthday, we exhibit one of these tight

relationships by looking at the class of regular languages recognized by circuits of very low

complexity.

BMichaël Cadilhac

michael@cadilhac.name

BCharles Paperman

charles@paperman.name

1School of Computing, DePaul University, 243 S. Wabash Ave., Chicago, IL 60604, USA

2Inria LINKS Team & University of Lille, 40 avenue Halley, 59000 Villeneuve d’Ascq, France

123

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The regular languages of wire linear AC^0

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