Abstract

We develop an algebraic theory for languages of data words. We prove that, under certain conditions, a language of data words is definable in first-order logic if and only if its syntactic monoid is aperiodic.

Highlights

  • This paper is an attempt to combine two fields.The first field is the algebraic theory of regular languages

  • A regular language of finite words is definable in first-order logic if and only if its syntactic monoid is aperiodic

  • We develop the algebraic theory of orbit-finite monoids, and show that it resembles the theory of finite monoids

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Summary

Introduction

This paper is an attempt to combine two fields. The first field is the algebraic theory of regular languages. A regular language of finite words is definable in first-order logic if and only if its syntactic monoid is aperiodic. The main result of the paper is Theorem 9.1, which shows that the Schützenberger-McNaughton-Papert characterization holds for languages of data words with orbit-finite syntactic monoids. They provide some decidable characterizations, including a characterization of first-order logic with local data comparisons within the class of languages recognized by deterministic register automata This result is incomparable to the one in this paper, because we characterize a different logic (data comparisons are not necessarily local), and inside a weaker class of recognizers (nominal monoids have less expressive power than deterministic register automata). We use the more general notion of nominal sets, from [3], which allows more structure on data values, such as order instead of equality only

Nominal Sets
Background
Free Monoid
Syntactic Monoid
Nominal vs Standard Logic
Local Finiteness
Local Finiteness for MSO
Green’s Relations for Nominal Monoids
First-Order Definable Functions
Formulas as a Nominal Set
First-Order Logic
From Aperiodic to First-Order Definable
Findings
10 Further Work

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