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Simple Classes of Automatic Structures

Achim Blumensath

TL;DR

This work delivers algebraic characterisations for two natural subfamilies of automatic structures—polynomial-growth and Presburger structures—focusing on automatic groups and equivalence structures. By developing FO-interpretation frameworks and exploiting semilinearity and vector-partition phenomena, it provides precise classifications: poly-growth automatic groups are finite, Presburger equivalence structures correspond to $\mathfrak E(g)+\mathfrak C$ with $g$ a generalized vector partition function, and poly-growth automatic equivalence structures decompose into finite unions of $\mathfrak E(p)$ alongside infinite-class components. The results illuminate the boundary between automatic and Presburger definability, establish closure properties, and refine previous (and sometimes flawed) proofs with new, robust arguments. Overall, the paper advances a cohesive theory linking model-theoretic interpretability with concrete algebraic decompositions in automatic structures.

Abstract

We study two subclasses of the class of automatic structures: automatic structures of polynomial growth and Presburger structures. We present algebraic characterisations of the groups and the equivalence structures in these two classes.

Simple Classes of Automatic Structures

TL;DR

This work delivers algebraic characterisations for two natural subfamilies of automatic structures—polynomial-growth and Presburger structures—focusing on automatic groups and equivalence structures. By developing FO-interpretation frameworks and exploiting semilinearity and vector-partition phenomena, it provides precise classifications: poly-growth automatic groups are finite, Presburger equivalence structures correspond to with a generalized vector partition function, and poly-growth automatic equivalence structures decompose into finite unions of alongside infinite-class components. The results illuminate the boundary between automatic and Presburger definability, establish closure properties, and refine previous (and sometimes flawed) proofs with new, robust arguments. Overall, the paper advances a cohesive theory linking model-theoretic interpretability with concrete algebraic decompositions in automatic structures.

Abstract

We study two subclasses of the class of automatic structures: automatic structures of polynomial growth and Presburger structures. We present algebraic characterisations of the groups and the equivalence structures in these two classes.

Paper Structure

This paper contains 6 sections, 26 theorems, 112 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Automatic structures of polynomial growth
  4. Presburger Structures
  5. Growth Arguments
  6. Equivalence Structures

Key Result

Theorem 3

Given an automatic structure $\mathfrak A$ (represented by a tuple of automata) and an $\smaller{\mathrm{FOC}}(\mathsf U)$-formula $\varphi(\bar{x})$ (without free second-order variables), one can effectively compute an automaton recognising the relation $\varphi^\mathfrak A$ defined by $\varphi$.

Theorems & Definitions (67)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Definition 4
  • Proposition 5: Blumensath99
  • Theorem 6: Blumensath99ColcombetLoeding07
  • Definition 7
  • Example 1
  • Theorem 8
  • proof
  • ...and 57 more