Cardinality and counting quantifiers on omega-automatic structures

TL;DR

This paper proves that certain logical relations involving countability and modularity on omega-automatic structures are omega-regular, supporting automata-based representations of countable structures and addressing conjectures in the field.

Contribution

It establishes omega-regularity of relations involving countability and modularity on omega-automatic structures, confirming Blumensath's conjecture and advancing automata-theoretic representations.

Findings

Relations with countability are omega-regular.

Omega-regular equivalence relations have omega-regular representatives.

Supports automata-based representation of countable structures.

Abstract

We investigate structures that can be represented by omega-automata, so called omega-automatic structures, and prove that relations defined over such structures in first-order logic expanded by the first-order quantifiers `there exist at most $\aleph_0$ many', 'there exist finitely many' and 'there exist $k$ modulo $m$ many' are omega-regular. The proof identifies certain algebraic properties of omega-semigroups. As a consequence an omega-regular equivalence relation of countable index has an omega-regular set of representatives. This implies Blumensath's conjecture that a countable structure with an $\omega$-automatic presentation can be represented using automata on finite words. This also complements a very recent result of Hj\"orth, Khoussainov, Montalban and Nies showing that there is an omega-automatic structure which has no injective presentation.