Cardinality and counting quantifiers on omega-automatic structures

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.

💡 Research Summary

The paper investigates the expressive power of first‑order logic extended with counting quantifiers over ω‑automatic structures, i.e., structures whose domain and relations are represented by automata reading infinite words. The authors consider three families of counting quantifiers: “there exist at most ℵ₀ many”, “there exist finitely many”, and “there exist k modulo m many”. Their main result is that any relation definable in this enriched logic (denoted FOC) is ω‑regular, meaning it can be recognized by a Büchi, parity, or Muller automaton operating synchronously on the encoded tuples.

To achieve this, the authors move away from the traditional automata‑centric proofs and instead employ the algebraic theory of ω‑semigroups. An ω‑semigroup consists of a finite semigroup for finite words together with a set of infinite‑word elements and two products (· for finite concatenation, ∗ for mixing finite and infinite parts) together with an infinite product π satisfying natural associativity laws. By the classical theorem of Perrin and Pin, a language of infinite words is ω‑regular iff it is recognized by a finite ω‑semigroup. This correspondence allows the authors to reason about the behaviour of automata through algebraic properties of the underlying semigroups.

The core technical contribution is Proposition 3.1, which provides a uniform bound C (computable from the presentation) such that for any tuple of parameters (\bar z) the following are equivalent: (1) the set of solutions of a given FO‑formula (\varphi(x,\bar z)) has only countably many ≈‑equivalence classes (where ≈ is the congruence induced by the presentation), and (2) there exist at most C representatives (x_1,\dots,x_C) such that every solution is ≈‑equivalent to one of them and, moreover, lies in the same ∼ₑ‑class (the “equal‑ends” relation) as that representative. The proof proceeds in three stages. First, a Ramsey‑theoretic argument extracts an infinite homogeneous factorisation of the candidate solutions with respect to the finite semigroups that recognise the formula and the congruence. Second, a “coarsening” step exploits a property of finite semigroups that recognise transitive relations: they contain idempotent elements that absorb other elements, enabling the construction of new words with controlled semigroup images. Third, a shuffling construction interleaves these refined factorisations to generate continuum many pairwise non‑≈‑equivalent solutions whenever condition (2) fails, thereby proving that the solution set must be uncountable.

From this characterisation the authors derive Theorem 2.4: the standard fundamental fact for automatic structures (that FO‑definable relations are regular and the FO‑theory is decidable) extends to all ω‑automatic presentations, even when they are not injective. Consequently, every counting quantifier of the three families can be eliminated by an effective construction of an ω‑automaton recognizing the corresponding relation. This yields decidability of the FO + counting‑quantifier theory for arbitrary ω‑automatic structures.

A further consequence concerns equivalence relations of countable index. Using Lemma 2.5 (originally due to Kuske and Lohrey) the authors show that an ω‑regular equivalence relation with only countably many classes admits an ω‑regular set of representatives. This resolves a conjecture of Blumensath: any countable structure that has an ω‑automatic presentation also has a (finite‑word) automatic presentation. The transformation is effective: one first obtains an injective ω‑automatic presentation by selecting the ω‑regular representatives, and then “packs” the infinite‑word encoding into a finite‑word encoding, as described in Proposition 2.7.

Finally, the paper situates its results with respect to recent work by Hjorth, Khoussainov, Montalbán, and Nies, who exhibited ω‑automatic structures lacking any injective presentation. The authors’ findings show that such counter‑examples necessarily involve uncountable domains; for countable ω‑automatic structures injective presentations always exist.

In summary, the paper makes three major contributions: (1) it extends the decidability and regularity results of automatic structures to the richer logic FOC over arbitrary ω‑automatic presentations, (2) it provides an algebraic (ω‑semigroup) framework that yields a clean characterisation of when a formula has only countably many solutions, and (3) it settles the open question whether countable ω‑automatic structures are automatically presentable, confirming that they are. These results deepen our understanding of the interplay between infinite‑word automata, algebraic semigroup theory, and model‑theoretic definability, and open avenues for further exploration of counting extensions in other infinite‑structure settings.