Multi-dimensional sets recognizable in all abstract numeration systems

We prove that the subsets of N^d that are S-recognizable for all abstract numeration systems S are exactly the 1-recognizable sets. This generalizes a result of Lecomte and Rigo in the one-dimensional setting.

💡 Research Summary

The paper investigates which subsets of the d‑dimensional lattice ℕⁿ are recognizable in every abstract numeration system (ANS). An ANS is a triple S = (L, Σ, <) where L is an infinite regular language over a totally ordered finite alphabet Σ, and rep_S maps each natural number n to the (n + 1)‑st word of L in genealogical order. A set X ⊆ ℕᵈ is S‑recognizable if the padded representation rep_S(X)# forms a regular language.

In one dimension, Lecomte and Rigo showed that a set is recognizable in all ANS if and only if it is 1‑recognizable, i.e., its unary language {aⁿ : n∈X} is regular, which is equivalent to being ultimately periodic. Extending to higher dimensions, Cobham–Semenov’s theorem states that a set recognizable in two multiplicatively independent integer bases is exactly semi‑linear. However, semi‑linear sets are not sufficient for the ANS‑universality: the set {(n, 2n)} is semi‑linear but not 1‑recognizable.

The authors’ main contribution is Theorem 9: a subset X ⊆ ℕᵈ is S‑recognizable for every ANS S if and only if X is 1‑recognizable. The proof relies on the classical decomposition theorem of Eilenberg, Elgot, and Shepherdson. By padding d‑tuples of words into a single word over the alphabet (Σ ∪ {#})ᵈ, the theorem characterizes regularity of the padded language as a finite union of products of regular languages each confined to a sub‑alphabet Σ_A. Because |Σ| = 1 in the unary case, each factor reduces to a language of the form {x·p + q : i∈ℕ}, which corresponds to linear sets with a single period. Lemma 16 translates this into an explicit description (formula (1)) of all 1‑recognizable sets as finite unions of linear expressions with nested constraints on the coordinates.

Lemma 18 further refines these expressions into a finite combination of simple linear inequalities and equalities (forms (2)–(5)). Each such component is easily seen to be S‑recognizable for any ANS, because the corresponding padded language is regular by construction. Consequently, any set fitting formula (1) is S‑recognizable for all S, establishing the “if” direction.

Conversely, if X is S‑recognizable for every ANS, then in particular it is 1‑recognizable (the unary ANS built on a*). Using the decomposition theorem, the authors show that the regularity of rep_S(X)# forces X to have the structure described in Lemma 16, and thus X must be 1‑recognizable.

The paper includes concrete examples, notably Example 10, where a set defined by mixed linear constraints is shown to be S‑recognizable for an arbitrary ANS by constructing automata for each component and combining them via product and union operations.

Overall, the work clarifies the landscape of multidimensional automatic sets: while semi‑linear sets capture base‑k recognizability, the class of sets recognizable in all abstract numeration systems collapses precisely to the 1‑recognizable (unary regular) sets. This result generalizes the one‑dimensional theorem of Lecomte and Rigo and provides a clean, algebraic characterization for the multidimensional case.