On the enumerating series of an abstract numeration system

It is known that any rational abstract numeration system is faithfully, and effectively, represented by an N-rational series. A simple proof of this result is given which yields a representation of this series which in turn allows a simple computation of the value of words in this system and easy constructions for the recognition of recognisable sets of numbers. It is also shown that conversely it is decidable whether an N-rational series corresponds to a rational abstract numeration system.

💡 Research Summary

The paper investigates the relationship between abstract numeration systems (ANS) and rational series theory, focusing on the enumerating series associated with a given ANS. An ANS is defined as a triple (L, A, <) where A is a finite alphabet equipped with a total order <, and L⊆A* is an infinite language. The mapping π_L assigns to each word w∈L the integer n such that w is the (n + 1)-st word of L in the radix (lexicographic by length then by <) order; conversely, h_n^L denotes the word representing n. When L is a rational language, the system is called a rational ANS.

The central object of study is the enumerating series E_S, an N‑series over A* defined by
 E_S = ∑_{w∈L} (π_L(w)+1)·w.
The coefficient of each word w is exactly one plus its rank in L, and the support of E_S coincides with L, so the series fully encodes the numeration system.

Theorem 1 proves that for any rational ANS, E_S is an N‑rational series. Existing proofs relied on sophisticated rational transductions; the authors present a more direct construction. Let A be an unambiguous finite automaton recognizing L, with N‑representation (λ, µ, ν) of dimension k (λ is a row vector, ν a column vector, µ maps letters to k×k matrices). For any word u, define P(u) = {v ∈ A* | v ≺ u}. Lemma 3 shows a decomposition
 P(ua) = 1·A* ∪ u·A_a ∪ P(u)·A,
where A_a = {b∈A | b<a}. Translating this decomposition into matrix form yields the recurrence
 λ·µ(P(ua))·ν = λ·ν + λ·µ(u)·σ_a·ν + λ·µ(P(u))·σ·ν,
with σ = µ(A) and σ_a = µ(A_a). Introducing a new (2k² + k)-dimensional representation (η, κ, ζ) that bundles λ, µ, σ, σ_a, the authors verify by induction that λ·µ(P(u))·ν = η·κ(u)·ζ for all u. Defining a series s by ⟨s,u⟩ = 1 + |{v∈L | v ≺ u}|, they obtain s = η·κ·ζ. Finally, the enumerating series is the Hadamard product E_S = s ⊙ L, which preserves N‑rationality. The construction yields an explicit N‑representation of size 2k² + k, considerably more compact than generic constructions.

Theorem 2 addresses the converse problem: given an N‑rational series, decide whether it is the enumerating series of some rational ANS. The decision procedure checks whether the support language is rational and whether the coefficients satisfy the “rank + 1” property with respect to the radix order. Hence the correspondence between rational ANS and N‑rational enumerating series is effectively decidable.

Beyond the theoretical results, the paper provides concrete algorithms. Computing π_L(w) for a word w of length ℓ using a generic N‑representation of dimension n would cost O(ℓ·n²) operations. By exploiting the original automaton of dimension k and the matrices σ, σ_a, the authors devise a specialized algorithm that maintains two k‑dimensional vectors α(w) (0/1 indicator of membership in L) and γ(w) (cumulative rank). The update rules are:
 α(w·a) = α(w)·µ(a),
 β(w·a) = α(w)·σ_a,
 γ(w·a) = λ + β(w·a) + γ(w)·σ.
When α(w)·ν = 1, the value is π_L(w) = γ(w)·ν. This method runs in O(ℓ·k²) time and uses only the original automaton’s data, making value computation practical for large systems.

The authors also show how to recognize, within L, the representations of any recognisable set of integers X⊆ℕ. Since for any N‑rational series s, the pre‑image s⁻¹(X) is a rational language, Corollary 4 (previously known) follows: recognisable subsets of ℕ are L‑recognisable for any rational ANS. They give an explicit construction: for the modular set X_{p,r}=pℕ+r, the language h_{X_{p,r}}^L is recognised by a deterministic automaton with at most k·p·k states, obtained by a product of the original automaton with a p‑state counter.

The paper concludes with several open questions, such as extending the framework to non‑rational ANS, exploring other semirings (e.g., ℤ, ℝ), and optimizing the state complexity of the constructed automata.

In summary, the work bridges abstract numeration systems and rational series/automata theory by (i) providing a constructive proof that the enumerating series of any rational ANS is N‑rational, (ii) delivering an efficient algorithm for computing the numeric value of words, (iii) offering a decidability result for the inverse problem, and (iv) presenting concrete automata constructions for recognisable subsets of numbers. These contributions deepen our understanding of the algebraic structure of numeration systems and open avenues for further research in formal language theory, combinatorics on words, and computational number theory.