Enumeration and Decidable Properties of Automatic Sequences

We show that various aspects of k-automatic sequences – such as having an unbordered factor of length n – are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or k-regular. These include many sequences previously studied in the literature, such as the recurrence function, the appearance function, and the repetitivity index. We also give some new characterizations of the class of k-regular sequences. Many results extend to other sequences defined in terms of Pisot numeration systems.

💡 Research Summary

This paper establishes a unified, algorithmic framework for deciding and enumerating a wide variety of properties of k‑automatic sequences. An infinite word x = (a(n))_{n≥0} over a finite alphabet is k‑automatic if a finite automaton, fed with the base‑k representation of n, reaches a state whose label is a(n). The authors build on earlier results (Honkala’s decidability of ultimate periodicity, Leroux’s polynomial‑time algorithm, and Allouche‑Rampersad‑Shallit’s logical characterization) to show that any property of a k‑automatic sequence that can be expressed in the first‑order structure ⟨ℕ,+,V_k⟩—where V_k(n) denotes the largest power of k dividing n—is decidable. Existential and universal quantifiers are simulated by nondeterministic and deterministic finite automata (NFA/DFA) constructions, while arithmetic operations (addition, subtraction, comparison) are performed on the fly within the automaton.

A central example is the existence of an unbordered factor of length n. The property “x has an unbordered factor of length n” is formalized as ∃j≥0 ∀1≤ℓ≤⌊n/2⌋ ∃i<ℓ (a_{j+i} ≠ a_{j+n−ℓ+i}). The authors construct an NFA that, given the encoding of (j,ℓ,n), guesses i, checks i<ℓ, computes the two positions j+i and j+n−ℓ+i, and verifies the inequality of the letters. After determinization and complement, a DFA recognises exactly those n for which the property holds, yielding a new k‑automatic sequence b(n) that is 1 when an unbordered factor exists and 0 otherwise. This method extends to many other properties: square‑freeness, r‑power‑freeness, presence of arbitrarily large powers, critical exponent, recurrence, uniform recurrence, and equality of factor sets between two automatic sequences.

The paper also addresses enumeration problems. For a given automatic sequence, the function f(n) counting distinct factors of length n is shown to be k‑regular. The authors use the notion of the k‑kernel: the set of subsequences (a(k^e n + c))_{n≥0} for all e≥0 and 0≤c<k^e. They prove that the counting function can be expressed as a linear combination of a finite set of sequences in the kernel, which is precisely the definition of a k‑regular sequence. Consequently, several classical combinatorial quantities—such as the recurrence function, the appearance function, and the repetitivity index—are all k‑regular. The paper provides new characterizations of (R,k)‑regular sequences via recognizable formal series, showing that operations like ignoring leading zeros preserve recognizability.

An important extension is to numeration systems based on Pisot numbers. By replacing the standard base‑k representation with a Pisot numeration system, the same logical‑automata translation works, because the necessary arithmetic predicates remain regular. Thus, many of the decidability and enumeration results hold for sequences defined over these more general numeration systems.

In summary, the authors present a powerful, systematic technique: express a property in the logical language of ⟨ℕ,+,V_k⟩, translate it into automata operations, and then either decide the property (by checking emptiness, finiteness, or inclusion of a regular language) or obtain an enumerating k‑regular sequence. This unifies and extends a large body of scattered results in the literature, provides effective algorithms for previously non‑constructive proofs, and opens the door to further applications in combinatorics on words, formal language theory, and dynamical systems.