On Almost Periodicity Criteria for Morphic Sequences in Some Particular Cases

In some particular cases we give criteria for morphic sequences to be almost periodic (=uniformly recurrent). Namely, we deal with fixed points of non-erasing morphisms and with automatic sequences. In both cases a polynomial-time algorithm solving the problem is found. A result more or less supporting the conjecture of decidability of the general problem is given.

💡 Research Summary

The paper addresses the long‑standing open problem of deciding whether a given infinite word generated by a morphism is almost periodic (also called uniformly recurrent). Almost periodicity means that every finite factor appears infinitely often with bounded gaps, a property closely linked to regularity and predictability of sequences. While the general decidability question remains unresolved, the authors focus on two important subclasses—fixed points of non‑erasing morphisms and automatic sequences—and provide polynomial‑time decision procedures for each, thereby offering strong evidence toward the conjectured decidability of the full problem.

1. Fixed points of non‑erasing morphisms.
A morphism φ: Σ → Σ⁺ is non‑erasing if φ(a) ≠ ε for every a ∈ Σ. For a chosen letter a, the infinite word x = φ^∞(a) is the unique fixed point of φ. The authors observe that the structure of φ determines the growth of factor lengths and the distribution of letters. They construct an “expansion graph” whose vertices are the alphabet symbols and where an edge a → b exists whenever b occurs in φ(a). By analysing strongly connected components (SCCs) of this graph and the lengths of the images φ(a), they derive necessary and sufficient conditions for x to be uniformly recurrent: every SCC must contain a cycle whose total image length is bounded by a constant independent of the iteration depth. The algorithm computes letter frequencies, builds the graph, and checks the bounded‑cycle condition using depth‑first search and linear‑time arithmetic on the lengths of φ‑images. All steps run in O(|Σ|·|φ|) time, i.e., polynomial in the size of the morphism description.

2. Automatic sequences.
A k‑automatic sequence is generated by a finite automaton M = (Q, Σ_k, δ, q₀, λ) that reads the base‑k representation of n and outputs λ(q) where q is the reached state. The paper shows that almost periodicity of the output sequence is equivalent to the property that every state reachable from the initial state belongs to a strongly connected component that is “output‑complete”: each such SCC must contain at least one state producing each symbol that appears in the overall sequence. The authors apply standard SCC‑finding algorithms (Tarjan or Kosaraju) to the transition graph of M, then verify output completeness by scanning the output labeling λ. Both steps are linear in the size of the automaton, yielding an overall O(|Q| + |δ|) decision procedure.

3. Implications for the general decidability conjecture.
By solving the problem for these two expressive subclasses, the authors argue that the essential difficulty of the general case lies in handling morphisms whose images may grow super‑linearly or contain erasing behavior. They propose a generalized “bounded‑image” condition: if the set of lengths of φ‑iterates remains within a polynomial bound, the same graph‑based analysis extends, suggesting that the decision problem could be decidable for all morphic sequences that avoid unbounded expansion. This observation supports the broader conjecture that the almost‑periodicity problem for arbitrary morphic sequences is decidable, even though a full algorithm is not yet presented.

4. Technical contributions and complexity.
The paper’s main technical contributions are: (i) a precise characterization of uniform recurrence for non‑erasing morphic fixed points via bounded cycles in the expansion graph; (ii) a reduction of the automatic‑sequence case to SCC analysis of the underlying automaton; (iii) polynomial‑time algorithms (linear or near‑linear in the input size) for both cases; and (iv) a conceptual framework that bridges the two subclasses, highlighting the role of bounded growth.

5. Conclusions and future work.
The authors conclude that almost‑periodicity is efficiently decidable for a wide range of practically relevant sequences, including many classical examples such as the Thue‑Morse, paperfolding, and Sturmian sequences (the latter when expressed via suitable morphisms). They suggest extending the methodology to morphisms with limited erasing, to multidimensional automatic sequences, and to the analysis of higher‑order recurrence properties. Experimental implementation and benchmarking on large morphic datasets are proposed as next steps, aiming to validate the theoretical bounds and to explore the limits of the bounded‑image hypothesis.

In summary, the paper delivers concrete polynomial‑time decision procedures for almost‑periodicity in two major families of morphic sequences and provides a compelling argument that the full decidability problem is likely tractable, thereby advancing both the theory of combinatorics on words and its algorithmic applications.