A new proof for the decidability of D0L ultimate periodicity

We give a new proof for the decidability of the D0L ultimate periodicity problem based on the decidability of p-periodicity of morphic words adapted to the approach of Harju and Linna.

💡 Research Summary

The paper addresses the long‑standing decision problem of whether the infinite word generated by a D0L system is ultimately periodic. A D0L system is a pair (h, u) where h : A* → A* is a morphism over a finite alphabet A and u ∈ A* is a seed word. The system is assumed to be prolongable on u, i.e. h(u)=u y and hⁿ(y)≠ε for all n≥0, which guarantees that the sequence of iterates hⁿ(u) forms an increasing chain of prefixes whose limit is the infinite word h^ω(u)=u y h(y) h²(y)… . The word is ultimately periodic if it can be written as uv^ω for some finite words u and v (v≠ε).

The authors first introduce a classification of letters into finite letters (those whose set of iterates {hⁿ(b)} is finite) and infinite letters (the complement). Among infinite letters they further distinguish recurrent letters, i.e. those occurring infinitely often in h^ω(a). The corresponding sets are denoted A_F, A_I, A_R and A₁=A_I∩A_R. They show how to compute these sets effectively: mortal letters (those that eventually map to ε) are removed, a non‑erasing morphism g is built by erasing mortal letters, and the finiteness of {gⁿ(b)} characterises finite letters. A directed graph G with vertices A and an edge b→c whenever c occurs in h(b) is used to decide recurrence: a letter b is recurrent iff there are infinitely many paths from the letters occurring in the suffix y to b.

The second major component is the p‑periodicity problem for morphic words. For a fixed integer p≥1, the infinite word x is examined modulo p: for each residue class k (0≤k<p) the k‑set consists of letters that appear infinitely often at positions congruent to k modulo p. The authors recall from earlier work that these k‑sets can be constructed effectively for words of the form h^ω(u). The key observation is that the incidence matrix M of h (M_{i,j}=#a_i in h(a_j)) satisfies that the sequence (Mⁿ mod p) is ultimately periodic; consequently the sequence of lengths |hⁿ(a_j)| mod p is ultimately periodic for every a_j. Using this periodicity they find integers r and q such that |h^r(b)|≡|h^{r+q}(b)| (mod p) for all b∈A. They then build a directed graph G_h whose vertices are pairs (a,i) with a∈A and 0≤i<p, and where an edge ((c,i)→(d,j)) exists if a letter c occurs in h^r(b) at a position congruent to i (mod p) and d occurs in h^q(c) at a position congruent to j (mod p). Traversing this graph yields the k‑sets.

The crucial Theorem 1 states that, given p, a coding g and a morphism h prolongable on a, the morphic word g(h^ω(a)) is ultimately p‑periodic iff for every pair of letters (b,c) belonging to the same k‑set we have g(b)=g(c). Since the k‑sets are computable, the theorem provides an algorithmic decision procedure for p‑periodicity.

The authors also recall Theorem 2, a classic result: if there exists n such that hⁿ(u)=hⁿ(v) for words u,v, then the restriction of h to the non‑mortal alphabet satisfies h|A|(u)=h|A|(v). The proof proceeds by induction on the alphabet size, using the fact that elementary morphisms are injective and that any morphism can be factored into elementary ones. This theorem is later used to lift equalities of finite iterates to equalities of the whole infinite word.

With these tools, the paper presents Theorem 3, the main result: the ultimate periodicity problem for D0L sequences is decidable. The proof proceeds as follows.

1. Reduction to a single seed letter: By a standard transformation, one may assume the seed word is a single letter a.

2. Eliminate non‑recurrent letters: If a letter never appears in h^ω(a) it can be removed; the algorithm first checks whether every letter of A occurs infinitely often.

3. Case analysis on the set A₁:

   • If A₁ is empty, the limit word consists only of finite letters after a finite prefix, hence it is trivially ultimately periodic.
   • If A₁ contains exactly one letter b, then h(a)=a y with y containing only finite letters except possibly b. One can find n and p such that h^{n+p}(y)=hⁿ(y); the concatenation hⁿ(y)h^{n+1}(y)… yields a period.
   • If |A₁|≥2, pick a distinguished recurrent letter b∈A₁ and write the limit word as u₀ b u₁ b u₂… with each u_i∈(A{b})*. Define U={u_i | i≥0}. If U is infinite, the word cannot be ultimately periodic (because the gaps between successive b’s never repeat). The algorithm checks finiteness of U by examining whether any infinite letter c can generate arbitrarily long finite blocks without b; this is decidable because one only needs to explore up to |A| iterations.

4. When U is finite: Enumerate the finitely many distinct blocks u_i. For each pair (u_i,u_j) there exists a bound m (in fact m=|A| by Theorem 2) such that h^m(b u_i b u_j)=h^m(b u_j b u_i). This equality implies that the images h^ℓ(b u_i) and h^ℓ(b u_j) commute for all ℓ≥m, and by a classical commutation argument there exists a primitive word z such that every h^ℓ(b u_i) belongs to z*. Consequently the whole limit word is ultimately |z|-periodic.

5. Final verification: The period length |z| is known, so the algorithm applies Theorem 1 with p=|z| (and the identity coding) to confirm that the word is indeed ultimately p‑periodic.

All steps are effective: constructing A_F, A_I, A_R, building the graphs G and G_h, checking finiteness of U, and applying Theorem 1 are all computable procedures. Hence the D0L ultimate periodicity problem is decidable.

The contribution of the paper lies in simplifying the original Harju‑Linna proof by replacing the intricate combinatorial analysis of morphism powers with the more modular p‑periodicity framework. This not only yields a cleaner conceptual proof but also highlights the utility of the p‑periodicity decision algorithm as a reusable tool for other problems concerning morphic words. Moreover, the paper demonstrates that the entire decision process can be carried out using elementary operations on finite automata, matrices modulo p, and graph reachability, making it amenable to implementation.

In summary, the authors provide a constructive, algorithmic proof that the ultimate periodicity of D0L systems is decidable, leveraging recent advances on p‑periodicity of morphic words to streamline the argument and to obtain a procedure that can be directly implemented.