Decidability of the HD0L ultimate periodicity problem

1. Introduction 1.1. The HD0L ultimate periodicity problem. In this paper we prove the decidability of the following problem :

Input: Two finite alphabets A and B, an endomorphism σ : A * → A * , a word w ∈ A * and a morphism φ : A * → B * . Question: Do there exist two words u and v in B * , with v non-empty, such that the sequence (φ(σ n (w))) n converges to uv ω (i.e., is ultimately periodic)?

(The convergence of the sequence (φ(σ n (w))) n meaning that (|φ(σ n (w))|) n goes to +∞ and that (φ(σ n (www • • • ))) n converges in B N endowed with the usual product topology.) We will refer to it as the HD0L ultimate periodicity problem. Observe that it is slightly more general than the classical statement where it is assumed in the input that the sequence (φ(σ n (w))) n converges.

Theorem 1. The HD0L ultimate periodicity problem is decidable.

This result was announced in [Durand 2012]. While we were ending the writing of this paper, I. Mitrofanov put on Arxiv [Mitrofanov preprint 2011] another solution of this problem. This problem was open for about 30 years. In 1986, positive answers were given independently for D0L systems (or purely substitutive sequences) in both [Harju and Linna 1986] and [Pansiot 1986], and, for automatic sequences (which are particular HD0L sequences) in [Honkala 1986]. Other proofs have been given for the D0L case in [Honkala 2008] and for automatic sequences in [Allouche, Rampersad and Shallit 2009]. Recently in [Durand 2012] the primitive case has been solved. In [Honkala and Rigo 2004] is given an equivalent statement of the HD0L ultimate periodicity problem in terms of recognizable sets of integers and abstract numeration systems. In fact, J. Honkala already gave a positive answer to this question in [Honkala 1986] but in the restricted case of the usual integer bases, i.e., for k-automatic sequences or constant length substitutive sequences. Recently, in [Bell, Charlier, Fraenkel, and Rigo 2009], a positive answer has been given for a (large) class of numeration systems including for instance the Fibonacci numeration system. Let us point out that the characterization of recognizable sets of integers for abstract numeration systems in terms of substitutions given in [Maes and Rigo 2002] (see also [Lecomte and Rigo 2010]), together with Theorem 1, provides a decision procedure to test whether a recognizable set of integers in some abstract numeration system is a finite union of arithmetic progressions. 1.2. Organization of the paper. In Section 2 are the classical definitions. In Section 3 we prove the HD0L ultimate periodicity problem for substitutive sequences. These sequences are such that (σ n (w)) n converges. This avoids to test the existence of the limit. Indeed, there are examples where (φ(σ n (w))) n converges and (σ n (w)) n does not: for σ and φ, defined by σ(a) = cb, σ(b) = ba, σ(c) = ab, φ(a) = φ(c) = 0 and φ(b) = 1, the sequence (σ n (a)) n does not converge but (φ(σ n (a))) n does (to the Thue-Morse sequence). Under these assumptions the proof could be sketched as follows. First we recall some primitivity arguments about matrices and substitutions. The “best or easiest situation” is when we deal with growing substitutions and codings (letter-to-letter morphisms). It is known that we can always consider we are working with codings (see [Cobham 1968, Pansiot 1983, Allouche and Shallit 2003, Cassaigne and Nicolas 2003]). In [Honkala 2009] it is shown this can be algorithmically realized. We propose a different algorithm using the proof of [Cassaigne and Nicolas 2003] where we replace some (non-algorithmic) arguments (Lemma 2, Lemma 3 and Lemma 4 of this paper) by algorithmic ones. We treat separately growing and non-growing substitutions. For growing substitutions we look at their primitive components and we use the decidability result established in [Durand 2012] about periodicity for primitive substitutions. Indeed, these primitive components should generate periodic sequences. Hence, we check it is the case (if not, then the sequence is not ultimately periodic). From there, Lemma 11 allows us to conclude. For the non-growing case we use a result of Pansiot [Pansiot 1984] saying that we can either consider we are in the growing case or there are longer and longer periodic words with the same period in the sequence. We again conclude with Lemma 11. In Section 4 we show how to use the substitutive case to solve the general HD0L case. This concludes the proof of Theorem 1. 1.3. Questions and comments. We did not compute the complexity of the algorithm provided by our proof of the HD0L ultimate periodicity problem. Looking at Proposition 4 and the results in [Durand 2012] we use here, our approach provides a high complexity.

Our result is for one-dimensional sequences. What can be said about multidimensional sequences generated by substitution rules ? or self-similar tilings ? It seems hopeless to generalize our method to tilings, although the main and key result we

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