# Binary Codes That Do Not Preserve Primitivity

January 3, 2023

### Abstract

A code $X$ is not primitivity preserving if there is a primitive list $\mathtt{ws} \in \mathtt{lists} \ X$ whose concatenation is imprimitive. We formalize a full characterization of such codes in the binary case in the proof assistant Isabelle/HOL. Part of the formalization, interesting on its own, is a description of $\{x,y\}$-interpretations of the square $xx$ if $\mathtt{length}\ y \leq \mathtt{length} \ x$. We also provide a formalized parametric solution of the related equation $x^jy^k = z^\ell$. The core of the theory is an investigation of imprimitive words which are concatenations of copies of two noncommuting words (such a pair of words is called a binary code). We follow the article [Barbin-Le Rest, Le Rest, 85] (mainly Théorème 2.1 and Lemme 3.1), while substantially optimizing the proof. See also [J.-C. Spehner. Quelques problèmes d’extension, de conjugaison et de présentation des sous-monoïdes d’un monoïde libre. PhD thesis, Université Paris VII, 1976] for an earlier result on this question, and [Maňuch, 01] for another proof.

### Related publications

• Rest, E. B., & Rest, M. L. (1985). Sur la combinatoire des codes à deux mots. Theoretical Computer Science, 41, 61–80. https://doi.org/10.1016/0304-3975(85)90060-x
• Maňuch, J. (2001). Defect Effect of Bi-infinite Words in the Two-element Case. Discrete Mathematics & Theoretical Computer Science, Vol. 4 no. 2. https://doi.org/10.46298/dmtcs.279
• J.-C. Spehner. Quelques problèmes d’extension, de conjugaison et de présentation des sous-monoïdes d’un monoïde libre. PhD thesis, Université Paris VII, 1976
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