### 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.

### License

BSD License

### History

August 17, 2023
  Updated to version v1.10.1.

### Topics

- Computer science/Automata and formal languages

### Related publications

- Holub, Š., Raška, M., & Starosta, Š. (2023). Binary Codes that do not Preserve Primitivity. Journal of Automated Reasoning, 67(3). https://doi.org/10.1007/s10817-023-09674-2
- 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
- Development repository

### Session Binary_Code_Imprimitive

- Binary_Square_Interpretation
- Binary_Code_Imprimitive
- Binary_Imprimitive_Decision

### Depends on

- Combinatorics on Words Basics
- Graph Lemma

### Auto-related entries

- Verification of the UpDown Scheme

## Cite

×

Binary_Code_Imprimitive-AFP

    @article{Binary_Code_Imprimitive-AFP,
      author  = {Štěpán Holub and Martin Raška},
      title   = {Binary codes that do not preserve primitivity},
      journal = {Archive of Formal Proofs},
      month   = {January},
      year    = {2023},
      note    = {\url{https://isa-afp.org/entries/Binary_Code_Imprimitive.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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- Sep 13, 2023 : Isabelle2023
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