Domino Problem Under Horizontal Constraints

Nathalie Aubrun (1, 2) , Mathieu Sablik (3) , Julien Esnay (2, 1, 3)

1 MC2 - Modèles de calcul, Complexité, Combinatoire
2 LIP - Laboratoire de l'Informatique du Parallélisme
3 IMT - Institut de Mathématiques de Toulouse UMR5219

The Domino Problem on Z² asks if it is possible to tile the plane with a given set of Wang tiles; it is a classical decision problem which is known to be undecidable. The purpose of this article is to parameterize this problem to explore the frontier between decidability and undecidability. To do so we fix some horizontal constraints H on the tiles and consider a new Domino Problem DPH : given a vertical constraint, is it possible to tile the plane? We characterize the nearest-neighbor horizontal constraints where DPH is decidable using graphs combinatorics.

Mots clés: Dynamical Systems; Symbolic Dynamics; Subshifts; Wang tiles; Undecidability; Domino Problem; Combinatorics; Tilings; Subshifts of Finite Type

Domaines: Mathématique discrète [cs.DM]; Théorie et langage formel [cs.FL]; Combinatoire [math.CO]

Nathalie Aubrun, Mathieu Sablik, Julien Esnay. Domino Problem Under Horizontal Constraints. STACS 2020 37th International Symposium on Theoretical Aspects of Computer Science, 2020, Montpellier, France. ⟨10.4230/LIPIcs.STACS.2020.26⟩. ⟨hal-02380657⟩