Decidability and Periodicity of Low Complexity Tilings

International audienceIn this paper we study colorings (or tilings) of the two-dimensional grid Z 2. A coloring is said to be valid with respect to a set P of n × m rectangular patterns if all n × m sub-patterns of the coloring are in P. A coloring c is said to be of low complexity with respect to a rectangle if there exist m, n ∈ N and a set P of n × m rectangular patterns such that c is valid with respect to P and |P | ≤ nm. Open since it was stated in 1997, Nivat's conjecture states that such a coloring is necessarily periodic. If Nivat's conjecture is true, all valid colorings with respect to P such that |P | ≤ mn must be periodic. We prove that there exists at least one periodic coloring among the valid ones. We use this result to investigate the tiling problem, also known as the domino problem, which is well known to be undecidable in its full generality. However, we show that it is decidable in the low-complexity setting. Then, we use our result to show that Nivat's conjecture holds for uniformly recurrent configurations. These results also extend to other convex shapes in place of the rectangle. After that, we prove that the nm bound is multiplicatively optimal for the decidability of the domino problem, as for all ε > 0 it is undecidable to determine if there exists a valid coloring for a given m, n ∈ N and set of rectangular patterns P of size n×m such that |P | ≤ (1 + ε)nm. We prove a slightly better bound in the case where m = n, as well as constructing aperiodic SFTs of pretty low complexity. This paper is an extended version of a paper published in STACS 2020 (Kari and Moutot 2020)

Decidability and Periodicity of Low Complexity Tilings

In this paper we study colorings (or tilings) of the two-dimensional grid Z(2). A coloring is said to be valid with respect to a set P of n x m rectangular patterns if all n x m sub-patterns of the coloring are in P. A coloring c is said to be of low complexity with respect to a rectangle if there exist m, n is an element of N and a set P of n x m rectangular patterns such that c is valid with respect to P and vertical bar P vertical bar 0 it is undecidable to determine if there exists a valid coloring for a given m, n is an element of N and set of rectangular patterns P of size n x m such that vertical bar P vertical bar <= (1 + epsilon)nm. We prove a slightly better bound in the case where m = n, as well as constructing aperiodic SFTs of pretty low complexity. This paper is an extended version of a paper published in STACS 2020 (Kari and Moutot 2020).</p

Algebrallinen näkökulma peittokoodeihin

Tässä tutkielmassa käsitellään algebrallista symbolidynamiikkaa ja sovelletaan sitä peittokoodien tutkimiseen. Algebrallisessa symbolidynamiikassa kombinatoriset ja topologiset ongelmat muutetaan polynomeja koskeviksi kysymyksiksi, jolloin ongelmaan saadaan algebrallinen näkökulma. Algebrallisella lähestymistavalla saadaan helppoja todistuksia neliöhilan ja kuningasgraafin peittokoodituloksille. Tutkielma alkaa symbolidynamiikan perinteisten käsitteiden määrittelyllä ja perustulosten esittelyllä. Tämän jälkeen määritellään kommutatiivisen algebran ja algebrallisen geometrian peruskäsitteet, joilla saadaan uusi näkökulman symbolidynamiikan tutkimukseen. Näin saadaan perinteisen topologisen rakenteen lisäksi myös algebrallista rakennetta käyttöön. Perustietojen jälkeen esitellään työkaluiksi erilaisia polynomihajotelmia ja todistetaan tutkielman kannalta olennaisten ihanteiden rakennetuloksia. Tutkielmassa pyritään rakentamaan tarvittava teoria mahdollisimman suppeilla esitietovaatimuksilla. Kun teoriapohja on rakennettu, sovelletaan polynomihajotelmia peittokooditulosten todistamiseen. Lopuksi annetaan vielä algoritmi peittokoodien etsimiseen tietyssä erikoistapauksessa

Cutting corners

We define a class of subshifts defined by a family of allowed patterns of the same shape where, for any contents of the shape minus a corner, the number of ways to fill in the corner is the same. For such a subshift, a locally legal pattern of convex shape is globally legal, and there is a measure that samples uniformly on convex sets. We show by example that these subshifts need not admit a group structure by shift-commuting continuous operations. Our approach to convexity is axiomatic, and only requires an abstract convex geometry that is “midpointed with respect to the shape”. We construct such convex geometries on several groups, in particular strongly polycyclic groups and free groups. We also show some other methods for sampling finite patterns, and show a link to conjectures of Gottshalk and Kaplansky.</p