Nivat's conjecture and pattern complexity in algebraic subshifts

Kari, Jarkko; Moutot, Etienne

doi:10.1016/j.tcs.2018.12.029

Cited by 34 publications

(9 citation statements)

References 13 publications

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“…If the considered shape is not convex the situation becomes more difficult. Theorem 5 is not true for an arbitrary shape in place of the rectangle but all counter examples we know are based on periodic sublattices [6,11]. For example, even lattice cells may form a configuration that is horizontally but not vertically periodic while the odd cells may have a vertical but no horizontal period.…”

Section: Discussionmentioning

confidence: 99%

Decidability and Periodicity of Low Complexity Tilings

Kari

Moutot

2021

Theory Comput Syst

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In this paper we study colorings (or tilings) of the two-dimensional grid ${\mathbb {Z}}^{2}$ 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 $m,n\in \mathbb {N}$ 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 $m,n\in \mathbb {N}$ 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 12).

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Section: Discussionmentioning

confidence: 99%

Decidability and Periodicity of Low Complexity Tilings

Kari

Moutot

2021

Theory Comput Syst

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“…Example Recall that, by convention, the alphabet

scriptA

only contains letters which occur in the language of

X

. Polygonal systems arise naturally in studying the Nivat Conjecture, and in this direction, it follows immediately from [, Corollary 2.6] that (note our terminology differs, and related results appear in ):…”

Section: Defining the Class Of Shiftsmentioning

confidence: 99%

Polygonal Z2‐subshifts

Franks

Kra

2020

Proc. London Math. Soc.

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Let P⊂double-struckZ2 be a convex polygon with each vertex in it labeled by an element from a finite set and such that the labeling of each vertex v∈P is uniquely determined by the labeling of all other points in the polygon. We introduce a class of Z2‐shift systems, the polygonal shifts, determined by such a polygon: These are shift systems such that the restriction of any x∈X to some polygon scriptP has this property. These polygonal systems are related to various well‐studied classes of shift systems, including subshifts of finite type and algebraic shifts, but also many other systems. We give necessary conditions for a Z2‐system X to be polygonal, in terms of the nonexpansive subspaces of X, and under further conditions can give a complete characterization for such systems.

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“…In [7] we considered low complexity configurations in algebraic subshifts where the alphabet A is a finite field F p . As Lemma 1 works as well in this setup, we have that every low complexity configuration c is annihilated by a non-zero polynomial f ∈ F p [x ±1 ].…”

Section: Low Complexity Configurations In Algebraic Subshiftsmentioning

confidence: 99%

“…We first proposed this approach in [9,10] to study Nivat's conjecture. It led to a number of subsequent results [6,7,8,14]. In this presentation we review the main results without proofs -the given references can be consulted for more details.…”

Section: Introductionmentioning

confidence: 99%

Low-Complexity Tilings of the Plane

Kari

2019

Descriptional Complexity of Formal Systems

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A two-dimensional configuration is a coloring of the infinite grid Z 2 with finitely many colors. For a finite subset D of Z 2 , the D-patterns of a configuration are the colored patterns of shape D that appear in the configuration. The number of distinct D-patterns of a configuration is a natural measure of its complexity. A configuration is considered having low complexity with respect to shape D if the number of distinct D-patterns is at most |D|, the size of the shape. This extended abstract is a short review of an algebraic method to study periodicity of such low complexity configurations.

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Nivat's conjecture and pattern complexity in algebraic subshifts

Cited by 34 publications

References 13 publications

Decidability and Periodicity of Low Complexity Tilings

Decidability and Periodicity of Low Complexity Tilings

Polygonal Z2‐subshifts

Low-Complexity Tilings of the Plane

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