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Low-Complexity Tilings of the Plane

Abstract : A two-dimensional configuration is a coloring of the infinite grid $$\mathbb {Z}^2$$ with finitely many colors. For a finite subset D of $$\mathbb {Z}^2$$, 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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Submitted on : Friday, November 29, 2019 - 4:35:47 PM
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Jarkko Kari. Low-Complexity Tilings of the Plane. 21th International Conference on Descriptional Complexity of Formal Systems (DCFS), Jul 2019, Košice, Slovakia. pp.35-45, ⟨10.1007/978-3-030-23247-4_2⟩. ⟨hal-02387291⟩

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