On Perfect Coverings of Two-Dimensional Grids

Heikkilä, Elias; Pyry, Herva,; Kari, Jarkko

doi:10.1007/978-3-031-05578-2_12

Cited by 4 publications

(5 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We have discussed a method to infer strong periodicity of a multidimensional configuration from its annihilators or periodizers. The method generalizes the two-dimensional technique used in [5,6,4] to arbitrary dimensions d > 2. The new method is in fact based on a more general condition on the annihilators or periodizers that implies expansivity of a multidimensional subshift in a given direction.…”

Section: Discussionmentioning

confidence: 99%

“…If c has a periodizer f that has no line polynomial factors in any direction then from f ∈ φ 1 φ 2 • • • φ m we conclude that m = 0 so that Per(c) = 1 , implying that c is strongly periodic. In [4] it was noted that this fact can also be proved without referring to the structure of Per(c) simply by noting that f and the special annihilator g(x) = (x t1 − 1) • • • (x tm − 1) guaranteed by Theorem 1 do not have any common factors as f has no line polynomial factors while all irreducible factors of g are line polynomials. It follows that there are non-zero linear combinations of f and g where either one of the two variables has been eliminated.…”

Section: Fibersmentioning

confidence: 96%

“…Low-complexity configurations are the object of interest in the unsolved Nivat's conjecture [2], and also in the recently solved periodic tiling problem [3] where tilings of Z d by translates of a single tile are low-complexity configurations [1]. Also so-called perfect colorings of grid graphs have annihilators [4].…”

Section: Introductionmentioning

confidence: 99%

“…Configurations with annihilators have global rigidity, although they are not necessarily periodic. In the two-dimensional case, periodicity in all directions is known to be enforced if the annihilator has no line polynomial factors, that is, an annihilating polynomial does not have a non-monomial factor whose monomials are on a single line [5,6,4]. In this paper we present a similar condition that works in all dimensions d. More generally, we provide a condition on the annihilator that enforces expansivity: this is a directional determinism property studied in multidimensional symbolic dynamics.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Expansivity and periodicity in algebraic subshifts

Kari¹

2023

Preprint

View full text Add to dashboard Cite

A d-dimensional configuration c : Z d −→ A is a coloring of the ddimensional infinite grid by elements of a finite alphabet A ⊆ Z. The configuration c has an annihilator if a non-trivial linear combination of finitely many translations of c is the zero configuration. Writing c as a d-variate formal power series, the annihilator is conveniently expressed as a d-variate Laurent polynomial f whose formal product with c is the zero power series. More generally, if the formal product is a strongly periodic configuration, we call the polynomial f a periodizer of c. A common annihilator (periodizer) of a set of configurations is called an annihilator (periodizer, respectively) of the set. In particular, we consider annihilators and periodizers of d-dimensional subshifts, that is, sets of configurations defined by disallowing some local patterns. We show that a (d − 1)-dimensional linear subspace S ⊆ R d is expansive for a subshift if the subshift has a periodizer whose support contains exactly one element of S. As a subshift is known to be finite if all (d − 1)-dimensional subspaces are expansive, we obtain a simple necessary condition on the periodizers that guarantees finiteness of a subshift or, equivalently, strong periodicity of a configuration. We provide examples in terms of tilings of Z d by translations of a single tile.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Fibersmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Expansivity and periodicity in algebraic subshifts

Kari¹

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…This article is an extended version of the conference paper [12] where we considered forced periodicity of perfect coverings, that is, perfect colorings with only two colors.…”

Section: Introductionmentioning

confidence: 99%

On forced periodicity of perfect colorings

Pyry¹,

Kari²

2023

Preprint

View full text Add to dashboard Cite

We study forced periodicity of two-dimensional configurations under certain constraints and use an algebraic approach to multidimensional symbolic dynamics in which ddimensional configurations and finite patterns are presented as formal power series and Laurent polynomials, respectively, in d variables. We consider perfect colorings that are configurations such that the number of points of a given color in the neighborhood of any point depends only on the color of the point for some fixed relative neighborhood, and we show that by choosing the alphabet suitably any perfect coloring has a non-trivial annihilator, that is, there exists a Laurent polynomial whose formal product with the power series presenting the perfect coloring is zero. Using known results we obtain a sufficient condition for forced periodicity of two-dimensional perfect colorings. As corollaries of this result we get simple new proofs for known results of forced periodicity on the square and the triangular grids. Moreover, we obtain a new result concerning forced periodicity of perfect colorings in the king grid. We also consider perfect colorings of a particularly simple type: configurations that have low abelian complexity with respect to some shape, and we generalize a result that gives a sufficient condition for such configurations to be necessarily periodic. Also, some algorithmic aspects are considered.

show abstract

On Forced Periodicity of Perfect Colorings

Pyry

Kari

2023

Theory Comput Syst

View full text Add to dashboard Cite

We study forced periodicity of two-dimensional configurations under certain constraints and use an algebraic approach to multidimensional symbolic dynamics in which d-dimensional configurations and finite patterns are presented as formal power series and Laurent polynomials, respectively, in d variables. We consider perfect colorings that are configurations such that the number of points of a given color in the neighborhood of any point depends only on the color of the point for some fixed relative neighborhood, and we show that by choosing the alphabet suitably any perfect coloring has a non-trivial annihilator, that is, there exists a Laurent polynomial whose formal product with the power series presenting the perfect coloring is zero. Using known results we obtain a sufficient condition for forced periodicity of two-dimensional perfect colorings. As corollaries of this result we get simple new proofs for known results of forced periodicity on the square and the triangular grids. Moreover, we obtain a new result concerning forced periodicity of perfect colorings in the king grid. We also consider perfect colorings of a particularly simple type: configurations that have low abelian complexity with respect to some shape, and we generalize a result that gives a sufficient condition for such configurations to be necessarily periodic. Also, some algorithmic aspects are considered.

show abstract

On Perfect Coverings of Two-Dimensional Grids

Cited by 4 publications

References 9 publications

Expansivity and periodicity in algebraic subshifts

Expansivity and periodicity in algebraic subshifts

On forced periodicity of perfect colorings

On Forced Periodicity of Perfect Colorings

Contact Info

Product

Resources

About