Periodicity in Tilings

Jeandel, Emmanuel; Vanier, Pascal

doi:10.1007/978-3-642-14455-4_23

Cited by 6 publications

(10 citation statements)

References 22 publications

(17 reference statements)

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“…Subsequent to the preliminary version of our paper [12], a similar result about periodic tilings was proven independently in [17]. However, their results are slightly different than ours.…”

Section: Tiling Resultssupporting

confidence: 48%

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Gottesman¹,

Irani²

2013

Theory of Comput.

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We study the complexity of a class of problems involving satisfying constraints which remain the same under translations in one or more spatial directions. In this paper, we show hardness of a classical tiling problem on an N × N 2-dimensional grid and a quantum problem involving finding the ground state energy of a 1-dimensional quantum system of N particles. In both cases, the only input is N, provided in binary. We show that the classical problem is NEXP-complete and the quantum problem is QMA EXP-complete. Thus, an algorithm for these problems which runs in time polynomial in N (exponential in the input size) would imply that EXP = NEXP or BQEXP = QMA EXP , respectively. Although tiling in general is already known to be NEXP-complete, the usual approach is to require that either the set of tiles and their constraints or some varying boundary conditions be given as part of the input. In the problem considered here, these are fixed, constant-sized parameters of the problem. Instead, the problem instance is encoded solely in the size of the system.

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“…Subsequent to the preliminary version of our paper [12], a similar result about periodic tilings was proven independently in [17]. However, their results are slightly different than ours.…”

Section: Tiling Resultssupporting

confidence: 48%

Untitled

Gottesman¹,

Irani²

2013

Theory of Comput.

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“…However, practically speaking, it is not at all simple to turn information about roots of p(z) into information about the set of exponents with positive coefficients for the power series expansion of 1 p(z) . Finally, we note that the possible sets for a generalized notion of least periods for multidimensional SFTs (which consist of Z d -indexed arrays of letters rather than sequences) were recently characterized in [2]. As is often the case for multidimensional SFTs, their characterization is in recursion-theoretic terms and much more complicated than the ones we derive in one dimension.…”

Section: Introductionmentioning

confidence: 86%

“…However, as our results show, not all semi-linear sets are realizable in this way; for instance, the set of positive odd integers is not the set of least periods of any (one-dimensional) SFT. It is strange that the much more complicated and difficult results of [2] appeared even though the one-dimensional characterization does not seem to be present anywhere in the literature; we hope that our results fill this gap.…”

Section: Introductionmentioning

confidence: 88%

“…As is often the case for multidimensional SFTs, their characterization is in recursion-theoretic terms and much more complicated than the ones we derive in one dimension. It is noted in [2] that the set of least periods for a (onedimensional) SFT must be semi-linear, and this is true. However, as our results show, not all semi-linear sets are realizable in this way; for instance, the set of positive odd integers is not the set of least periods of any (one-dimensional) SFT.…”

Section: Introductionmentioning

confidence: 99%

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A characterization of the sets of periods within shifts of finite type

Doering

Pavlov

2019

Involve

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In this paper, we characterize precisely the possible sets of periods and least periods for the periodic points of a shift of finite type (SFT). We prove that a set is the set of least periods of some mixing SFT iff it is either {1} or cofinite, and the set of periods of some mixing SFT iff it is cofinite and closed under multiplication by arbitrary natural numbers. We then use these results to derive similar characterizations for the class of irreducible SFTs and the class of all SFTs. Specifically, a set is the set of (least) periods for some irreducible SFT iff it can be written as a natural number times the set of (least) periods for some mixing SFT, and a set is the set of (least) periods for an SFT iff it can be written as the finite union of the sets of (least) periods for some irreducible SFTs. Sharkovsky's Theorem ([6]). For any interval I in R, if f : I → I is continuous and has a point of least period k, then there exist points of all least periods less than k in the Sharkovsky ordering, where the ordering is as follows:

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“…Some of the results of this paper were announced at the DLT conference in the extended abstract [JV10]. A pattern P of support D appears in a pattern P of support D if there exists a position v ∈ D so that v + D ⊆ D and P (v + z) = P (z) for all z ∈ D. We will write P ∈ P to say that P appears in P .…”

Section: Introductionmentioning

confidence: 99%