Simulation of Effective Subshifts by Two-dimensional Subshifts of Finite Type

Aubrun, Nathalie; Sablik, Mathieu

doi:10.1007/s10440-013-9808-5

Cited by 79 publications

(95 citation statements)

References 15 publications

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“…• For all r > max(r F , r X ), there exist two r-blocks M 1 And a corollary of the proof of Theorem 2.11, we obtain the 0 3 -hardness for effective subshifts.…”

Section: Effective and Sofic Shiftsmentioning

confidence: 68%

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Hardness of conjugacy, embedding and factorization of multidimensional subshifts

Jeandel

Vanier

2015

Journal of Computer and System Sciences

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Subshifts of finite type are sets of colorings of the plane defined by local constraints. They can be seen as a discretization of continuous dynamical systems. We investigate here the hardness of deciding factorization, conjugacy and embedding of subshifts in dimensions d > 1 for subshifts of finite type and sofic shifts and in dimensions d ≥ 1 for effective shifts. In particular, we prove that the conjugacy, factorization and embedding problems are 0 3 -complete for sofic and effective subshifts and that they are 0 1 -complete for SFTs, except for factorization which is also 0 3 -complete.A d-dimensional subshift is the set of colorings of Z d by a finite set of colors in which a family of forbidden patterns never appear. These are shift-invariant spaces, hence the name. If the family of forbidden patterns is finite, then it is a subshift of finite type (SFT). If the family of forbidden patterns is recursively enumerable, then the subshift is called effective. Another class of subshifts can be defined by the help of local maps, namely the class of sofic shifts: they are the letter by letter projections of SFTs.One can also see SFTs as tilings of Z d , and in dimension 2 they are equivalent to the usual notion of tilings introduced by Wang [17]. Subshifts are a way to discretize continuous dynamical systems: if X is a compact space and φ : X → X a continuous map, we can partition X in a finite number of parts A = {1, . . . , n} and transform the orbit of a pointConjugacy is the right notion of isomorphism between subshifts, and plays a major role in their study: when two subshifts are conjugate they code each other and hence have the same dynamical properties. Conjugacy is an equivalence relation and allows to separate SFTs into equivalence classes. Deciding whether two SFTs are conjugate is called the classification problem. It is a long standing open problem in dimension one [5], although has been proved decidable in the particular case of one-sided SFTs on N, see [18]. It has been known for a long time that in higher dimensions the problem is undecidable when given two SFTs, since it can be reduced to the emptiness problem which is 0 1 -complete [2]. However, ✩ This work was sponsored by grants ANR-09-BLAN-0164, EQINOCS ANR 11 BS02 004 03, TARMAC ANR 12 BS02 007 01.we prove here a slightly stronger result: even by fixing the class in advance, it is still undecidable to decide whether some given SFT belongs to it:Theorem 0.1. For any fixed SFT X, given some SFT Y as an input, it is 0 1 -complete to decide whether X and Y are conjugate (resp. equal).As for the classes of sofic and effective shifts, the complexity is higher:Theorem 0.2. Given two sofic/effective shifts X, Y , it is 0 2 -complete to decide whether X and Y are equal.Theorem 0.3. Given two sofic/effective subshifts X, Y , it is 0 3 -complete to decide whether X and Y are conjugate.An interesting open question for higher dimension that would probably help solve the one dimensional problem would be is conjugacy of subshifts decidable when provided an ...

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“…• For all r > max(r F , r X ), there exist two r-blocks M 1 And a corollary of the proof of Theorem 2.11, we obtain the 0 3 -hardness for effective subshifts.…”

Section: Effective and Sofic Shiftsmentioning

confidence: 68%

“…We reduce the problem from 0 , the halting problem. Given a Turing machine M we construct a SFT Y M such that Y M is conjugate to X iff M halts.Let R M be Robinson's SFT[15] encoding computations of M: R M is empty iff M halts 1. …”

mentioning

confidence: 99%

Hardness of conjugacy, embedding and factorization of multidimensional subshifts

Jeandel

Vanier

2015

Journal of Computer and System Sciences

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“…The main problem is to prove a reverse statement: given a measure satisfying the computational obstructions, we want to construct a cellular automaton which, starting from any simple initial measure, reaches this measure asymptotically. Similar computational obstructions appear in topological dynamics, when characterizing possible properties of subshifts of finite type or cellular automata: possible entropies [29], possible growth-type invariants [38], possible sub-actions [1,28]... However, the construction is quite different here, since starting from a random configuration, the construction requires an ability to self-organize the space.…”

Section: Limit Measures Of Cellular Automata: a Computational Approachmentioning

confidence: 88%

“…Afterward, arcs in T ransit are updated in accordance with t 4,2 , that corresponds to the set T ransit 1 . Finally, T 4,2 pDq is the diagram that contains arcs in StayYFullYT ransit 1 .…”

Section: Transition In a Diagrammentioning

confidence: 99%

Probability and algorithmics: a focus on some recent developments

Cénac¹,

Marcovici²,

Rovetta³

et al. 2017

ESAIM: Procs

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Abstract.This article presents different recent theoretical results illustrating the interactions between probability and algorithmics. These contributions deal with various topics: cellular automata and calculability, variable length Markov chains and persistent random walks, perfect sampling via coupling from the past. All of them involve discrete dynamics on complex random structures.Résumé. Cet article présente différents résultats récents de nature théorique illustrant les interactions entre probabilités et algorithmique. Ces contributions traitent de sujets variés : automates cellulaires et calculabilité, chaînes de Markov à mémoire variable et marches aléatoires persistantes, échantillonage parfait par la méthode de couplage par le passé. Leur point commun est de faire intervenir des dynamiques discrètes sur des structures aléatoires complexes.

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“…Let L ⊆ N * d be a co-recursively enumerable set, there exists a ddimensional sofic subshift X such that P r X = L. In order to prove the lemma, we will need the following result: Theorem 7.1 (Aubrun and Sablik [AS10]; Durand, Romashchenko, and Shen [DRS10]). Let X be a d dimensional subshift, and X be the d+1 dimension subshift obtained by adding a dimension to X and keeping symbols identical on it.…”

Section: Periodicity In Sofic and Effective Subshiftsmentioning

confidence: 99%