Tiling Problem and Undecidability in Cellular Automata

Kari, Jarkko

doi:10.1007/978-1-4939-8700-9_552

Cited by 2 publications

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References 23 publications

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“…Kari and Ollinger proved in [38] that the periodicity problem for one-dimensional cellular automata is undecidable. We refer to Kari [34,35,36,37] for more related results for cellular automata.…”

Section: Introductionmentioning

confidence: 99%

An automaton group with undecidable order and Engel problems

Gillibert

2018

Journal of Algebra

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Abstract. For every Turing machine, we construct an automaton group that simulates it. Precisely, starting from an initial configuration of the Turing machine, we explicitly construct an element of the group such that the Turing machine stops if, and only if, this element is of finite order. If the Turing machine is universal, the corresponding automaton group has an undecidable order problem. This solves a problem raised by Grigorchuk.The above group also has an undecidable Engel problem: there is no algorithm that, given g, h in the group, decides whether there exists an integer n such that the n-iterated commutator [. . . [[g, h], h], . . . , h] is the identity or not. This solves a problem raised by Bartholdi.

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Section: Introductionmentioning

confidence: 99%

An automaton group with undecidable order and Engel problems

Gillibert

2018

Journal of Algebra

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“…In some of these settings, generalizations of Williams' decomposition theorem have been established [16,3], showing that conjugacies between shifts of finite type can be expressed as a sequence of splittings followed by amalgamations, suitably defined. Conjugacy of higher-dimensional SFTs has been shown to be undecidable (in fact, Σ 0 1 -complete) by Berger [5] who phrased the problem in terms of tilings (see Kari [19] for a general discussion of such problems), and recently Jeandel and Vanier [15] have shown that the simpler problem of determining conjugacy to any given fixed SFT is still undecidable (and Σ 0 1 -complete). The same authors also give the recursion-theoretic complexity of related problems, such as deciding factorization and embedding.…”

Section: Introductionmentioning

confidence: 99%

Optimal state amalgamation is NP-hard

Frongillo¹

2017

Ergod. Th. Dynam. Sys.

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A state amalgamation of a directed graph is a node contraction which is only permitted under certain configurations of incident edges. In symbolic dynamics, state amalgamation and its inverse operation, state splitting, play a fundamental role in the theory of subshifts of finite type (SFT): any conjugacy between SFTs, given as vertex shifts, can be expressed as a sequence of symbol splittings followed by a sequence of symbol amalgamations. It is not known whether determining conjugacy between SFTs is decidable. We focus on conjugacy via amalgamations alone and consider the simpler problem of deciding whether one can perform $k$ consecutive amalgamations from a given graph. This problem also arises when using symbolic dynamics to study continuous maps, where one seeks to coarsen a Markov partition in order to simplify it. We show that this state amalgamation problem is NP-complete by reduction from the hitting set problem, thus giving further evidence that classifying SFTs up to conjugacy may be undecidable.

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Tiling Problem and Undecidability in Cellular Automata

Cited by 2 publications

References 23 publications

An automaton group with undecidable order and Engel problems

An automaton group with undecidable order and Engel problems

Optimal state amalgamation is NP-hard

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