Turing degrees of multidimensional SFTs

Jeandel, Emmanuel; Vanier, Pascal

doi:10.1016/j.tcs.2012.08.027

Cited by 17 publications

(21 citation statements)

References 17 publications

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“…Nevertheless, we may obtain: Theorem 6 [JV13]. In the proof of the embedding theorem, the subshift S is uncountable, and this cannot be corrected.…”

Section: Peculiarities Of Subshifts Of Finite Typementioning

confidence: 99%

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Pursuit of the Universal

Beckmann¹,

Bienvenu²,

Jonoska³

2016

Lecture Notes in Computer Science

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Abstract.A general correction grammar for a language L is a program g that, for each (x, t) ∈ N 2 , issues a yes or no (where when t = 0, the answer is always no) which is g's t-th approximation in answering "x∈L?"; moreover, g's sequence of approximations for a given x is required to converge after finitely many mind-changes. The set of correction grammars can be transfinitely stratified based on O, Kleene's system of notation for constructive ordinals. For u ∈ O, a u-correction grammar's mind changes have to fit a count-down process from ordinal notation u; these u-correction grammars capture precisely the Σ −1 u sets in Ershov's hierarchy of sets for Δ 0 2 . Herein we study the relative succinctness between these classes of correction grammars. Example: Given u and v, transfinite elements of O with u H(iv). We also exhibit relative succinctness progressions in these systems of grammars and study the "information-theoretic" underpinnings of relative succinctness. Along the way, we verify and improve slightly a 1972 conjecture of Meyer and Bagchi.

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“…Nevertheless, we may obtain: Theorem 6 [JV13]. In the proof of the embedding theorem, the subshift S is uncountable, and this cannot be corrected.…”

Section: Peculiarities Of Subshifts Of Finite Typementioning

confidence: 99%

“…The situation when S has a computable point is completely understood and subsumed in the following theorem: Theorem 3 [JV13]. For any set S with a computable point, there is a subshift T s.t.…”

Section: Definition 3 Let S S Two Subsets Ofmentioning

confidence: 99%

Pursuit of the Universal

Beckmann¹,

Bienvenu²,

Jonoska³

2016

Lecture Notes in Computer Science

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“…In particular, the theorem cannot be used to prove results on countable subshifts of finite type. Nevertheless, we may obtain: Theorem 6 ( [JV13]). Let S be a countable Π 0 1 set.…”

Section: Peculiarities Of Subshifts Of Finite Typementioning

confidence: 99%

Computability in Symbolic Dynamics

Jeandel

2016

Pursuit of the Universal

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“…The second component in the computable homeomorphism cannot easily be taken out: it is pointed in [21] that all aperiodic subshifts admit a cone of Turing degrees (that is one degree and all degrees above it).…”

Section: Synchronizing Computationmentioning

confidence: 99%

Hierarchy and Expansiveness in 2D Subshifts of Finite Type

Zinoviadis

2015

Lecture Notes in Computer Science

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Subshifts are sets of configurations over an infinite grid defined by a set of forbidden patterns. In this thesis, we study two-dimensional subshifts of finite type (2D SFTs), where the underlying grid is Z 2 and the set of forbidden patterns is finite. We are mainly interested in the interplay between the computational power of 2D SFTs and their geometry, examined through the concept of expansive subdynamics. 2D SFTs with expansive directions form an interesting and natural class of subshifts that lie between dimensions 1 and 2. An SFT that has only one non-expansive direction is called extremely expansive. We prove that in many aspects, extremely expansive 2D SFTs display the totality of behaviours of general 2D SFTs.For example, we construct an aperiodic extremely expansive 2D SFT and we prove that the emptiness problem is undecidable even when restricted to the class of extremely expansive 2D SFTs. We also prove that every Medvedev class contains an extremely expansive 2D SFT and we provide a characterization of the sets of directions that can be the set of non-expansive directions of a 2D SFT. Finally, we prove that for every computable sequence of 2D SFTs with an expansive direction, there exists a universal object that simulates all of the elements of the sequence. We use the so called hierarchical, self-simulating or fixed-point method for constructing 2D SFTs which has been previously used by Gács, Durand, Romashchenko and Shen.

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Turing degrees of multidimensional SFTs

Cited by 17 publications

References 17 publications

Pursuit of the Universal

Pursuit of the Universal

Computability in Symbolic Dynamics

Hierarchy and Expansiveness in 2D Subshifts of Finite Type

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