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Effect of quantified irreducibility on the computability of subshift entropy

The second author was supported by Basal PFB-03 CMM, Universidad de Chile, and did this work in part at in part in the Departamento de Matématicas, Universidad Andrés Bello, Republica 220, Santiago, Chile and Centro de Modelamiento Matematico, Beauchef 851, Santiago, Chile

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  • We study the algorithmic computability of topological entropy of subshifts subjected to a quantified version of a strong condition of mixing, called irreducibility. For subshifts of finite type, it is known that this problem goes from uncomputable to computable as the rate of irreducibility decreases. Furthermore, the set of possible values for the entropy goes from all right-recursively computable numbers to some subset of the computable numbers. However, the exact nature of the transition is not understood.

    In this text, we characterize a computability threshold for subshifts with decidable language (in any dimension), expressed as a summability condition on the rate function. This class includes subshifts of finite type under the threshold, and offers more flexibility for the constructions involved in the proof of uncomputability above the threshold. These constructions involve bounded density subshifts that control the density of particular symbols in all subwords.

    Mathematics Subject Classification: Primary: 37B50, 37B40; Secondary: 37B10, 68Q17.

    Citation:

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  • Figure 1.  Every pattern $v$ on $D_m$ appearing in some locally admissible pattern of $ \mathcal A ^{C_{N_0}}$ appears jointly with $u$ in some other locally admissible pattern

    Figure 2.  Illustration of Definition 3.12

    Figure 3.  Illustration of the definition of the function $ {\delta}_N $

    Figure 4.  Illustration of the definition of the algorithm. The sequence is already defined up to $ F^{n+1}(1) $. The number $ m^{*}_n $ is the smallest one such that the mixing condition is verified

    Table 1.  (First line) Computational difficulty of computing the entropy; (Second line) Set of possible entropies. "Weak" and "Strong" mixing stand for irreducibility rates above or below the threshold, respectively; "Very strong" stands for constant irreducibility rates, or similar properties. "$ \Pi_1 $-comp." means that the problem is $ \Pi_1 $-computable, but not computable; "$ \Pi_1 $ reals" stands for the set of $ \Pi_1 $-computable reals; $ \dagger $ symbols indicate the contribution of the present article

    Subshift class Mixing properties
    None Weak Strong Very strong
    SFT $\Pi_1$-comp. [8] ? computable $\dagger$ computable [8]
    $d\geq 2$ all $\Pi_1$ reals [8] ? ? partial char. [19]
    Decidable $\Pi_1$-comp. [20] $\Pi_1$-comp. $\dagger$ computable $\dagger$ computable [21]
    $d\geq 1$ all $\Pi_1$ reals [6] all $\Pi_1$ reals $\dagger$ ? ?
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  • [1] M. D'amicoG. Manzini and L. Margara, On computing the entropy of cellular automata, Theoretical Computer Science, 290 (2003), 1629-1646.  doi: 10.1016/S0304-3975(02)00071-3.
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