Subshift class | Mixing properties | |||
None | Weak | Strong | Very strong | |
SFT | ? | computable |
computable [8] | |
all |
? | ? | partial char. [19] | |
Decidable | computable |
computable [21] | ||
all |
all |
? | ? |
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.
Citation: |
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. "
Subshift class | Mixing properties | |||
None | Weak | Strong | Very strong | |
SFT | ? | computable |
computable [8] | |
all |
? | ? | partial char. [19] | |
Decidable | computable |
computable [21] | ||
all |
all |
? | ? |
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Every pattern
Illustration of Definition 3.12
Illustration of the definition of the function
Illustration of the definition of the algorithm. The sequence is already defined up to