Computability of Countable Subshifts in One Dimension

Abstract

We investigate the computability of countable subshifts in one dimension, and their members. Subshifts of Cantor-Bendixson rank two contain only eventually periodic elements. Any rank two subshift in 2 Z is decidable. Subshifts of rank three may contain members of arbitrary Turing degree. In contrast, effectively closed ( 0 1 ) subshifts of rank three contain only computable elements, but 0 1 subshifts of rank four may contain members of arbitrary 0 2 degree. There is no subshift of rank ω + 1.

Decidability and universality of quasiminimal subshifts

“…For a subshift X ⊂ S Z , the following are equivalent: We call a point x ∈ S Z computable if there is an algorithm that, given i ∈ Z, computes x i . It is well-known that not all Π 0 1 subshifts contain computable points, see for example [6].…”

“…This implies X (2) = ∅, so the CB-rank of X is 2. Every countable subshift with CB-rank 2 is generated by a single point which is eventually periodic in both directions [6,21], and since X has only one minimal subshift X ′ , the repeating patterns in both tails must be equal. Decidability of such a subshift is easy to show.…”

“…The question for us is then whether a recursive quasiminimal subshift can be universal. One might guess that if X has only finitely many subshifts, then it must be a quite simple extension 6 of the one or more minimal subshifts it necessarily contains, or perhaps even essentially just a union. It turns out that, unlike in Proposition 1, whether this is true depends on whether the acting monoid is N or Z, that is, on whether our subshifts are one-or two-sided.…”

“…The classes of essentially minimal systems and quasiminimal systems are incomparable. 6. In Proposition 9 of [7], it is shown that if the limit set of a symbolic dynamical system is a finite union of minimal systems, then it is decidable.…”

Turing degrees of multidimensional SFTs

“…Π 0 1 classes occur naturally in various areas in computer science and recursive mathematics, see e.g. [Cenzer and Remmel(1998), Simpson(2011a)] and the upcoming book [Cenzer and Remmel(2011)]. It is easy to see that any SFT is a Π 0 1 class (up to a computable coding of Σ Z 2 into {0, 1} N ).…”