Medvedev degrees of 2-dimensional subshifts of finite type

Stephen G. Simpson^{∗} Department of Mathematics Pennsylvania State University http://www.math.psu.edu/simpson simpson@math.psu.edu

Submitted: May 1, 2007 This draft: April 30, 2014

This paper has been published in Ergodic Theory and Dynamical Systems, 34, 2014, pp. 665–674, http://dx.doi.org/10.1017/etds.2012.152.

Abstract

In this paper we apply some fundamental concepts and results from recursion theory in order to obtain an apparently new example in symbolic dynamics. Two sets X and Y are said to be Medvedev equivalent if there exist partial computable functionals from X into Y and vice versa. The Medvedev degree of X is the equivalence class of X under Medvedev equivalence. There is an extensive recursion-theoretic literature on the lattices E_{s} and E_{w} of Medvedev degrees and Muchnik degrees of nonempty effectively closed subsets of {0, 1}^{ℕ}. We now prove that E_{s} and E_{w} consist precisely of the Medvedev degrees and Muchnik degrees of 2-dimensional subshifts of finite type. We apply this result to obtain an infinite collection of 2-dimensional subshifts of finite type which are, in a certain sense, mutually incompatible.

1 Background

Definition 1.1. Let A be a finite set of symbols. The full 2-dimensional shift on A is the dynamical system consisting of the natural action of ℤ^{2} on the compact set A^{ℤ^{2}}. A 2-dimensional subshift is a nonempty closed set X ⊆ A^{ℤ^{2}} which is invariant under the action of ℤ^{2}. A 2-dimensional subshift X is said to be of finite type if it is defined by a finite set of forbidden configurations. An interesting paper on 2-dimensional subshifts of finite type is Mozes [26]. A standard reference for the 1-dimensional case is the book of Lind/Marcus [22], which also includes an appendix [22, §13.10] on the 2-dimensional case.

∗The author’s research was partially supported by the United States National Science Foundation, grant DMS-0600823, and by the Cada and Susan Grove Mathematics Enhancement Endowment at the Pennsylvania State University. In addition, the author thanks Ali Dashti for telling him of the results in Cenzer/Dashti/King [9], and Benjamin Weiss for calling his attention to the papers of Hochman/Meyerovitch [19] and Mozes [26], and the referee for helpful remarks.

Definition 1.2. If X is a 2-dimensional subshift, the shift operators S_{1}, S_{2} : X → X are defined by S_{1}(x)(m, n) = x(m+1, n) and S_{2}(x)(m, n) = x(m, n+1) for all x ∈ X and (m, n) ∈ ℤ^{2}. If X and Y are 2-dimensional subshifts, a shift morphism^{1} of X into Y is a continuous function f : X → Y which commutes with the shift operators, i.e., f(S_{1}(x)) = S_{1}(f(x)) and f(S_{2}(x)) = S_{2}(f(x)) for all x ∈ X. It follows that f commutes with the action of ℤ^{2} on X and Y. A shift isomorphism of X onto Y is a shift morphism of X one-to-one onto Y, i.e., a homeomorphism of X onto Y which commutes with S_{1} and S_{2}. We say that X and Y are shift isomorphic if there exists a shift isomorphism of X onto Y.

Remark 1.3. In the study of 2-dimensional subshifts of finite type, it has been useful to note that they are essentially the same thing as tiling problems in the sense of Wang [49].

On the one hand, if F is a finite set of Wang tiles, let T_{F} be the set of tilings of the plane by F. Identifying T_{F} as a subset of F^{ℤ^{2}}, it is clear that T_{F} is a 2dimensional subshift of finite type. Conversely, given a 2-dimensional subshift of finite type, X, it is easy to construct a finite set of Wang tiles, F, such that T_{F} is computably isomorphic to X. Namely, let r be a positive integer which is greater than or equal to the diameters of all of the forbidden configurations defining X, and let F ⊆ A^{{1,...,r}^{2}} be the set of r × r configurations which do not contain any of the forbidden configurations. Our isomorphism of X onto T_{F} associates to each point x ∈ X a tiling t ∈ T_{F} defined by t(m, n)(i, j) = x(m + i, n + j) for all (m, n) ∈ ℤ^{2} and (i, j) ∈ {1, …, r}^{2}. The adjacency rules for τ_{1}, τ_{2} ∈ F are: (a) τ_{1} is allowed to occur immediately to the left of τ_{2} if and only if τ_{1}(i + 1, j) = τ_{2}(i, j) for all 1 ≤ i ≤ r − 1 and 1 ≤ j ≤ r, and (b) τ_{1} is allowed to occur immediately below τ_{2} if and only if τ_{1}(i, j + 1) = τ_{2}(i, j) for all 1 ≤ i ≤ r and 1 ≤ j ≤ r − 1.

Mozes [26, 27] and recently Hochman/Meyerovitch [19] have used the tiling methods of Wang [49] and Robinson [31] to construct 2-dimensional subshifts of finite type with interesting dynamical properties.

Remark 1.4. In this paper we shall examine 2-dimensional subshifts of finite type from the viewpoint of recursion theory, also known as computability theory, the branch of mathematics which grew out of Turing’s celebrated theory of computability and unsolvability. Some classical references for recursion/computability theory are [14, 32, 47]. Two recent references are [15, 30]. The purpose of the next few definitions and remarks is to briefly explain the recursion-theoretic concepts which we shall need in §§2 and 3 below.

1Note that a shift morphism f : X → Y need not be onto Y. In this respect our terminology may differ from that of other authors.

Definition 1.5 (Turing oracles).

1. Let ℕ = {1, 2, …} be the set of positive integers. Let A = {a_{1}, …, a_{k}} be a finite set of symbols. Let A^{ℕ} = {x | x : ℕ → A} be the space of all functions from ℕ into A. We endow A with the discrete topology and A^{ℕ} with the product topology.

2. Let M be a deterministic Turing machine program with tape symbols a_{1}, …, a_{k}, 0, 1, b. Here b is a blank symbol, and we assume that A ∩ {0, 1, b} = ∅. Given x ∈ A^{ℕ} and n ∈ ℕ, we denote by M(x, n) the output of the unique run of M with oracle x and input n provided this run eventually halts. More precisely, M(x, n) = a means that the program M, starting in its initial state with x inscribed using elements of A at tape positions −1, −2, −3, … and n inscribed in binary notation using 0’s and 1’s at tape positions 0, 1, 2, …, j and blanks at tape positions j + 1, j + 2, … where 2^{j} ≤ n < 2^{j+1}, runs for a finite number of deterministic computational steps and then reaches a halting state with the symbol a at tape position 0. For a fuller explanation of Turing’s oracle concept, see Davis [14] and Rogers [32, Chapter 9].

Definition 1.6 (Medvedev and Muchnik degrees).

1. Let Φ : ⊆A^{ℕ} → A^{ℕ} be a partial functional from A^{ℕ} into A^{ℕ}, i.e., a function whose domain and range are subsets of A^{ℕ}. The domain of Φ is denoted dom(Φ). We say that Φ is partial recursive (see Rogers [32, §15.3]) or partial computable if there exists a deterministic Turing machine program M which computes Φ in the following sense: for all x ∈ A^{ℕ} we have x ∈ dom(Φ) if and only if for all n ∈ ℕ there exists a ∈ A such that M(x, n) = a, in which case Φ(x)(n) = a. Note that each partial computable functional Φ : ⊆A^{ℕ} → A^{ℕ} is continuous on its domain.

2. Let X and Y be subsets of A^{ℕ}. We say that X and Y are computably homeomorphic if there exist partial computable functionals Φ, Ψ : ⊆A^{ℕ} → A^{ℕ} such that X ⊆ dom(Φ) and Y ⊆ dom(Ψ) and Φ maps X one-to-one onto Y and Ψ(Φ(x)) = x for all x ∈ X.

3. We say that Y is Medvedev reducible to X (see Rogers [32, §13.7] and Medvedev [23]) if there exists a partial computable functional Φ : ⊆A^{ℕ} → A^{ℕ} such that X ⊆ dom(Φ) and Φ(x) ∈ Y for all x ∈ X. We say that X and Y are Medvedev equivalent if X is Medvedev reducible to Y and Y is Medvedev reducible to Y.

4. We say that Y is Muchnik reducible to X (see Muchnik [28]) if for each x ∈ X there exists a partial computable functional Φ such that x ∈ dom(Φ) and Φ(x) ∈ Y. We say that X and Y are Muchnik equivalent if X is Muchnik reducible to Y and Y is Muchnik reducible to X.

5. Obviously computable homeomorphism implies Medvedev equivalence, and Medvedev equivalence implies Muchnik equivalence. However, the converses do not hold. The equivalence classes under Medvedev equivalence and Muchnik equivalence are known as Medvedev degrees and Muchnik degrees respectively.

Remark 1.7. The idea behind Medvedev and Muchnik degrees is to regard each set X ⊆ A^{ℕ} as the solution set of a so-called “mass problem,” prob(X), the problem of finding at one least element of X, where “to find” means “to compute in the sense of Turing.” Accordingly, one says that the problem prob(X) is “solvable” if it has at least one computable solution, i.e., the set X contains at least one point which is Turing computable. Similarly, one could say that prob(Y) is “reducible” to prob(X) if each solution of prob(X) can be used as a Turing oracle to compute at least one solution of prob(Y), or in other words, Y is Muchnik reducible to X. Thus for instance prob(X) is solvable if and only if X is Muchnik reducible to Y for all Y, if and only if X is Medvedev reducible to Y for all Y. In general, the Medvedev or Muchnik degree of a mass problem is a measure of the “degree of unsolvability” or “degree of difficulty” which is inherent in the problem. In this way one can use Medvedev and Muchnik degrees to classify unsolvable mathematical problems.

Remark 1.8. For |A| ≥ 2 it is straightforward to construct a computable homeomorphism of A^{ℕ} onto {0, 1}^{ℕ}. Therefore, the Medvedev degrees and Muchnik degrees of subsets of A^{ℕ} are the same as those of subsets of {0, 1}^{ℕ}.

Definition 1.9 (effectively closed sets).

1. A subset of A^{ℕ} is said to be effectively closed or Π_{1}^{0} (see Rogers [32, Chapter 15]) if it is the complement of the union of a computable sequence of basic open neighborhoods in A^{ℕ}. Equivalently, X ⊆ A^{ℕ} is effectively closed if there exists a deterministic Turing machine program M such that for all x ∈ A^{ℕ}, x ∈ X if and only if M(x, 1) never halts.

2. In view of Remark 1.8, each effectively closed set X ⊆ A^{ℕ} is computably homeomorphic to an effectively closed set P ⊆ {0, 1}^{ℕ}. It follows that P has the same Medvedev degree as X and the same Muchnik degree as X.

3. Let E_{s} (respectively E_{w}) be the partial ordering of Medvedev degrees (respectively Muchnik degrees) of nonempty effectively closed subsets of {0, 1}^{ℕ}. Here the partial ordering of E_{s} is given by defining the Medvedev degree of P ⊆ {0, 1}^{ℕ} to be ≤ the Medvedev degree of Q ⊆ {0, 1}^{ℕ} if and only if P is Medvedev reducible to Q, and similarly for E_{w}.

4. The partial orderings E_{s} and E_{w} have the same bottom element, namely, the equivalence class consisting of all subsets of {0, 1}^{ℕ} which contain at least one computable point. The common bottom element of E_{s} and E_{w} is denoted 0.

Remark 1.10 (the lattices E_{s} and E_{w}).

1. The partial orderings E_{s} and E_{w} in Definition 1.9 are distributive lattices. Moreover, there is a natural lattice homomorphism of E_{s} onto E_{w} obtained by mapping the Medvedev degree of P to the Muchnik degree of P.

2. The lattices E_{s} and E_{w} are mathematically rich and have been studied extensively. See Alfeld [1], Binns [2, 3, 4, 5], Binns/Simpson [6], Cenzer/Hinman [10], Cole/Simpson [12], Hudelson [20], and Simpson [34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45].

Remark 1.11. For our purposes in this paper, we shall need to apply the above concepts not only to subsets of A^{ℕ} but also to subsets of A^{ℤ^{2}}. In order to do this, we shall use a simple one-to-one correspondence between ℤ^{2} and ℕ. For instance, let θ : ℤ^{2} → ℕ be the one-to-one correspondence given by

θ(m, n) = \frac{(ϕ(m) + ϕ(n) − 1)(ϕ(m) + ϕ(n) − 2)}{2} + ϕ(m)

where

ϕ(m) = \{ \begin{matrix}2m & if m > 0, \\ 1 − 2m & if m ≤ 0. \end{matrix}

We may then identify subsets of A^{ℤ^{2}} with corresponding subsets of A^{ℕ}. In this way Definitions 1.6 and 1.9 and Remarks 1.7, 1.8 and 1.10 for subsets of A^{ℕ} apply equally well to subsets of A^{ℤ^{2}}.

Remark 1.12. Recent research has revealed that E_{w} contains a large number of specific, interesting Muchnik degrees associated with particular mass problems which are of foundational interest. For example, the top degree in E_{w} is the Muchnik degree of the problem of finding a complete, consistent theory which contains the usual axioms for first-order arithmetic. Another interesting degree in E_{w} is the Muchnik degree of the problem of finding an infinite sequence of 0’s and 1’s which is algorithmically random in the sense of Martin-Löf [39, 41]. Other interesting degrees in E_{w} are closely related to other interesting topics in the foundations of mathematics. Among these topics are reverse mathematics [36, 39], almost everywhere domination [42], measure-theoretic regularity [44], the hyperarithmetical hierarchy [12, 44], effective Hausdorff dimension [24, 46] and Kolmogorov complexity [20]. A new survey of this research area is [45].

2 Medvedev degrees of subshifts

Remark 2.1. Let X be a 2-dimensional subshift. In the following theorem we see that the Medvedev degree of X, the Muchnik degree of X, and the computable homeomorphism type of X depend only on the shift isomorphism type of X. This suggests the possibility that Medvedev degrees, Muchnik degrees, and computable homeomorphism types may be useful as invariants for the problem of classifying 2-dimensional subshifts up to shift isomorphism. More evidence for such a possibility is presented in Theorems 2.5, 2.9 and 3.4 below.

Theorem 2.2. All shift morphisms f : X → Y are given by partial computable functionals. If X and Y are shift isomorphic, then X and Y are computably homeomorphic, hence Medvedev equivalent, hence Muchnik equivalent.

Proof. Let A and B be finite sets of symbols such that X and Y are subshifts of the full 2-dimensional shifts A^{ℤ^{2}} and B^{ℤ^{2}} respectively. Let f : X → Y be a shift morphism. By the Curtis/Hedlund/Lyndon Theorem (see Boyle [8] or Lind/Marcus [22]) f is a block code, i.e., we can find an integer r ≥ 0 and a projection π : A^{{−r,...,r}^{2}} → B such that

f(x)(m, n) = π(S_{1}^{m}S_{2}^{n}(x) ↾ {−r, …, r}^{2})

for all x ∈ X and (m, n) ∈ ℤ^{2}. In particular f is given by a partial computable functional. If moreover f is a shift isomorphism of X onto Y, it follows that f is a computable homeomorphism of X onto Y. This completes the proof.

Remark 2.3. Let X be a 2-dimensional subshift. If X contains a periodic point, one can easily show (see for instance the second and third paragraphs of [31, §1]) that X contains a point which is Turing computable, so by part 4 of Definition 1.9 the Medvedev degree of X is 0, the degree of solvable problems. Thus Medvedev degrees and Muchnik degrees are of no interest in the case of subshifts with periodic orbits. However, there are many 2-dimensional subshifts of finite type which do not have periodic points [18, 26, 27, 29, 31], and in this case the Medvedev degrees and Muchnik degrees could well serve as a useful classification tool. See also Remarks 1.12 and 2.10.

Remark 2.4. An open problem is to characterize the computable homeomorphism types of 2-dimensional subshifts of finite type. Clearly any 2-dimensional subshift of finite type is an effectively closed subset of A^{ℤ^{2}} and is therefore computably homeomorphic to, hence of the same Medvedev degree and Muchnik degree as, a nonempty effectively closed subset of {0, 1}^{ℕ}. We shall now prove a theorem in the opposite direction. Namely, every nonempty effectively closed subset of {0, 1}^{ℕ} is of the same Medvedev degree, hence of the same Muchnik degree, as some 2-dimensional subshift of finite type. This characterization of the Medvedev degrees and Muchnik degrees of 2-dimensional subshifts of finite type appears to be new, but see Remark 2.7 below.

Theorem 2.5. Given a nonempty effectively closed set P ⊆ {0, 1}^{ℕ}, we can find a 2-dimensional subshift of finite type which is of the same Medvedev degree as P, hence of the same Muchnik degree as P.

Proof. Our proof uses the tiling constructions of Robinson [31] and Hanf [18] and Myers [29].

Let F be a finite set of Wang tiles, and let τ ∈ F be a distinguished tile in F. We write T_{F}^{τ} = {t ∈ T_{F} | t(0, 0) = τ}. The tilings in T_{F}^{τ} are said to be origin-constrained. Given an effectively closed set P ⊆ {0, 1}^{ℕ}, Hanf [18] shows how to construct an origin-constrained tiling system F, τ and a projection π : F → {0, 1} such that

P = {⟨π(t(n, 0)) | n ∈ ℕ⟩ | t ∈ T_{F}^{τ} }.

Namely, F and τ are such that each origin-constrained tiling t ∈ T_{F}^{τ} describes a non-halting run of a particular deterministic Turing machine M program starting with

⟨π(t(−n, 0)) | n ∈ ℕ⟩ ∈ {0, 1}^{ℕ}

inscribed at positions −1, −2, −3, … of the Turing machine tape. The distinguished tile τ represents the initial internal state of M. We can then construct a fixed partial computable functional Φ : ⊆{0, 1}^{ℕ} → F^{ℤ^{2}} which is independent of t and maps ⟨π(t(−n, 0)) | n ∈ ℕ⟩ to t. Thus T_{F}^{τ} and P are computably homeomorphic.

Now, starting with F and τ as above, Myers [29] shows how to construct a finite set of tiles F′ and a projection π′ : F′ → F with the following properties:

1. F′ is a Robinson set of tiles.

(Unexplained terms are defined in Myers’ paper [29].)

2. For each tiling t′ ∈ T_{F′}, the π′-images of the center rows of the finite boards of t′ are synchronized.

3. For each tiling t′ ∈ T_{F′}, the result of piecing together the π′-images of the center rows of the finite boards of t′ is ⟨t(n, 0) | n ∈ ℤ⟩ for some t ∈ T_{F}^{τ}. Using this piecing-together procedure, we can construct a fixed computable functional which maps t′ to t.

4. Given an origin-constrained tiling t ∈ T_{F}^{τ}, we can find a (nonunique) tiling t′ ∈ T_{F′} such that ⟨t(n, 0) | n ∈ ℤ⟩ is the result of piecing together the π′-images of the center rows of the finite boards of t′. Moreover, we can construct a fixed partial computable functional which maps t to such a t′.

Properties 3 and 4 imply that T_{F′} is Medvedev equivalent to T_{F}^{τ}. Therefore, T_{F′} is Medvedev equivalent to P. Since T_{F′} is a 2-dimensional subshift of finite type, our theorem is proved.

Remark 2.6. Actually Hanf [18] and Myers [29] dealt only with effectively closed sets P ⊆ {0, 1}^{ℕ} of the form

P = S(I, J) = {x ∈ {0, 1}^{ℕ} | x separates I, J}

where I and J are recursively enumerable subsets of ℕ. However, their constructions work just as well for arbitrary effectively closed sets P ⊆ {0, 1}^{ℕ}.

Remark 2.7. Leonid A. Levin has kindly informed us that our Theorem 2.5 was already implicit in his remark [21, last paragraph of §3] concerning tilings of the plane with a 2-adic coordinate system.

Definition 2.8. Let S^{2} be the set of all 2-dimensional subshifts. For X, Y ∈ S^{2} we write X ≥ Y if there exists a shift morphism f : X → Y. Obviously ≥ is transitive and reflexive. We write X ≡ Y if X ≥ Y and Y ≥ X. Obviously ≡ is an equivalence relation on S^{2}. Obviously X ≡ Y whenever X and Y are shift isomorphic, but the converse does not hold. Let S^{2}/≡ be the set of equivalence classes of S^{2} modulo ≡. There is an obvious partial ordering of S^{2}/≡ induced by ≥. It is easy to verify that, under this partial ordering, S^{2}/≡ is a distributive lattice. Let S_{fin}^{2} be the subset of S^{2} consisting of the 2-dimensional subshifts of finite type. It is easy to verify that S_{fin}^{2}/≡ is a sublattice of S^{2}/≡.

Theorem 2.9. There is a natural lattice homomorphism of S_{fin}^{2}/≡ onto E_{s}. Namely, for each X ∈ S_{fin}^{2} we map the ≡-equivalence class of X to the Medvedev degree of X.

Proof. The greatest lower bound and least upper bound operations in each of the lattices S^{2}/≡, S_{fin}^{2}/≡, E_{s}, E_{w} are given by disjoint union and Cartesian product respectively. In particular, our mapping of S_{fin}^{2}/≡ to E_{s} is a lattice homomorphism. By Theorem 2.5 this homomorphism is onto E_{s}.

Remark 2.10. A consequence of Theorem 2.9 is that there is a natural lattice homomorphism of S_{fin}^{2}/≡ onto E_{w}, obtained by mapping the ≡-equivalence class of X ∈ S_{fin}^{2} to the Muchnik degree of X. In particular we have a new invariant, the Muchnik degree, associated to (shift isomorphism types of) 2-dimensional subshifts of finite type. Obviously this invariant is qualitatively different from other such invariants which have been considered previously in the symbolic dynamics literature. Moreover, this new invariant is of clear interest, inasmuch as E_{w} is known to contain many specific, natural Muchnik degrees which are closely related to significant topics in the foundations of mathematics. See also Remark 1.12 above.

3 An application

Remark 3.1. We shall now present an application which is stated purely in terms of 2-dimensional subshifts, with no reference to Medvedev degrees or Muchnik degrees. Namely, we shall construct an infinite collection of 2dimensional subshifts of finite type which are, in a certain sense, mutually incompatible.

Definition 3.2. If X and Y are 2-dimensional subshifts on k and l symbols respectively, let X +Y and X ×Y be the disjoint union and Cartesian product of X and Y. These are 2-dimensional subshifts on k+l and kl symbols respectively. If X = (X, S_{1}, S_{2}) is a 2-dimensional subshift on k symbols, and if a, b, c, d are integers such that ad − bc ≠ 0, let X[a, b, c, d] = (X, S_{1}^{a}S_{2}^{b}, S_{1}^{c}S_{2}^{d}). This is a 2-dimensional subshift on k^{|ad−bc|} symbols.

Definition 3.3. If U is a collection of 2-dimensional subshifts, let cl(U) be the closure of U under the operations of Definition 3.2. In other words, cl(U) is the smallest collection of 2-dimensional subshifts with the following properties:

1. For all X ∈ U, X ∈ cl(U).

2. For all X ∈ cl(U) and Y ∈ cl(U), X + Y ∈ cl(U) and X × Y ∈ cl(U).

3. For all X ∈ cl(U) and all a, b, c, d ∈ ℤ with ad−bc ≠ 0, X[a, b, c, d] ∈ cl(U).

Theorem 3.4. We can find an infinite collection of 2-dimensional subshifts of finite type, W, such that for all partitions of W into two subcollections, U and V, there is no shift morphism of X into Y for any X ∈ cl(U) and Y ∈ cl(V).

Proof. By Binns/Simpson [6] let P_{i}, i = 1, 2, …, be nonempty effectively closed subsets of {0, 1}^{ℕ} whose Muchnik degrees are independent, i.e., P_{i_{i}} +⋯+P_{i_{m}} is not Muchnik reducible to P_{j_{1}} ×⋯×P_{j_{n}} provided {i_{1}, …, i_{m}}∩{j_{1}, …, j_{n}} = ∅. By Theorem 2.5, for each i = 1, 2, … let X_{i} be a 2-dimensional subshift of finite type which is Medvedev equivalent, to P_{i}, hence Muchnik equivalent to P_{i}. Let W be the collection X_{i}, i = 1, 2, …, and let U, V be a partition of W. By induction on X ∈ cl(U) and Y ∈ cl(V) we can easily show that neither of X and Y is Muchnik reducible to the other. Hence by Theorem 2.2 there is no shift morphism of X into Y or vice versa. Q.E.D.

Remark 3.5. In our proof of Theorem 3.4, all of the subshifts in W are of different Muchnik degrees, hence of different Medvedev degrees. The referee has kindly informed us that, using classical methods, one can obtain a similar result in Medvedev degree 0. Namely, for each prime number p choose an almost one-to-one extension of the p-adic odometer.

Remark 3.6. In this paper we have dealt only with 2-dimensional subshifts. What about the 1-dimensional case? Clearly Theorem 2.2 holds in this context. On the other hand, Theorems 2.5 and 2.9 and 3.4 fail, because all 1-dimensional subshifts of finite type contain periodic points and are therefore of Medvedev degree zero. (We thank the referee for pointing out the failure of Theorem 3.4 in this context.) What about effectively closed 1-dimensional subshifts? Cenzer/Dashti/King [9] have constructed a 1-dimensional effectively closed subshift which is of nonzero Medvedev degree. Recently Miller [25] improved this result by showing that Theorems 2.5 and 2.9 hold for 1-dimensional effectively closed subshifts. And from this plus Binns/Simpson [6] it follows that Theorem 3.4 holds in the same context.

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