Computability of Countable Subshifts - Semantic Scholar
Computability of Countable Subshifts? Douglas Cenzer, Ali Dashti, Ferit Toska and Sebastian Wyman Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, Florida 32611 Phone: 352-392-0281 Fax: 352-392-8357 [email protected] l.edu
Abstract. The computability of countable subshifts and their members is examined. Results include the following. Subshifts of CantorBendixson rank one contain only eventually periodic elements. Any rank one subshift, in which every limit point is periodic, is decidable. Subshifts of rank two may contain members of arbitrary Turing degree. In contrast, effectively closed (Π10 ) subshifts of rank two contain only computable elements, but Π10 subshifts of rank three may contain members of arbitrary c. e. degree. There is no subshift of rank ω. Keywords: Computability, Symbolic Dynamics, Π10 Classes
1
Introduction
There is a long history of interaction between computability and dynamical systems. A Turing machine may be viewed as a dynamical system which produces a sequence of configurations or words before possibly halting. The reverse notion of using an arbitrary dynamical system for general computation has generated much interesting work. See for example [1, 7]. In this paper we will consider computable aspects of certain dynamical systems over the Cantor space 2IN . The study of computable dynamical systems is part of the Nerode program to study the effective content of theorems and constructions in analysis. Weihrauch [16] has provided a comprehensive foundation for computability theory on various spaces, including the space of compact sets and the space of continuous real functions. Computable analysis is related as well to the so-called reverse mathematics of Friedman and Simpson [13], where one studies the proof-theoretic content of various mathematical results. The study of reverse mathematics leads in turn to the concept of degrees of difficulty [9, 15] Here we say that P ≤M Q if there is a Turing computable functional F which maps Q into P ; thus the problem of finding an element of P can be uniformly reduced to that of finding an element of Q, so that P is “less difficult” than Q. The degrees of difficulty of effectively closed sets (also known as Π10 classes) have been intensively investigated in several recent papers [5, 12]. ?
This research was partially supported by NSF grants DMS 0532644 and 0554841 and 652372.
2
The computability of Julia sets in the reals has been studied by Cenzer [3] and Ko [8]; the computability of complex dynamical systems has been investigated by Rettinger and Weihrauch [11] and by Braverman and Yampolski [2]. The connection between dynamical systems and subshifts is the following. Certain dynamical systems may be given by a continuous function F on a symbolic space X (one with a basis of clopen sets). For each X ∈ X , the sequence (X, F (X), F (F (X)), . . . ) is the trajectory of X. Given a fixed partition U0 , . . . , Uk−1 of X into clopen sets, the itinerary It(X) of a point X is the sequence (a0 , a1 , . . . ) ∈ k IN where an = i iff F n (X) ∈ Ui . Let It[F ] = {It(X) : X ∈ X }. Note that It[F ] will be a closed set. We observe that, for each point X with itinerary (a0 , a1 , . . . ), the point F (X) has itinerary (a1 , a2 , . . . ). Now the shift operator σ on k IN is defined by σ(a0 , a1 , . . . ) = (a1 , a2 , . . . ). It follows that It[F ] is closed under the shift operator, that is, It[F ] is a subshift. Computable subshifts and the connection with effective symbolic dynamics were investigated by Cenzer, Dashti and King [4] in a recent paper. A total, Turing computable functional F : 2IN → 2IN is always continuous and thus will be termed computably continuous or just computable. Effectively closed sets (also known as Π10 classes) are a central topic in computability theory; see [6] and Section 2 below. It was shown for any computably continuous function F : 2IN → 2IN , It[F ] is a decidable Π10 class and, conversely, any decidable Π10 subshift P is It[F ] for some computable map F . In this paper, Π10 subshifts are constructed in 2IN and in 2Z which have no computable elements and are not decidable. Thus there is a Π10 subshift with non-trivial Medvedev degree. J. Miller [10] showed that every Π10 Medvedev degree contains a Π10 subshift. Simpson [14] studied Π10 subshifts in two dimensions and obtained a similar result there. Now every nonempty countable Π10 class contains a computable element, so that all countable Π10 classes have Medvedev degree 0, and many uncountable classes also have degree 0. In the present paper, we will consider more closely the structure of countable subshifts, using the Cantor-Bendixson (CB) derivative. We will compare and contrast countable subshifts of finite CB rank with Π10 subshifts of finite CB rank as well as with arbitrary Π10 classes of finite rank. The outline of this paper is as follows. Section 2 contains definitions and preliminaries. Section 3 focuses on subshifts of rank one and has some general results about periodic and eventually periodic members of subshifts. We show that every member of a subshift of rank one is eventually periodic (and therefore computable) but there are rank one subshifts of arbitrary Turing degree and rank one Π10 subshifts of arbitrary c. e. degree, so that rank one undecidable Π10 subshifts exist. We give conditions under which a rank one subshift must be decidable. We show that there is no subshift of rank ω. In section 4, we study subshifts of rank two and three. We show that subshifts of rank two may contain members of arbitrary Turing degree. In contrast, we show that Π10 subshifts of rank two contain only computable elements, but Π10 subshifts of rank three may contain members of arbitrary c. e. degree.
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2
Preliminaries
We begin with some basic definitions. Let IN = {0, 1, 2, ...} denote the set of natural numbers. For any set Σ and any i ∈ IN, Σ i denotes the strings of length i from Σ, Σ ∗ denotes the set of all finite strings from Σ, andSΣ IN denotes the set of countably infinite sequences from Σ. We write Σ 1. u t We note here that there are Π10 classes of rank one with noncomputable elements [6]. Hence we have the following. Corollary 1. There is a Π10 class Q ⊆ 2IN of rank one such that, for any Π10 subshift P of rank one, there is no degree-preserving homeomorphism between P and Q. Next we consider the decidability of rank one subshifts. Theorem 2. (a) For any Turing degree d, there is a subshift Q of rank one such that TQ has degree d (b) For any c. e. degree d, there is a Π10 subshift Q of rank one such that TQ has degree d. Proof. Let A be any set of natural numbers of degree d and let Q contain limit points 0ω and 1_ 0ω , along with isolated points 0n 10ω , for n > 0 and 1_ 0n 10ω for n ∈ A. Then Q is a rank one subshift and TQ ≡T A. For (b), just take a c. e. set B of degree d and let A = IN − B. u t We will next consider a special case in which rank one subshifts are decidable. The following lemma is needed. Lemma 5. Let Q be a subshift, let X = v ω be a periodic element of Q with period k and, for each i < k and each n, let Qi,n = {Z : v n_ (v i)_ (1−v(i))_ Z ∈ Q}. Then Qi,n+1 ⊆ Qi,n for each n. Proof. If Z ∈ Qi,n+1 , then v n+1 (v i)(1 − v(i))Z ∈ Q, so that, since Q is a subshift, v n (v i)(1 − v(i))Z ∈ Q and therefore Z ∈ Qi,n . u t Theorem 3. Let Q be a subshift of rank one such that every element of D(Q) is periodic. Then Q is decidable and every element of Q is computable. Proof. Every element of D(Q) is eventually periodic, hence computable by Theorem 1. Let X = v ω be a periodic element of D(Q) with period k and let v = X k. Let Qi,n be defined as in Lemma 5. Since X has rank one, there exists, for each i < k, some n such that Qi,n is finite for all m ≥ n; let ni be the least such n. Since Qi,n+1 ⊆ Qi,n for all i and n, it follows that the sequence {Qi,n : n ∈ IN} converges to some fixed finite subset Pi of 2IN . Let Di be the decidable set TPi . Now let S(v) = {(i, n) : i < k & Qi,n is finite} and let A(v) be the set of strings of the form v n (v i)w for some (i, n) ∈ S(v). Then S(v) is computable since it is a
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cofinite set. It follows that A(v) is computable, since it the union of finitely many computable sets together with {v n (v i)(1 − v(i))w : (i, n) ∈ S(v) & w ∈ Pi }. For each of the finitely many limit paths v ω ∈ Q, we may similarly define the set A(v) of strings in TQ which branch off from v ω where the appropriately defined set Qi,n is finite. We claim that TQ is the union of the finitely many computable sets A(v) together with the words of the form v n (v i) for some i and n and is therefore decidable. Certainly each such string is in TQ . Now suppose that u is some string in TQ which is not an initial segment of any of the limit paths. Choose v so that u has the longest agreement with v ω of the limit paths and choose i and n so that v n (v i)(1 − v(i)) @ u. Then v n (v i)(1 − v(i)) disagrees with every limit path so that Qi,n is finite and hence u ∈ A(v). u t Corollary 2. For any subshift Q of rank one, there is some finite n such that σ n (Q) is decidable. Proof. By Theorem 1, D(Q) is a finite set of eventually periodic points. For each X ∈ D(Q), σ m (X) is periodic for some m; just let n be the maximumum over X ∈ D(Q). Then by Lemma 3, D(σ n (Q)) = σ n (D(Q)) and thus contains only periodic points, so that Theorem 3 applies. u t There is another interesting consequence of Lemma 5 Theorem 4. There is no subshift of rank ω. Proof. Suppose by way of contradiction that Q has rank ω. Then Dω+1 (Q) = ∅ and Dω (Q) is finite. Then there is a periodic element X of rank ω by Proposition 2. Let X have period k and let the sets Qi,n be defined as in Lemma 4. Since X has rank ω, there is some n such that for all i and all m ≥ n, Qi,m has rank < ω. Suppose that Qi,n has rank ri < ω and let r = max{ri : i > k}. Then by Lemma 5, rk(Qi,m ) ≤ r for all m > n. But this implies that rk(X) ≤ r + 1, which is the desired contradiction. u t It follows from the proof that there is no subshift of rank λ, for any limit ordinal λ. In the next section, we will show that a rank two subshift can have members which are not eventually periodic and indeed not even computable.
4
Subshifts of Rank Two and Three
Proposition 4. For any increasing sequence n0 < n1 < . . . , there is a subshift Q of rank two which contains the element 0n0 10n1 1 . . . Proof. The subshift Q will have the following elements: (0) For each k > 0 and each n ≤ nk , the isolated element 0n 10nk+1 10nk+2 1 . . . (1) For every n, the element 0n 10ω which will have rank one in Q. (2) The unique element 0ω of rank 2 in Q. u t
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Theorem 5. For any Turing degree d, there is a rank two subshift Q which contains a member of Turing degree d and such that TQ has Turing degree d. Proof. Let A = {a0 , a1 , . . . } be any infinite set of degree d and let ni = a0 + a1 + · · · + ai for each i. Now apply Proposition 4. u t This result can be improved as follows. For Q ⊆ 2IN , let D(Q) be the set of Turing degrees of members of Q. Theorem 6. For any countable set D of Turing degrees containing 0, there is a rank two subshift Q such that D(Q) = D. For effectively closed subshifts, the result is quite different. Theorem 7. If Q is a Π10 subshift of rank two, then all of its members are computable. Proof. D(Q) is a subshift of rank one and hence all of its members are eventually periodic and therefore computable. The remaining members of Q are isolated and therefore computable by Theorem 3.12 of [6]. u t Finally, we consider Π10 subshifts of rank three with noncomputable elements. Theorem 8. For any c. e. set A, there exists B ≡T A and a Π10 subshift Q of rank three with B ∈ Q. S Proof. Let A = s As where A0 is empty and for each s, As+1 − As contains at most one number. Let B ≡T A have the form 0n0 10n1 1 · · · where n0 < n1 < · · · represent a modulus of convergence for A. That is, 1. n0 = 0 if 0 ∈ / A and otherwise n0 is the least s such that 0 ∈ As . 2. For each k, nk+1 = nk + 1 if k + 1 ∈ / A and otherwise nk+1 is the least s > nk such that k ∈ As . For any stage s and k < s, we have other sequences m0 , . . . , mk which are candidates to be the actual modulus sequence. That is, m0 , . . . , mk is a candidate for the modulus at stage s if the following holds. 1. m0 is the least t ≤ s such that 0 ∈ At if such s exists and otherwise m0 = 0. 2. For each k, mk+1 is the least t ≤ s such that k ∈ As and t > mk , if such t exists. Otherwise mk+1 = mk + 1. Note that the set of candidate sequences at stage s is uniformly computable. Since the sequence As converges to A, it follows that if (m0 , . . . , mk ) is a candidate sequence for all large enough stages s, then mi = ni for all i ≤ k. Define the tree T to contain all finite strings of length s of the following form, 0a0 _ 1_ 0a1 _ · · · _ 0ak−1 _ 1_ 0ak ,
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such that there exists j < s and m0 < m1 < · · · < mk−1 , a candidate sequence at stage s, such that a0 ≤ mj and for all i < k − 1, aj+i+1 = mj+1 . Here we only require a0 ≤ mj since we are trying to construct a subshift. Observe that there is no restriction on ak . This is in case j + k ∈ / A, so that any m > mk−1 is a candidate to be mj+k . T is a computable tree so that Q = [T ] is a Π10 class. It can be verified that Q is a subshift and has the following elements. 1. 0ω has rank 3 in Q. 2. For each n > 0, 0n 1ω has rank 2 in Q. 3. For all j and all n ≤ nj , X = 0n_ 1_ 0nj+1 _ 0nj+2 _ 1_ · · · has rank 1 in Q. Each of these elements is a shift of B and is Turing equivalent to A. These are the elements of Q with infinitely many 1’s. 4. For each j and k and each n ≤ nj , X = 0n_ 1_ 0nj+1 _ 1 · · · _ 0nj+k _ 1_ 0ω is an isolated element of Q. These are the elements of Q with more than one but only finitely many 1’s. u t This result can probably be improved to obtain elements of arbitrary ∆02 degree in a rank three Π10 subshift. We can also modify the proof as in the proof of Theorem 6 to obtain elements of different degrees in the same Π10 subshift. Proposition 5. For any Π10 subshift Q of rank three, every element of Q is ∆03 . Proof. Let Q be a Π10 subshift of rank three. Then D2 (Q) has rank one, so that its members are all eventually periodic . Thus any element of rank two or three in Q is computable. The isolated members of Q are also computable. Finally, suppose that X has rank one in Q. Then X is ∆03 by Theorem 3.15 of [6]. u t On the other hand, an arbitrary Π10 class of rank three may contain members which are not Σ60 and even a Π10 class of rank two may contain members which are not Σ40 .
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Conclusions and Future Research
In this paper, we have investigated subshifts of finite Cantor-Bendixson rank and compared them with Π10 subshifts and with arbitrary Π10 classes of finite rank. We showed that every member of a subshift of rank one is eventually periodic (and therefore computable) but there are rank one subshifts of arbitrary Turing degree and rank one Π10 subshifts of arbitrary c. e. degree, so that rank one undecidable Π10 subshifts exist. We gave conditions under which a rank one subshift must be decidable. We showed that there is no subshift of rank ω. We showed that subshifts of rank two may contain members of arbitrary Turing degree, that Π10 subshifts of rank two contain only computable elements, but Π10 subshifts of rank three may contain members of arbitrary c. e. degree. Future investigation includes the possible improvement of Theorem 8 to show that Π10 subshifts of rank three may contain members of arbitrary ∆02 degree and possibly elements which are ∆03 and not ∆02 . The observed complexity of
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the members of a rank three Π10 subshift corresponds to the complexity of the members of a Π10 class of rank one. We want to develop this connection further. One conjecture would be that for any Π10 class P of rank one, there is a Π10 subshift Q of rank three such that D(P ) = D(Q). Another area for investigation is the nature of the functions F for which the subshift It[F ] of itineraries is countable.
References 1. O. Bournez and M. Cosnard, On the computational power of dynamical systems and hybrid systems, Theoretical Computer Science 168 (1996), 417–459. 2. M. Braverman and M. Yampolski, Non-computable Julia sets, J. Amer. Math. Soc. 19 (2006), 551-578. 3. D. Cenzer, Effective dynamics, in Logical Methods in honor of Anil Nerode’s Sixtieth Birthday (J. Crossley, J. Remmel, R. Shore and M. Sweedler, eds.), Birkhauser (1993), 162-177. 4. D. Cenzer, Ali Dashti and J.L.F. King, Computable Symbolic Dynamics, Math. Logic Quarterly 54 (2008), 524–533. 5. D. Cenzer and P. G. Hinman, Degrees of difficulty of generalized r. e. separating classes, Arch. for Math. Logic 45 (2008), 629–647. 6. D. Cenzer and J. B. Remmel, Π10 classes, in Handbook of Recursive Mathematics, Vol. 2: Recursive Algebra, Analysis and Combinatorics, editors Y. Ersov, S. Goncharov, V. Marek, A. Nerode, J. Remmel, Elsevier Studies in Logic and the Foundations of Mathematics, Vol. 139 (1998) 623–821. 7. J-Delvenne, P. Kurka, V. Blondel, Decidability and Universality in Symbolic Dynamical Systems, Fund. Informaticae (2005). 8. K. Ko, On the computability of fractal dimensions and Julia sets, Ann. Pure Appl. Logic 93 (1998), 195-216. 9. Medvedev, Yu., Degrees of difficulty of the mass problem, Dokl. Akad. Nauk SSSR, 104 (1955) 501–504 10. J. Miller, Two notes on subshifts, preprint. 11. R. Rettinger and K. Weihrauch, The computational complexity of some Julia sets, in Proc. 35th ACM Symposium on Theory of Computing (San Diego, June 2003) (M.X. Goemans, ed.), ACM Press, New York (2003), 177-185. 12. S. G. Simpson, Mass problems and randomness, Bull. Symbolic Logic 11 (2005), 1–27. 13. S. G. Simpson, Subsystems of Second Order Arithmetic, 2d Edition, Cambridge U. Press (2009). 14. S. G. Simpson, Medvedev degrees of two-dimensional subshifts of finite type, Ergodic Theory and Dynamical Systems, to appear. 15. Sorbi, A., The Medvedev lattice of degrees of difficulty, “Computability, Enumerability, Unsolvability: Directions in Recursion Theory”, Ed. S.B. Cooper et.al., London Mathematical Society Lecture Notes 224, Cambridge University Press (1996) 289–312, ISBN 0-521-55736-4 16. K. Weihrauch, Computable Analysis, Springer (2000).
Abstract. The computability of countable subshifts and their members is examined. Results include the following. Subshifts of CantorBendixson rank one contain only eventually periodic elements. Any rank one subshift, in which every limit point is periodic, is decidable. Subshifts of rank two may contain members of arbitrary Turing degree. In contrast, effectively closed (Π10 ) subshifts of rank two contain only computable elements, but Π10 subshifts of rank three may contain members of arbitrary c. e. degree. There is no subshift of rank ω. Keywords: Computability, Symbolic Dynamics, Π10 Classes
1
Introduction
There is a long history of interaction between computability and dynamical systems. A Turing machine may be viewed as a dynamical system which produces a sequence of configurations or words before possibly halting. The reverse notion of using an arbitrary dynamical system for general computation has generated much interesting work. See for example [1, 7]. In this paper we will consider computable aspects of certain dynamical systems over the Cantor space 2IN . The study of computable dynamical systems is part of the Nerode program to study the effective content of theorems and constructions in analysis. Weihrauch [16] has provided a comprehensive foundation for computability theory on various spaces, including the space of compact sets and the space of continuous real functions. Computable analysis is related as well to the so-called reverse mathematics of Friedman and Simpson [13], where one studies the proof-theoretic content of various mathematical results. The study of reverse mathematics leads in turn to the concept of degrees of difficulty [9, 15] Here we say that P ≤M Q if there is a Turing computable functional F which maps Q into P ; thus the problem of finding an element of P can be uniformly reduced to that of finding an element of Q, so that P is “less difficult” than Q. The degrees of difficulty of effectively closed sets (also known as Π10 classes) have been intensively investigated in several recent papers [5, 12]. ?
This research was partially supported by NSF grants DMS 0532644 and 0554841 and 652372.
2
The computability of Julia sets in the reals has been studied by Cenzer [3] and Ko [8]; the computability of complex dynamical systems has been investigated by Rettinger and Weihrauch [11] and by Braverman and Yampolski [2]. The connection between dynamical systems and subshifts is the following. Certain dynamical systems may be given by a continuous function F on a symbolic space X (one with a basis of clopen sets). For each X ∈ X , the sequence (X, F (X), F (F (X)), . . . ) is the trajectory of X. Given a fixed partition U0 , . . . , Uk−1 of X into clopen sets, the itinerary It(X) of a point X is the sequence (a0 , a1 , . . . ) ∈ k IN where an = i iff F n (X) ∈ Ui . Let It[F ] = {It(X) : X ∈ X }. Note that It[F ] will be a closed set. We observe that, for each point X with itinerary (a0 , a1 , . . . ), the point F (X) has itinerary (a1 , a2 , . . . ). Now the shift operator σ on k IN is defined by σ(a0 , a1 , . . . ) = (a1 , a2 , . . . ). It follows that It[F ] is closed under the shift operator, that is, It[F ] is a subshift. Computable subshifts and the connection with effective symbolic dynamics were investigated by Cenzer, Dashti and King [4] in a recent paper. A total, Turing computable functional F : 2IN → 2IN is always continuous and thus will be termed computably continuous or just computable. Effectively closed sets (also known as Π10 classes) are a central topic in computability theory; see [6] and Section 2 below. It was shown for any computably continuous function F : 2IN → 2IN , It[F ] is a decidable Π10 class and, conversely, any decidable Π10 subshift P is It[F ] for some computable map F . In this paper, Π10 subshifts are constructed in 2IN and in 2Z which have no computable elements and are not decidable. Thus there is a Π10 subshift with non-trivial Medvedev degree. J. Miller [10] showed that every Π10 Medvedev degree contains a Π10 subshift. Simpson [14] studied Π10 subshifts in two dimensions and obtained a similar result there. Now every nonempty countable Π10 class contains a computable element, so that all countable Π10 classes have Medvedev degree 0, and many uncountable classes also have degree 0. In the present paper, we will consider more closely the structure of countable subshifts, using the Cantor-Bendixson (CB) derivative. We will compare and contrast countable subshifts of finite CB rank with Π10 subshifts of finite CB rank as well as with arbitrary Π10 classes of finite rank. The outline of this paper is as follows. Section 2 contains definitions and preliminaries. Section 3 focuses on subshifts of rank one and has some general results about periodic and eventually periodic members of subshifts. We show that every member of a subshift of rank one is eventually periodic (and therefore computable) but there are rank one subshifts of arbitrary Turing degree and rank one Π10 subshifts of arbitrary c. e. degree, so that rank one undecidable Π10 subshifts exist. We give conditions under which a rank one subshift must be decidable. We show that there is no subshift of rank ω. In section 4, we study subshifts of rank two and three. We show that subshifts of rank two may contain members of arbitrary Turing degree. In contrast, we show that Π10 subshifts of rank two contain only computable elements, but Π10 subshifts of rank three may contain members of arbitrary c. e. degree.
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2
Preliminaries
We begin with some basic definitions. Let IN = {0, 1, 2, ...} denote the set of natural numbers. For any set Σ and any i ∈ IN, Σ i denotes the strings of length i from Σ, Σ ∗ denotes the set of all finite strings from Σ, andSΣ IN denotes the set of countably infinite sequences from Σ. We write Σ 1. u t We note here that there are Π10 classes of rank one with noncomputable elements [6]. Hence we have the following. Corollary 1. There is a Π10 class Q ⊆ 2IN of rank one such that, for any Π10 subshift P of rank one, there is no degree-preserving homeomorphism between P and Q. Next we consider the decidability of rank one subshifts. Theorem 2. (a) For any Turing degree d, there is a subshift Q of rank one such that TQ has degree d (b) For any c. e. degree d, there is a Π10 subshift Q of rank one such that TQ has degree d. Proof. Let A be any set of natural numbers of degree d and let Q contain limit points 0ω and 1_ 0ω , along with isolated points 0n 10ω , for n > 0 and 1_ 0n 10ω for n ∈ A. Then Q is a rank one subshift and TQ ≡T A. For (b), just take a c. e. set B of degree d and let A = IN − B. u t We will next consider a special case in which rank one subshifts are decidable. The following lemma is needed. Lemma 5. Let Q be a subshift, let X = v ω be a periodic element of Q with period k and, for each i < k and each n, let Qi,n = {Z : v n_ (v i)_ (1−v(i))_ Z ∈ Q}. Then Qi,n+1 ⊆ Qi,n for each n. Proof. If Z ∈ Qi,n+1 , then v n+1 (v i)(1 − v(i))Z ∈ Q, so that, since Q is a subshift, v n (v i)(1 − v(i))Z ∈ Q and therefore Z ∈ Qi,n . u t Theorem 3. Let Q be a subshift of rank one such that every element of D(Q) is periodic. Then Q is decidable and every element of Q is computable. Proof. Every element of D(Q) is eventually periodic, hence computable by Theorem 1. Let X = v ω be a periodic element of D(Q) with period k and let v = X k. Let Qi,n be defined as in Lemma 5. Since X has rank one, there exists, for each i < k, some n such that Qi,n is finite for all m ≥ n; let ni be the least such n. Since Qi,n+1 ⊆ Qi,n for all i and n, it follows that the sequence {Qi,n : n ∈ IN} converges to some fixed finite subset Pi of 2IN . Let Di be the decidable set TPi . Now let S(v) = {(i, n) : i < k & Qi,n is finite} and let A(v) be the set of strings of the form v n (v i)w for some (i, n) ∈ S(v). Then S(v) is computable since it is a
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cofinite set. It follows that A(v) is computable, since it the union of finitely many computable sets together with {v n (v i)(1 − v(i))w : (i, n) ∈ S(v) & w ∈ Pi }. For each of the finitely many limit paths v ω ∈ Q, we may similarly define the set A(v) of strings in TQ which branch off from v ω where the appropriately defined set Qi,n is finite. We claim that TQ is the union of the finitely many computable sets A(v) together with the words of the form v n (v i) for some i and n and is therefore decidable. Certainly each such string is in TQ . Now suppose that u is some string in TQ which is not an initial segment of any of the limit paths. Choose v so that u has the longest agreement with v ω of the limit paths and choose i and n so that v n (v i)(1 − v(i)) @ u. Then v n (v i)(1 − v(i)) disagrees with every limit path so that Qi,n is finite and hence u ∈ A(v). u t Corollary 2. For any subshift Q of rank one, there is some finite n such that σ n (Q) is decidable. Proof. By Theorem 1, D(Q) is a finite set of eventually periodic points. For each X ∈ D(Q), σ m (X) is periodic for some m; just let n be the maximumum over X ∈ D(Q). Then by Lemma 3, D(σ n (Q)) = σ n (D(Q)) and thus contains only periodic points, so that Theorem 3 applies. u t There is another interesting consequence of Lemma 5 Theorem 4. There is no subshift of rank ω. Proof. Suppose by way of contradiction that Q has rank ω. Then Dω+1 (Q) = ∅ and Dω (Q) is finite. Then there is a periodic element X of rank ω by Proposition 2. Let X have period k and let the sets Qi,n be defined as in Lemma 4. Since X has rank ω, there is some n such that for all i and all m ≥ n, Qi,m has rank < ω. Suppose that Qi,n has rank ri < ω and let r = max{ri : i > k}. Then by Lemma 5, rk(Qi,m ) ≤ r for all m > n. But this implies that rk(X) ≤ r + 1, which is the desired contradiction. u t It follows from the proof that there is no subshift of rank λ, for any limit ordinal λ. In the next section, we will show that a rank two subshift can have members which are not eventually periodic and indeed not even computable.
4
Subshifts of Rank Two and Three
Proposition 4. For any increasing sequence n0 < n1 < . . . , there is a subshift Q of rank two which contains the element 0n0 10n1 1 . . . Proof. The subshift Q will have the following elements: (0) For each k > 0 and each n ≤ nk , the isolated element 0n 10nk+1 10nk+2 1 . . . (1) For every n, the element 0n 10ω which will have rank one in Q. (2) The unique element 0ω of rank 2 in Q. u t
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Theorem 5. For any Turing degree d, there is a rank two subshift Q which contains a member of Turing degree d and such that TQ has Turing degree d. Proof. Let A = {a0 , a1 , . . . } be any infinite set of degree d and let ni = a0 + a1 + · · · + ai for each i. Now apply Proposition 4. u t This result can be improved as follows. For Q ⊆ 2IN , let D(Q) be the set of Turing degrees of members of Q. Theorem 6. For any countable set D of Turing degrees containing 0, there is a rank two subshift Q such that D(Q) = D. For effectively closed subshifts, the result is quite different. Theorem 7. If Q is a Π10 subshift of rank two, then all of its members are computable. Proof. D(Q) is a subshift of rank one and hence all of its members are eventually periodic and therefore computable. The remaining members of Q are isolated and therefore computable by Theorem 3.12 of [6]. u t Finally, we consider Π10 subshifts of rank three with noncomputable elements. Theorem 8. For any c. e. set A, there exists B ≡T A and a Π10 subshift Q of rank three with B ∈ Q. S Proof. Let A = s As where A0 is empty and for each s, As+1 − As contains at most one number. Let B ≡T A have the form 0n0 10n1 1 · · · where n0 < n1 < · · · represent a modulus of convergence for A. That is, 1. n0 = 0 if 0 ∈ / A and otherwise n0 is the least s such that 0 ∈ As . 2. For each k, nk+1 = nk + 1 if k + 1 ∈ / A and otherwise nk+1 is the least s > nk such that k ∈ As . For any stage s and k < s, we have other sequences m0 , . . . , mk which are candidates to be the actual modulus sequence. That is, m0 , . . . , mk is a candidate for the modulus at stage s if the following holds. 1. m0 is the least t ≤ s such that 0 ∈ At if such s exists and otherwise m0 = 0. 2. For each k, mk+1 is the least t ≤ s such that k ∈ As and t > mk , if such t exists. Otherwise mk+1 = mk + 1. Note that the set of candidate sequences at stage s is uniformly computable. Since the sequence As converges to A, it follows that if (m0 , . . . , mk ) is a candidate sequence for all large enough stages s, then mi = ni for all i ≤ k. Define the tree T to contain all finite strings of length s of the following form, 0a0 _ 1_ 0a1 _ · · · _ 0ak−1 _ 1_ 0ak ,
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such that there exists j < s and m0 < m1 < · · · < mk−1 , a candidate sequence at stage s, such that a0 ≤ mj and for all i < k − 1, aj+i+1 = mj+1 . Here we only require a0 ≤ mj since we are trying to construct a subshift. Observe that there is no restriction on ak . This is in case j + k ∈ / A, so that any m > mk−1 is a candidate to be mj+k . T is a computable tree so that Q = [T ] is a Π10 class. It can be verified that Q is a subshift and has the following elements. 1. 0ω has rank 3 in Q. 2. For each n > 0, 0n 1ω has rank 2 in Q. 3. For all j and all n ≤ nj , X = 0n_ 1_ 0nj+1 _ 0nj+2 _ 1_ · · · has rank 1 in Q. Each of these elements is a shift of B and is Turing equivalent to A. These are the elements of Q with infinitely many 1’s. 4. For each j and k and each n ≤ nj , X = 0n_ 1_ 0nj+1 _ 1 · · · _ 0nj+k _ 1_ 0ω is an isolated element of Q. These are the elements of Q with more than one but only finitely many 1’s. u t This result can probably be improved to obtain elements of arbitrary ∆02 degree in a rank three Π10 subshift. We can also modify the proof as in the proof of Theorem 6 to obtain elements of different degrees in the same Π10 subshift. Proposition 5. For any Π10 subshift Q of rank three, every element of Q is ∆03 . Proof. Let Q be a Π10 subshift of rank three. Then D2 (Q) has rank one, so that its members are all eventually periodic . Thus any element of rank two or three in Q is computable. The isolated members of Q are also computable. Finally, suppose that X has rank one in Q. Then X is ∆03 by Theorem 3.15 of [6]. u t On the other hand, an arbitrary Π10 class of rank three may contain members which are not Σ60 and even a Π10 class of rank two may contain members which are not Σ40 .
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Conclusions and Future Research
In this paper, we have investigated subshifts of finite Cantor-Bendixson rank and compared them with Π10 subshifts and with arbitrary Π10 classes of finite rank. We showed that every member of a subshift of rank one is eventually periodic (and therefore computable) but there are rank one subshifts of arbitrary Turing degree and rank one Π10 subshifts of arbitrary c. e. degree, so that rank one undecidable Π10 subshifts exist. We gave conditions under which a rank one subshift must be decidable. We showed that there is no subshift of rank ω. We showed that subshifts of rank two may contain members of arbitrary Turing degree, that Π10 subshifts of rank two contain only computable elements, but Π10 subshifts of rank three may contain members of arbitrary c. e. degree. Future investigation includes the possible improvement of Theorem 8 to show that Π10 subshifts of rank three may contain members of arbitrary ∆02 degree and possibly elements which are ∆03 and not ∆02 . The observed complexity of
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the members of a rank three Π10 subshift corresponds to the complexity of the members of a Π10 class of rank one. We want to develop this connection further. One conjecture would be that for any Π10 class P of rank one, there is a Π10 subshift Q of rank three such that D(P ) = D(Q). Another area for investigation is the nature of the functions F for which the subshift It[F ] of itineraries is countable.
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