EFFECTIVE SYMBOLIC DYNAMICS AND COMPLEXITY

Transcription

1 EFFECTIVE SYMBOLIC DYNAMICS AND COMPLEXITY By FERIT TOSKA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2013

2 c 2013 Ferit Toska 2

3 I dedicate this to existence. 3

4 ACKNOWLEDGMENTS I would like to recognize my advisor Dr. Douglas Cenzer for his support. Faculty, staff, and graduate students of the University of Florida Department of Mathematics, specifically the Logic group, and Sebastian Wyman and Thanos Gentimis have contributed to the completion of this project as well. I thank them all. I would also like to acknowledge the Toska family in Turkey as well as the Toska s in Gainesville for being with me during this odyssey. 4

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ABSTRACT CHAPTER 1 INTRODUCTION Summary Countable Subshifts Algorithmic Randomness Preliminaries COUNTABLE SUBSHIFTS Preliminaries Subshifts of Rank Two Subshifts of Rank Three and Four SPM COMPLEXITY OF INFINITE STRINGS Preliminaries Effective Hausdorff Dimension and SPM Complexity SPM Complexity of Subsets of Cantor Space COMPRESSIBILITY OF COUNTABLE SUBSETS OF CANTOR SPACE Preliminaries Rank Two Subsets of 2 N REFERENCES BIOGRAPHICAL SKETCH

6 Chair: Douglas Cenzer Major: Mathematics Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EFFECTIVE SYMBOLIC DYNAMICS AND COMPLEXITY By Ferit Toska May 2013 In the second chapter 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 (Π 0 1 ) subshifts of rank three contain only computable elements, but Π 0 1 subshifts of rank four may contain members of arbitrary 0 2 degree. There is no subshift of rank ω + 1. In the third chapter we introduce a notion of description for infinite sequences and their sets. Strict process machine complexity of a computable sequence is zero whereas there are sequences of arbitrary Turing degree with zero complexity. SPM complexity of a sequence or of a subset of Cantor space is equal to its effective Hausdorff dimension. In the fourth chapter we investigate the compressibility of countable subsets of Cantor space. 1-decidable sets of Cantor-Bendixson rank two describe each other where the rate of description is determined by the limit points. If P and Q are strongly homeomorphic then P 1-compresses Q and vice versa. 6

7 CHAPTER 1 INTRODUCTION 1.1 Summary We will present a summary of the following chapters along with a historical survey on the background theories Countable Subshifts The results of Chapter 2 appeared in [9]. An initial version of this paper was presented at [8]. 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, 16]. In Chapter 2 we will consider computable aspects of certain dynamical systems over the Cantor space 2 N and the related space 2 Z. The study of computable dynamical systems is part of the Nerode program to study the effective content of theorems and constructions in analysis. Weihrauch [44] has provided a comprehensive foundation for computability theory on various spaces, including the space of compact sets and the space of continuous real functions. The computability of Julia sets in the reals has been studied by Cenzer [5] and Ko [21]; the computability of complex dynamical systems has been investigated by Rettinger and Weihrauch [37] and by Braverman and Yampolski [2]. Computable subshifts and the connection with effective symbolic dynamics were investigated by Cenzer, Dashti and King [10]. A total, Turing computable functional F : 2 N 2 N is always continuous and thus will be termed computably continuous or just computable. Effectively closed sets (also known as Π 0 1 classes) are a central topic in computability theory; see [4]. It was shown for any computably continuous function F : 2 N 2 N, It[F ] is a decidable Π 0 1 class and, conversely, any decidable Π 0 1 subshift 7

8 P is It[F ] for some computable map F. In Chapter 2, Π 0 1 subshifts are constructed in 2 N and in 2 Z which have no computable elements and are not decidable. Thus there is a Π 0 1 subshift with non-trivial Medvedev degree. J. Miller [34] showed that every Π 0 1 Medvedev degree contains a Π 0 1 subshift. Simpson [41] studied Π 0 1 subshifts in two dimensions and obtained a similar result there. In Chapter 2, 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 Π 0 1 subshifts of finite CB rank as well as with arbitrary Π 0 1 classes of finite rank. The outline of Chapter 2 is as follows. Section 2.1 contains definitions and preliminaries. Section 2.2 focuses on subshifts of rank two and has some general results about periodic and eventually periodic members of subshifts. We show that if Q is a subshift of rank two, then every member of Q is eventually periodic (and therefore computable) and furthermore if Q 2 Z, then the members of rank two are periodic and Q is a decidable closed set. However, there are rank two subshifts in 2 N of arbitrary Turing degree and rank two Π 0 1 subshifts of arbitrary c. e. degree, so that rank two undecidable Π 0 1 subshifts exist in 2 N. We give conditions under which a rank two subshift in 2 N must be decidable. We show that there is no subshift of rank ω + 1. In section 2.3, we study subshifts of rank three and four. We show that subshifts of rank three may contain members of arbitrary Turing degree and that subshifts of rank three in 2 Z may have arbitrary Turing degree. In contrast, we show that Π 0 1 subshifts of rank three contain only computable elements, but Π 0 1 subshifts of rank four may contain members of arbitrary c. e. degree. More generally, we show that for any given Π 0 1 class P of rank two, there is a subshift Q of rank four such that the degrees of the members of P and the degrees of the members of Q are identical. 8

9 1.1.2 Algorithmic Randomness In Chapter 3 and Chapter 4, we will turn our attention to algorithmic randomness, an area of mathematics trying to define the notion of randomness using the tools of computability theory. Early in 20 th century von Mises [35] asserted that any random sequence should follow the law of large numbers. His search for other rules a random sequence should obey was addressed by Church [13]. He was the first making the connection between randomness and computability. These rules had to be computable. Later, Martin Löf [32] gave a characterization of random numbers using measure theoretical tests. There are two other approaches leading to the same notion of randomness. Kolmogorov [22] and Solomonoff [43] developed the notion of Kolmogorov complexity, following the intuition that a random sequence should be incompressible. The third notion is motivated by the idea that a random sequence should be unpredictable. Following this idea Levy [26] defined martingales and Schnorr [38, 40] effectivized it. The idea of finite strings describing other finite strings is the central aspect of Kolmogorov complexity. Different families of machines used for the description systems give rise to different notions of complexity. Levin [24, 25] used monotone machines to characterize randomness and Schnorr [39, 40] introduced process complexity. In Chapter 3 we will use another type of machines, strict process machines which can be found in the works of Zvonkin and Levin [45], and in Solomonoff [43]. The interplay between process machines and other notions of complexities is studied by Day [15]. In Section 3.1, we will use strict process machines to formalize a notion of description for infinite sequences. We will define strict process machine (SPM) complexity of infinite strings. We show that the SPM complexity of computable sequences are zero. On the other hand, there are sequences with zero complexity of any Turing degree. 9

10 In section 3.2 we prove that the SPM complexity of an infinite string is equal to its effective Hausdorff dimension. The classical definition of Hausdorff dimension goes back to works of Lebesgue [23], Caratheodory [3] and Hausdorff [18]. Effective version of Hausdorff dimension was given by Lutz [29 31] using martingales. Mayordomo [33] provided a characterization of effective Hausdorff dimension in terms of Kolmogorov complexity. In section 3.3 we extend the notion of description to the sets of infinite strings. The notions of compression, description space and representation space are introduced. We provide a definition of SPM complexity for subsets of Cantor space and prove that it is equal to effective Hausdorff dimension. In Chapter 4 we study the compressibility of countable subsets of Cantor space. In section 4.1 we define n-decidability of a closed set which is the backbone of strict process machines constructed in the following section. In section 4.2 we focus on rank two subsets. Given any two 1-decidable set P and Q, we construct a machine M so that P is an M-description of Q. We see that the rate of description does only depend on the limit points and that the isolated points can be compressed with any rate we desire. We observe that homeomorphism is not sufficient to entail 1-compressibility. The notion of strong homeomorphism is introduced to capture this idea. 1.2 Preliminaries In this section our goal is to introduce background theories that needed to present this dissertation. Our work can be considered as a part of mathematics contained in symbolic dynamics, computability theory and algorithmic randomness. While these areas cover a very broad range of interest our scope will be limited. We will also introduce some notational conventions in this section. Dynamical Systems: The theory of dynamical systems aims to understand the asymptotic behavior of an iterative process. These processes can be categorized as continuous dynamical 10

11 systems and discrete dynamical systems. Our interest, symbolic dynamics, falls in the discrete category. A process determined by the iteration of a function is an example of a discrete process. The theory investigates the eventual behavior of the points x, f (x), f 2 (x),..., f n (x),.... For more information on discrete dynamical systems see [17] or [14]. Symbolic Dynamics: Symbolic dynamics can be studied as a subfield of formal language theory, independent from dynamical systems. From the perspective of computability theory, both approaches have their own merits. Hence, we shall adopt and use both. We need to introduce some basic definitions and notations: For any set A, we let card(a) denote the cardinality of the set A. The set N = {0, 1, 2,...} denotes the set of natural numbers, and Z = {..., -2, -1, 0, 1,... } denotes the set of integers. An alphabet is a set of letters. A word is a finite string of letters. For any set Σ and any i N, Σ i denotes the strings of length i from Σ, Σ denotes the set of all finite strings from Σ. Σ N denotes the set of countably infinite sequences from Σ, that is, the set of functions mapping N into Σ, and Σ Z denotes the set of functions mapping Z into Σ. We write Σ <n for i<n Σi. We use the notation X = X (0), X (1),... to denote the members of Σ N where X (i) Σ for each i N. We write Z =..., Z( n),..., Z( 1), Z(0), Z(1),..., Z(n),... for Z Σ Z For a string w = (w(0), w(1),..., w(n 1)), w denotes the length n of w. The reverse of a string w = (w(0),..., w(n 1)) is the string w = (w(n 1),..., w(0)). λ denotes the empty string, that is λ is the unique string with λ = 0. For m w, w m is the string (w(0),..., w(m 1)). 11

12 Given two strings v and w, the concatenation v w is defined by v w = (v(0), v(1),..., v(m 1), w(0), w(1),..., w(n 1)), where v = m and w = n. For a Σ, we write w a (or just wa) for w (a) and we write a w (or just aw) for (a) w. We say w is an initial segment or prefix of v (written w v) if v = w x for some x Σ. This is equivalent to saying that w = v m for some m N. We also say w is a suffix of v if v = x w for some x Σ ; w is a factor of v if v = x w y for some x, y Σ. We also write w v if w v and w v. For any X Σ N and n N, the initial segment X n is (X (0),..., X (n 1)). For Z Σ Z and i j from Z, Z[i, j] denotes the finite string (Z(i), Z(i +1),..., Z(j)); Z[i, j), Z(i, j], and Z(i, j) are similarly defined. These definitions also apply to Z Σ N as well as to finite strings. We say that a word v is a factor of X Σ N or of X Σ Z if v = X [i, j] for some i and j. We say that X avoids v if v is not a factor of X. For a string w Σ and any X Σ N, we write w X if w = X n for some n. For any w Σ n and any X Σ N, we let w X = (w(0),..., w(n 1), X (0), X (1),...). For X Σ N, let X = (, X (2), X (1), X (0)); that is, for each i N, let X ( i 1) = X (i), so that X Σ Z, where Z = {i Z : i < 0}. For X, Y Σ N, let Z = X.Y Σ Z be defined so that, for all n N, Z(n) = Y (n) and Z( n 1) = X (n). We denote by v ω the infinite concatenation v v ; similarly v ω = v v and v = v ω.v ω. Next, we introduce the notion of formal language and language characterization of subshifts. Definition If Σ is a finite alphabet, then the full Σ-shift is the collection of all infinite sequences (or bi-infinite sequences) of symbols from Σ. 2. The shift map σ on Σ is defined by σ(w) = (w(1),..., w( w 1)). For X 2 N, σ(x ) = (X (1), X (2),... ). For Z 2 Z, Y = σ(z) is defined so that Y (i) = Z(i + 1). 12

13 3. For any subset S of Σ, let QS Z = {X 2Z : X avoids w for all w S} and let QS N = {X 2N : X avoids w for all w S}. We will write Q S when there is no confusion. The set S is also called the set of forbidden words over Σ. Using the set of forbidden words we define shift spaces as follows: Definition 1.2. A shift space is a subset of the full shift such that Q = Q S for some collection of forbidden words over Σ. We can also describe a shift space using the words that are allowed rather than forbidden. Definition A language L is a subset of Σ. 2. Let P be a subset of a full shift. The language of P, denoted by L(P) is {u Σ : u occurs in X for somex P} Proposition 1.1. Let L be a language. Then, L = L(P) for some shift space P if and only if L satisfies the following conditions: 1. if w L and u is a factor of v then u L. 2. if w L then there are nonempty words u, v L such that uwv L 3. Let P and Q be two shift spaces. Then, P = Q if and only if L(P) = L(Q). 4. P is a shift space if and only if whenever X [i, j] L(P) for all i, j then X P. For more information on symbolic dynamics see [27] and on formal languages see [42]. Cantor Space: For the rest of this dissertation we will set Σ = {0, 1}, and the full shift will be denoted by 2 N. A shift space P 2 N will be called a subshift. Cantor Space, the set of infinite binary sequences, is our main object of study. We list some basic facts about 2 N : Its topology generated by a basis of intervals, of the form [w] = {X : w X }. 13

14 Intervals are clopen, that is both open and closed. 2 N is compact. The same topology is generated by the metric d defined as follows: For all X, Y 2 N, let d(x, Y ) = 1/2 n if n is the largest k such that X k = Y k. Let U {0, 1}. The open set generated by U is [U] = {X 2 N : u X for some u U} A subset P of 2 N is clopen if and only if P = [U] for some finite subset U of {0, 1}. Lebesgue measure of an interval [w], denoted by µ([w]) is 2 w. Trees of {0, 1} and Π 0 1 classes: Definition 1.4. A tree T is a subset of {0, 1} which contains the empty string λ and closed under initial segments. That is, if v T and u v, then u T. We say that w T is an immediate successor of v T if w = va for some a {0, 1}. The elements of a tree are often referred as nodes. Definition For any tree T, X 2 N is said to be an infinite path through T if X n T for all n. We let [T ] denote the set of infinite paths through T. 2. For any tree T, we say that a node w T is extendible if there exists X [T ] such that w X. The set of extendible nodes of T is denoted by Ext(T ). 3. A node u such that v / T, but u T for all u v, is called a dead end of T. Space: We will list some useful facts about the interaction of trees and subsets of Cantor Proposition (König s Lemma) If T is an infinite tree then [T ]. 2. A subset Q of 2 N is closed if and only if Q = [T ] for some tree T. 3. For any closed set P, T P = {w {0, 1} : P [w] } is a tree such that P = [T P ]. 14

15 4. If P = [T ], then T P will equal the set of extendible nodes of T. Definition A subset P of 2 N is a Π 0 1 class (or effectively closed set) if P = [T ] for some computable tree T. 2. A Π 0 1 class P is decidable if T P is computable. We note the following facts about a Π 0 1 class P: There is a computable tree T with P = [T ] such that Ext(T ) is computable. There is a computable tree T with no dead ends such that P = [T ]. We also note that if T is computable, then the set of extendible nodes is a tree which is a co-c. e. subset of {0, 1} but is not in general computable. For more information on Π 0 1 classes see [6]. Subshifts of 2 Z will be studied in Chapter 2. We list here some definitions and notations needed: The topology on 2 Z has a basis of clopen sets of the form {Z 2 Z : Z[ n, n] = w}, where w has odd length. A bi-tree T is a set of finite strings of odd length which is closed under central segments, that is, (x( n 1), x( n),..., x(0), x(1),..., x(n), x(n + 1)) T implies that (x( n), x( n + 1),..., x(0), x(1),..., x(n 1), x(n)) T. For a bi-tree T, Z 2 Z is a bi-infinite path through T if Z[ n, n] T for all n. Here also [T ] is the set of bi-infinite paths through T Q 2 Z is closed if and only if Q = [T ] for some bi-tree T. The definitions of effectively closed sets, extendible nodes, and decidable closed is analogous to those given above for 2 N. Definition A tree T 2 N is said to be subsimilar if for every v and w, vw T implies w T. 15

16 2. A bi-tree T is said to be subsimilar if for every v of even length and every w, vw T or wv T implies that w T. Next we will point out some facts connecting formal language characterization of subshifts and subsimilar trees: Proposition A closed set P is a subshift if and only if P is shift-invariant, that is σ(x ) P whenever X P. 2. The closed set P is a subshift if and only if T P is subsmilar. 3. (Cenzer, Dashti, King [10]) For any S {0, 1}, Q Z S is a subshift in 2Z and Q N S is a subshift in 2 N. Furthermore, if S is a c. e. set, then the sets Q Z S and QN S are Π0 1 classes. 4. If the set Q in 2 N (or 2 Z ) is a subshift, then there is a set S of words such that Q = Q N S (or Q = QZ S ). Furthermore, if Q is a Π0 1 class, then S may be taken to be a c. e. set. Definition 1.8. Let X 2 N. The orbit of X, denoted by O(X ) is {Y : Y = σ n (X ) for some n N} From the definition it follows that for any X 2 N the closure of O(X ) is a subshift. We will often use orbits to construct subshifts. For instance, let A 2 N. Then, the smallest closed subset of 2 N containing O(X ) for each X A is a subshift. Cantor-Bendixson Derivative: In Chapter 2 and Chapter 4 we will focus on countable subshifts. Cantor-Bendixson derivative will be an important tool to characterize complexity of closed sets. We will first give a general definition for Cantor-Bendixson derivative. A separable completely metrizable space is called Polish. A point x of a topological space is called a limit point if for every open neighborhood U of x there is a point y U such that y x. An isolated point of a topological space is a point that is not a limit point. A subset P of a space X is called perfect in X if it is closed and all of its points are limit points. 16

17 Theorem 1.1. (Cantor-Bendixson) Let X be a Pollish space. Then X can be uniquely written as X = P C, where P is a perfect set and C is countable open. Definition 1.9. For any Polish space X, if X = P C, where P is perfect and C is countable with P C =, we call P the perfect kernel of X. Now we define the Cantor-Bendixson derivative: Definition For any topological space X define the the Cantor-Bendixson derivative, denoted by D(X ), to be the set of non-isolated elements of X. Using transfinite recursion we define the iterated Cantor-Bendixson derivatives D α (X ) for any ordinal α as follows: D 0 (X ) = X D α+1 (X ) = D(D α (X )) D λ (X ) = α<λ Dα (X ) for limit ordinals λ. Theorem 1.2. Let X be a Polish space. For some countable ordinal α, D β (X ) = D α (X ) for all β α and D α (X ) is the perfect kernel of X. Definition For any Polish space X, the least ordinal α as in Theorem 1.2 is called the Cantor-Bendixson rank of X and is denoted by rk(x ). For more information on Cantor-Bendixson derivative and rank see [20]. The spaces we will consider in this dissertation are subspaces of 2 N or 2 Z. Let P be a countable closed subset of Cantor space and X P. We have the following observations: X is isolated if there is a clopen set U such that P U = {X }, equivalently, if there exists i j such that P {Y : Y [i, j] = X [i, j]} = {X }. The perfect kernel of P is the empty set. If P is nonempty finite set then rk(p) = 1. Also, rk( ) = 0. follows: In the special case when P is countable we can define the rank of elements of P as 17

18 Definition Let P be a countable closed subset of 2 N (or 2 Z ) and X P. Then the Cantor-Bendixson rank rk P (X ) of X in P is defined to be the least integer n such that X / D n+1 (P). For countable P and X P we note the following: rk P (X ) = 0 if and only if X is isolated in P. rk(p) is the supremum of {rk P (X ) + 1 : X P} We note that in several articles on effectively closed sets [4, 7, 8, 11], a different definition is given for the Cantor-Bendixson rank of a countable closed set, namely the the least ordinal α such that D α+1 (P) =. This will always be one less than rk(p) as defined above. This alternative definition allows the rk(p) for P 2 N to be any countable ordinal and makes rk(p) the supremum of {rk P (X ) : X P}. We will use the classical definition. For more background on computability and on the Cantor-Bendixson derivative, see [4], which includes the following: Lemma 1. Let F be a continuous map from 2 N into 2 N and let P and Q be closed sets such that F [P] = Q. Then for any Y Q, rk Q (Y ) max{rk P (X ) : X P & F (X ) = Y }. Note that the lemma also holds for the continuous functions from 2 Z into 2 Z since 2 Z is homeomorphic to 2 N. Notation 1.3. For two closed sets P and Q in 2 N, let P Q = {X.Y : X P & Y Q}, which will be a closed set in 2 Z. Remark 1.4. Note that our version of P Q is computably homeomorphic to the usual definition (see [6]) as the set of sequences X Y = (X (0), Y (0), X (1), Y (1),... ) for X P and Y Q. We will also need the following (essentially Theorem 4.1 of [6]). Let α β denote the Hessenberg sum of ordinals α and β. For our purposes, it suffices to note that for natural numbers m and n, m n = m + n for finite ordinals and ω n = n ω = ω + n. 18

19 Lemma 2. For any closed sets P and Q in 2 N, any X P and any Y Q, the Cantor- Bendixson rank of X.Y in P Q equals rk P (X ) rk Q (Y ). Hence if rk(p) and rk(q) are finite, rk(p Q) = rk(p) + rk(q) 1. Kolmogorov Complexity: Next, we will introduce Kolmogorov complexity of finite strings. Although same notions can be studied for other finite objects, we will restrict our attention to binary strings, that is elements of {0, 1}. The notions machine and description play a central role in studying Kolmogorov complexity. A machine is nothing but a partial computable function that maps finite strings to finite strings. Using the word machine may not be needed to develop the theory of partial computable functions but it provides us a very useful heuristic device. We expect from our machines to perform a task. We like to feed in some input to our machines and wait for an output. A similar breakthrough happens when we imagine our machines as description systems. If a machine M on input u outputs v, that is M is a partial computable function defined on u {0, 1} where its image is v, we say that u is an M-description of v. A natural question arises: How good is a description?. The attempts addressing this question leads to the theory of Kolmogorov Complexity and algorithmic randomness. Intuitively, a description is good if it is short. For example, consider a machine that outputs 1 n where n = max(k, l) so that k is the number of occurrences of 0 and l is the number of occurrences of 1 in a given input. Then, both 0000 and describes the same output, namely Actually, 0000 is the better one, being one of the two shortest M-descriptions of Intuitively, a string is considered complex if it is hard to describe, in other words if it has no short descriptions. So, the question what is the length of the shortest description? plays a central role. We have the following definition. Definition Let M be a machine and v {0, 1}. 19

20 1. The Kolmogorov Complexity of v with respect to M is C M (v) = min{ u : M(u) = v} where min =. 2. The string v is said to be random relative to M if C M (v) v. The definition above is not sufficient to capture our intuition about complexity of a string, for it depends on a machine. In particular, if a string v has no M-description, then its Kolmogorov complexity would be. This difficulty is addressed by using universal machines defined below. Definition A machine M is called a universal machine if for any partial computable function N, there is a string u N, called coding string, such that ( v)m(u N v) = N(v). The existence of a universal machine is given by the enumeration theorem. We will fix a universal machine and denote it by V. The Kolmogorov complexity is defined to be C V (v), and denoted by C(v). The choice of a universal machine does not play an important role since for any machine M, there is a constant c = u M, the length of the coding constant, such that C(v) C M (v) + c for any v {0, 1}. As seen in the following theorem, Kolmogorov complexity has a drawback. This is an important problem, especially for the study of complexity of the infinite strings. For, it implies that any infinite string has non-complex initial segments of any size. Theorem 1.5. For any d N, there exists w {0, 1} such that w = uv and C(w) > C(u) + C(v) + d. One way of avoiding this difficulty is using prefix-free machines. Definition A set A {0, 1} is called prefix-free, if u, v A(u v u = v). 2. A machine M is called prefix-free if its domain is a prefix-free set. Theorem 1.6. There exists a prefix-free universal machine. We will fix a universal prefix-free machine and denote it by U. The prefix-free Kolmogorov complexity of a string v is defined to be C U (v) and it is denoted by K(v). 20

21 The following result relates Kolmogorov complexity and prefix-free Kolmogorov complexity. Theorem 1.7. There exists c 1, c 2 N such that for all v {0, 1}, C(v) K(v) + c 1 and K(v) C(v) + 2 log( v ) + c 2. Next, we will define another class of description systems. Definition Let M : {0, 1} {0, 1} be a partial computable machine. 1. M is called a process machine if for all u, v {0, 1} such that u v and M(u) and M(v), then M(u) M(v). 2. M is called a strict process machine if for all u, v {0, 1} such that u v and M(v), then M(u) and M(u) M(v). Our main interest will be on strict process machines (SPM). We have the following: Theorem 1.8. There exists a universal strict process machine. Once we fix a universal strict process machine, say S we can define the strict process complexity of a string u to be C S (u) and denote it by K s (u). In chapter 3, we will use universal strict process machines to define strict process complexity of an infinite string. Effective Hausdorff Dimension: In chapter 3, we show that the strict process complexity of an infinite string is equal to its effective Hausdorf dimension. Here, we present basic definitions and results needed. We start with the classical Hausdorff dimension defined on Cantor space. Definition Let P 2 N and A {0, 1}. 1. For 0 s 1, the s-measure of a clopen set [u] is µ s ([u]) = 2 s u. 2. A is an n-cover of P if P [A] and A {0, 1} n. 3. H s n(p) = inf{ u A µ s( u ) : A is an n-cover of P } 4. The s-dimensional outer Hausdorff measure of P is H s (P) = lim n H s n(p). The following result enables us to define Hausdorff dimension. 21

22 Theorem 1.9. Let P 2 N. Then, there is s [0, 1] such that 1. H t (P) = 0 for all t > s and 2. H r (P) = for all 0 r < s. Definition Let P 2 N. The Hausdorff dimension of P is defined to be inf{s : H s (P) = 0} and denoted by dim H (P). Martingales play an important role in the study of algorithmic randomness and has been studied extensively. Here we give some of the basic definitions used in the characterization of effective Hausdorff dimension. Definition An s-gale is a function d : {0, 1} R such that d(u) = 2 s (d(u0) + d(u1)) for all u {0, 1}. 2. An s-supergale is a function d : {0, 1} R such that d(u) 2 s (d(u0) + d(u1)) for all u {0, 1}. 3. The success set S[d] of an s-(super)gale is defined to be {X 2 N : lim sup n d(x n) = }. Lutz [29 31] characterized effective Hausdorff dimension using martingales. Definition Let P 2 N and let C be a complexity class of real-valued functions. 1. The C-dimension of P 2 N is defined to be inf{s Q : P S[d] for some s-gale d C } 2. The C-dimension of P is called effective Hausdorff dimension of P, denoted by dim(p) when C = Σ The effective Hausdorff dimension of X 2 N is defined to be dim({x }) and denoted by dim(x ). The following result by Hitchcock [19] relates gales and supergales. Theorem Let 0 r < t be rationals. There is a c. e. t-gale d such that for all c. e. r-supergales d, we have S[d] S[d ]. 22

23 The following result which will be used in Chapter 3 to show that the SPM complexity of a subset of 2 N is equal to its effective Hausdorff dimension. Proposition 1.4. Let P 2 N. Then, dim(p) = sup{dim(x ) : X P}. Proof. Let s {s Q : P S[d] for some c. e. s-gale d}. Then, for any X P, s {s Q : {X } S[d] for some c. e. s-gale d}. Thus, dim(x ) dim(p). Therefore, dim(p) sup{dim(x ) : X P}. To prove the other direction assume that dim(p) > r where r = sup{dim(x ) : X P}. Then, there exists t, r Q such that dim(p) > t > r > r. Now, for each X P there exists s X Q and an s X -gale d X where s X < r and {X } S[d X ]. We note that each d X is an r -supergale. Therefore, by Theorem 1.10 there exists a c. e. t gale d such that for all X P, we have S[d X ] S[d]. Thus, P S[d] follows. But this is a contradiction since d is a t-gale and t < dim(p). Hence, dim(p) sup{dim(x ) : X P}. Finally, we have the following theorem by Mayordomo [33] that characterizes effective Hausdorff dimension in terms of Kolmogorov complexity. Theorem For X 2 N, we have dim(x ) = lim inf n K(X n) n = lim inf n C(X n) n For more information on algorithmic randomness see [17] or [36]. 23

24 CHAPTER 2 COUNTABLE SUBSHIFTS 2.1 Preliminaries In this section we introduce the notion of periodicity and some basic results concerning periodic elements of Cantor space. Definition An element X of 2 N is said to be periodic if X = v ω for some finite string v; the period of X is the minimal length of v such that X = v ω. 2. X is said to be eventually periodic if for some strings u and v, X = u v ω. 3. An element Z of 2 Z is periodic if Z = v for some finite v; the period of Z is the minimal v such that Z = v. 4. Z is eventually periodic if for some finite u, v and w, Z = w ω.u v ω. The following facts about periodic sequences will be useful. Lemma 3. Let u and v be finite words and let X = v ω where v is minimal. If X = u v ω, then u = v m for some m. Proof. Suppose that X = u v ω = v ω. If u = λ, then u = v 0. Otherwise, u ω = v ω and therefore u v by the minimality of v and it follows that u = v m for some m. We will need the following simple connection between periodicity and the shift. Lemma X 2 N is periodic if and only if σ n (X ) = X for some n > X 2 N is eventually periodic if and only if σ m+n (X ) = σ m (X ) for some m and n > Z 2 Z is periodic if and only if σ n (Z) = Z for some n > 0. Proof. 1. If X is periodic then X = u ω for some u {0, 1}. Then, σ u (X ) = u ω = X. For the other direction assume that X = σ n (X ) for some n > 0. Let u be the prefix of X of length n, that is u = n and X = u Y for some Y 2 N. Then, 24

25 X = σ n (X ) = σ n (uy ) = Y. So, X = ux. An easy induction argument shows that X = u k X for all k. Hence, X = u ω. 2. If X is eventually periodic then X = v u ω for some v, u {0, 1}. Then, σ v (X ) = u ω. So, by the first part σ v + u (X ) = σ u (σ v (X )) = σ v (X ). For the other direction assume that σ m+n (X ) = σ m (X ) for some m and n. Let X = v{ Y where v = m. Then, by the first part we see that Y is periodic. So, Y = u ω for some u {0, 1}. Thus, X = v u ω. 3. Suppose first that Z is periodic and let Z = u for some finite string u of length n. Then we have Z[kn, (k + 1)n) = u for all k Z. So, σ n (Z[kn, (k + 1)n)]) = Z[(k + 1)n, (k + 2)n)] = u for all k Z. Thus, σ n (Z) = u = Z 2.2 Subshifts of Rank Two This section contains results on the computability and decidability of subshifts of rank two. We examine the connection between the shift operator and the Cantor-Bendixson derivative. This leads to the surprising result that there are no subshifts of Cantor-Bendixson rank exactly ω + 1. The following lemmas will be needed. Lemma 5. (a) (b) If Q 2 N is a finite subshift, then Q contains a periodic element and every element of Q is eventually periodic. If Q 2 Z is a finite subshift, then every element of Q is periodic. Proof. (a) Let Y Q, where Q is a finite subshift. Then for each i, σ i (Y ) Q. Since Q is finite, there must exist m and n such that σ m+n (Y ) = σ n (Y ), so that Y is eventually periodic. Then X = σ m (Y ) is a periodic member of Q. For (b), let Z Q. Since Q is finite, there exist m < n such that σ m (Z) = σ n (Z). Thus, Z = σ m (σ n (Z)) = σ n m (Z). Hence Z is periodic. Example 2.1. The results are different for 2 N and 2 Z since part (b) does not hold in 2 N, as seen by the example of {0 ω, 10 ω }. 25

26 Lemma 6. Let P 2 N (or P 2 Z ) be any closed set. Then D α σ(p) = σ(d α (P)) for any ordinal α. Hence, if P 2 N is a subshift, then, for all X P, rk P (X ) rk P (σ(x )). Furthermore, if P 2 Z is a subshift, then, for all X P, rk P (X ) = rk P (σ(x )). Proof. The key to the proof is the case when α = 1. Let P 2 N be a subshift. Suppose first that X D(σ(P)). Then there is a sequence {Y n } n N P of distinct members of P such that lim n σ(y n ) = X. For each n, either Y n (0) = 0 or Y n (0) = 1. Thus there exists i {0, 1} and an infinite subsequence n 0, n 1,... such that Y nk (0) = i for all k. It follows that lim k Y nk = i X and belongs to D(P). But then we have σ(i X ) = X and hence X σ(d(p)). Conversely, suppose that X σ(d(p)) and choose Y D(P) such that σ(y ) = X. Then there is a sequence {Y n } n N P of distinct members of P such that lim n Y n = Y. It follows from the continuity of the shift operator that σ lim n σ(y n ) = σ(y ) = X and hence X D(σ(P)). For the space 2 Z, the converse argument is the same, but the first direction is simpler, since lim n σ(y n ) = X implies that lim n Y n = σ 1 (X ), so that σ 1 (X ) D(P) and hence X σ(d(p)). The proof proceeds by induction on α. For the successor case, we have D α+1 (σ(p)) = D(D α (σ(p))) = D(σ(D α (P))) = σ(d(d α (P))) = σ(d α+1 (P)). For the limit case, we have D α (σ(p)) = β<α D β (σ(p)) = β<α σ(d β (P)) = σ( β<α D β (P)) = σ(d α (P)). For the third equality, observe that if Y = σ(x β ) with X β D β (P) for each β < α, then in 2 N, there exists i < 2 and a set B α cofinal in α such that X β (0) = i for all β B and hence i Y D β (P) for all β B. Thus i Y β<α Dβ (P) (since this is a decreasing intersection) and hence Y σ( β<α Dβ (P)). 26

27 For the final conclusion, suppose that rk P (X ) = α. Then X D α (P) and hence σ(x ) σ(d α (P)) = D α (σ(p)) D α (P). For P 2 Z, we note that the proof above can be modified to show that D α (σ 1 (P)) = σ 1 (D α (P)) and rk P (X ) rk P (σ 1 (X )). Proposition 2.1. For any subshift Q 2 N (or Q 2 Z ) and any ordinal α, D α (Q) is a subshift. Proof. Let Q 2 N (2 Z ). For any X D α (Q), it follows from Lemma 6 that σ(x ) D α (σ(q)) and hence σ(x ) D α (Q). We will need the following lemmas relating subshifts of 2 Z to the corresponding subshifts of 2 N. Let π 0 and π 1 be the two projection maps from 2 Z onto 2 N so that if Z = X.Y, then π 1 (Z) = Y and π 0 (Z) = X ; that is, for all n, Y (n) = Z(n) and X (n) = Z( n 1). Lemma 7. Let Q 2 Z be a subshift. (a) (b) (c) Let S be the set of finite strings u such that no element of Q has u as factor and let S = {u : u S}. Then Q = Q Z S, π 1[Q] = Q N S and π 0[Q] = Q N S. π 0 [Q] and π 1 [Q] are subshifts of 2 N and are effectively closed if Q is effectively closed. Q π 0 [Q] π 1 [Q]. Proof. Let S be as defined. It is clear that Q QS Z. On the other hand, we know by Proposition 1.3 that Q = Q Z R for some R and we must have R S, so that QZ S QZ R = Q. Observe next that π 1 [Q] is a closed set as the continuous image of the closed set Q and is effectively closed if Q is effectively closed, since π 1 is a computable mapping. Furthermore, π 1 [Q] is a subshift. That is, if Y = π 1 (Z), then σ(y ) = π 1 (σ(z)). Thus by Proposition 1.3, we have π 1 [Q] = Q N R for some R and we may assume that S R, since certainly π 1 [Q] QS N. On the other hand, if u / S then for some Z Q, u is a factor of 27

28 Z and by shifting Z if necessary we can obtain u as a factor of π 1 (Z), so that u / R. It follows that π 1 [Q] = QS N. A similar argument holds for π 0[Q]. For part (c), let Z = X.Y Q. Then Z avoids S, so that X avoids S and Y avoids S. Proposition 2.2. (a) (b) For any subshift Q 2 N of rank α + 1, D α (Q) has a periodic element and any element of Q having rank α is eventually periodic. If Q 2 Z has rank α + 1, then all elements of rank α are periodic. Proof. (a) (b) Let Q have rank α + 1, so that D α+1 (Q) = and D α (Q) is finite. Then D α (Q) is a subshift by Proposition 2.1 and the result follows by Lemma 5 (a). The same argument works as in (a). Next we consider some results on the decidability of subshifts. Lemma 8. For any Z 2 N (or in 2 Z ) which is eventually periodic, the set of factors of Z is decidable. Proof. We will give the proof for 2 N. The proof for 2 Z case is similar. Suppose that X = vw ω and let W be the set of factors of X. Then x W if and only if it has one of the following forms: (i) v[s, t) where 0 s t v (ii) v[s, v )w n w[0, t) where s < v, n 0 and t w (iii) w[s, w )w n w[0, t] where s < w, t < w, and n 0. The possible choices of n here are bounded by x, so that there is a finite algorithm for checking whether x W. This shows that W is decidable, as required. Note that if X is computable, then in general the set of factors of X is not decidable. 28

29 Example 2.2. Let E N be any set which is c. e. but not computable and let E = {n 0, n 1,... } be a computable enumeration without repetition. Let X = 10 n 0 10 n Then X is computable but the set of factors of X is not since 10 n 1 is a factor of X if and only if n E. Proposition 2.3. Given any natural number m, there is an at most countable decidable subshift P 2 N (2 Z ) such that its rank is equal to m and all of its elements are eventually periodic. Proof. Let P 0 = and for each n, let P n+1 be the set of elements of 2 N containing at most n ones. Then each P n+1 is clearly a subshift and is decidable since v T Pn+1 if and only if v contains at most n ones. Certainly P 0 = has rank 0 and P 1 = {0 ω } has rank one. For each n, we claim that P n P n+1 and D(P n+1 ) = P n. Suppose first that X P n and let X = v 0 ω where v has at most n 1 ones. Then X = lim i X i where X i = v 0 i 1 0 ω P n+1. Hence X D(P n+1 ). This shows that D(P n+1 ) = P n and it follows by induction that P n+1 has rank n. Next suppose that X / P n. If X / P n+1, then certainly X / D(P n+1 ), so we may assume that X P n+1. Then X must have exactly n ones, so that X = v 0 ω where v has exactly n ones. Then clearly P n+1 [v] = {X }, so that X / D(P n+1 ). The same construction also works for 2 Z. Note that for the sequence of sets P n defined above, n P n is not closed and in fact is dense in 2 N. The following lemma is well-known in the area of combinatorics on words. For example, it follows easily from Theorem of [28, p. 22]. We will present an independent proof. Lemma 9. Suppose X 2 N or X 2 Z is not eventually periodic. Then for any k N, there are at least k + 1 distinct factors of length k that occur infinitely often in X. 29

30 Proof. The proof proceeds by induction on k. Clearly the lemma is true for k = 1, since both 0 and 1 will occur infinitely often in a not eventually periodic sequence. Now suppose that the lemma holds for k, so that X has at least k + 1 different factors of length k which occur infinitely often in X. If there are more than k + 1 such factors u 1,..., u l where l k + 2, then clearly X is going to have at least k + 2 factors of length k + 1, since either u i 0 or u i 1 would occur infinitely often as a factor of X for all i l. Thus we may suppose there are exactly k + 1 distinct factors of length k that occur infinitely often in X ; denote them by u 0,..., u k. It suffices to show that for at least one i, both u i 0 and u i 1 occur infinitely often in X. Suppose, by way of contradiction, that this is not the case. Then for each i there exists e i {0, 1} and there exists n such that 1. For every m > n, the factor X (m),..., X (m + k 1) is in the set {u 0,..., u k }. 2. For every occurrence of u i which occurs past X n u i is followed in X by e i. Now we can show that X must be eventually periodic, which will provide the necessary contradiction. Now consider the sequence of factors x i = (X (n + i), x(n + i + 1),..., X (n + i + k 1)) and let j k be the least such that x j = x 0. Without loss of generality the sequence has the form u 0, u 1,..., u j, u 0 for some j k, so that x i = u i for i j and x j+1 = u 0. Since each u i must be followed by e i, it follows that for i > j, u i+i = σu i e)i and that u 0 = σu j e j. Thus x j+2 = u 1, x j+3 = u 2 and so on, so that X = X (n + k)(e 0 e 1... e j ) ω. This contradiction shows that at least one of the factors u i of length k must have two possible extensions of length k + 1 which occur infinitely often in X. The result for 2 Z from the corresponding result for 2 N. Let Z 2 Z and let Z = X.Y where X and Y are in 2 N. If Z is not eventually periodic, then at least one of the two elements X and Y of 2 N are not eventually periodic. The result now follows from the 2 N case above. 30

31 Theorem 2.3. For any subshift Q 2 N (2 Z ) of rank two, every member of Q is eventually periodic. Proof. Suppose Q 2 N is a subshift, and suppose, by way of contradiction, that X Q and that X is not eventually periodic. Let k be arbitrary and let w 0,..., w k be distinct factors of X of length k which occur infinitely often in X. Then for each i k, there are infinitely many n such that σ n (X ) [w i ] Q. Since X is not eventually periodic, m n implies that σ m (X ) σ n (X ). Thus [w i ] Q has a limit point for each i k. It follows that Q has at least k + 1 limit points. Since k was arbitrary, Q has infinitely many limit points and thus rk(q) > 2, a contradiction. For the 2 Z case, the argument is similar. Again, for a given k, we have k + 1-many distinct factors of Z of length k which occur infinitely often in Z, say w 0, w 1,..., w k. For each i k, this implies that there are an infinite number of distinct members of Q which have w i as a central block. Then by compactness Q must have a limit point having w i as a central block. Since each w i for 0 i k is distinct, we obtain k + 1 distinct limit points. Since k was arbitrary, Q has infinitely many limit points and thus rk(q) > 2. We note here that there are Π 0 1 classes of rank two with non-computable elements [4]. Hence we have the following. Corollary 1. There is a Π 0 1 class of rank two which is not degree-isomorphic to any subshift of rank two. Next we will discuss the decidability of rank two subshifts. Theorem 2.4. (a) (b) For any Turing degree d, there is a subshift Q 2 N of rank two such that T Q has degree d. For any c. e. degree d, there is a Π 0 1 subshift Q 2 N of rank two such that T Q has degree d. Proof. Let A be any set of natural numbers of degree d and let Q contain limit points 0 ω and 1 0 ω, along with isolated points 0 n 1 0 ω, for n > 0 and 1 0 n 1 0 ω for n A. Then Q 31

32 is a rank two subshift and we have 1 0 n 1 T Q if and only if n A. Thus A T T Q. For the other direction, just observe that no string with more than two 1 s belongs to T Q and every string with one or no 1 s belongs to T Q. For (b), just take a c. e. set B of degree d, let A = N B and construct Q as in (a). Our next theorem will show that we cannot achieve the result of the Theorem 2.4 for subshifts of 2 Z. However, in the next section we will present a similar result for rank three subshifts of 2 Z. Theorem 2.5. Let Q 2 Z be a subshift of rank two. Then Q can be decomposed as a finite number of periodic elements together with a finite number of elements of type u ω vw ω and their orbits under the shift map. Proof. Since Q is of rank two, D(Q) has finitely many points and they are periodic by Lemma 5. Next, consider the isolated points. By Theorem 2.3 they are eventually periodic. Thus every element of Q is eventually periodic and hence is computable. It should be noted that although every limit point is periodic, there may also be periodic points which are isolated. Nevertheless we will argue that Q has only finitely many periodic points. Assume for a contradiction that the set A of periodic points of Q is infinite. Then, by compactness, there exist a limit point of A, say X. Moreover, X must be periodic, since it belongs to the finite shift D(Q); let X = v. Our goal is to construct a sequence {Z k : k ω} of elements of A, which converges to a nonperiodic limit. We begin with a X = v = lim k uk, for a sequence {u k : k ω} of distinct u k, all different from v. We may assume without loss of generality that u k k v. Since lim k uk = v, we may assume furthermore that v k u k. 32

33 Now for each k, let n k k be the largest such that v n k u k and let u k = v n k wk where v w k. Now let Z k = σ n k v (u k ) A so that Z k = (w k v n k ) = (w k v n k ) ω.w k (v n k w k ) ω. It is important to recall here that when we write, in general Z = U.v W, this means that Z(i) = v(i) for i < v, that Z( v + j) = W (j) for all j, and that Z( j 1) = U(j) for all j. Since Z k has period k, it follows that {Z k : k ω} is infinite and hence has a limit point Z. We may assume without loss of generality that the for each i < j, Z i Z j. Since the sequence n k k<ω tends to infinity, it follows that Z = v ω.y for some Y. Since Z is a limit point of Q and hence must be periodic, it follows that Y = v ω. We will show that this leads to a contradiction, in two cases. Case I: Suppose that w k v for infinitely many k. Since lim k Z k = v, there is some K such that for k K, Z k begins with v ω.v and hence there is some k such that v w k, a contradiction. Case II: Suppose that w k < v for all but finitely many k. Then there is a fixed w with w < v such that Z k = (wv n k ) ω.w (v n k w) ω for infinitely many k. It follows that v = lim k Z k = v ω.w v ω. Then by Lemma 3, w = λ and therefore infinitely many of the Z k = v, again a contradiction. Thus Q has in fact only a finite number of periodic points. Next, we will analyze the set of nonperiodic (hence isolated) elements of Q. These are all of the form u ω.v w ω. Note that for every element of Q of this form, both u and w are limit points of Q. Since D(Q) is finite, there are only finitely many such u and w. Now let S be the set of strings v such that there exist u and w with u ω.v w ω Q and such that u is not a prefix of v and w is not a suffix of v. 33

34 We claim that S is finite. Otherwise, by the above, there are fixed u and w such that {v S : u ω.v w ω Q} is infinite. We may assume that u and w is minimal here, that is, there is no proper prefix u 0 of u such that u0 ω = u ω and similarly for w. Then there will be an infinite sequence {(v k ) : k N} such that, for each k, u is not a prefix of v k, w is not a suffix of v k, v k < v k+1 and Y k = u ω.v k w ω Q. It is easy to see that if j k, then Y j Y k. That is, suppose that j < k but Y j = Y k. Then v j w ω = v k w ω. Now by deleting the first v j terms of v k, we obtain w ω = v w ω, where v is a nonempty suffix of v k. It now follows from Lemma 3 that v = w m for some m > 0. But this implies that w is a suffix of v k, which is a contradiction. It now follows that this infinite set {Y i : i N} has a limit point Z Q of the form u ω.x with u X. But this means that Z is a nonperiodic limit point of Q, a contradiction. Hence S must be finite. It follows that Q consists of finitely many periodic points together with the shifts of a finite set C = {X i = u ω i.v i wi ω : i < n}. Corollary 2. Let Q 2 Z be a subshift of rank two. Then, Q is decidable and every element of Q is computable. Proof. It follows from Theorem 2.5 that Q consists of finitely many periodic points together with the shifts of a finite set C = {X i = u ω i.v i wi ω : i < n}. Then a finite string x belongs to the bi-tree T Q if and only if it is a factor of one of the elements of C. This now implies that T Q is decidable. That is, for each i, the set of factors of X i is decidable by Lemma 8 and there are only finitely many X i, so that T Q is a finite union of decidable sets. We will next consider a special case in which rank two subshifts of 2 N are decidable. The following lemma is needed. 34