Habilitationsschrift eingereicht bei der Fakultät für Mathematik und Informatik der Ruprecht-Karls-Universität Heidelberg

Transcription

1 Habilitationsschrift eingereicht bei der Fakultät für Mathematik und Informatik der Ruprecht-Karls-Universität Heidelberg Vorgelegt von Dr. rer. nat. Wolfgang Merkle aus Horkheim bei Heilbronn am Neckar 2004

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3 Effective randomness Wolfgang Merkle

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5 Contents 1 Introduction Introduction and overview Bibliographical notes Acknowledgements Notation Words and sequences Kolmogorov complexity Reducibilities Stochastic and random sequences Stochastic sequences Random sequences Effective measure Martin-Löf random sequences Relation between randomness notions Basic facts on effective randomness Kolmogorov-Loveland stochastic sequences Probabilistic construction of stochastic sequences Closure under selection of subsequences The complexity of random and stochastic sequences Compressibility of random sequences Positive results on the compressibility of random and stochastic sequences Negative results on the compressibility of random and stochastic sequences The construction of random sequences Every sequence is reducible to a Martin-Löf random sequence C.e. reals and Martin-Löf random sequences that are c.e.-selfreducible A computably random sequence that is wtt-autoreducible i

6 6 Infinitely often autoreductions of random sequences Guessing random bits The hat problem and error-correcting codes Autoreducibility Autoreductions of random sequences A sharp bound on the density of guessed bits The global power of additional queries Local and global separations Resource-bounded randomness Separations by p-random oracles Separations by computably random oracles Selection functions that do not preserve normality Regular stochastic sequences Normal sequences Normality is not preserved by deterministic one-counter languages Normality is not preserved by linear languages ii

7 Chapter 1 Introduction Summary of Chapter 1. Besides a brief historical sketch of the developments that led to the concepts of effectively stochastic and random sequences, this chapter contains bibliographical notes and a review of some standard notation. 1.1 Introduction and overview We start with a brief and rather incomplete sketch of the motivations and historical developments that led to the concepts of effectively stochastic and random sequences; formal definitions and basic properties of these concepts will then be given in Chapter 2. The historical sketch is partially based on the more comprehensive accounts by Ambos-Spies and Kučera [8] and by Li and Vitányi [58], see there for further material and references. For further details on the various concepts of stochastic and random sequences, we refer to the following chapters and to the corresponding survey papers and monographs cited in the bibliography [8, 9, 58, 63]. Around 1920, von Mises proposed a concept of randomness for infinite sequences that is based on considering the limits of the frequencies at which individual symbols occur in the prefixes of the sequence under consideration. In the setting of uniformly distributed binary sequences, von Mises proposed to call a sequence random if the sequence itself and all subsequences selected by admissible selection rules behave like a typical sequence behaves according to the strong law of large numbers, i.e., the frequencies of 1 s in the prefixes of the sequence converge to 1/2. Von Mises did not make precise which selection rules should be considered as being admissible, however, he postulated that whether a certain place is to be selected should depend only on the already observed bits of the sequence and, in particular, must not depend on the value of the sequence at this place. For the moment and until the concept is formally introduced in Definition 1, we use the following informal notion of a selection rule. A selection rule specifies 1

8 a process that scans bits of a given infinite binary sequence. Based on the already scanned bits of the sequence, the selection rule might determine a yet unscanned bit that is to be scanned next and, before actually inspecting this bit, specifies whether this bit is to be selected. A sequence S is called stochastic with respect to a given set of admissible selection rules if for any admissible selection rule the bits that are successively selected from S form a sequence that is either finite or such that the frequencies of 1 s in the prefixes of the sequence converge to 1/2. Around 1940, at a time when formalizations of the concept of computability had just been developed, Church [33] proposed to admit exactly the computable selection rules. According to custom at that time, Church actually considered only selection rules that are monotonic, i.e., where the place to be scanned next is always strictly larger than all the places scanned before. Church argued that according to a result of Wald there are sequences that are stochastic with respect to this class of selection rules. A quarter of a century after Church s proposal, Kolmogorov [48, 49, 50] and Loveland [60, 61] independently introduced a more liberal version of selection rule, leading to a more restricted concept of stochastic sequence. They proposed to admit partial computable selection rules that are not necessarily monotonic. Stochastic sequences may lack certain statistical properties that are associated with the intuitive understanding of randomness. For example, Ville [105] demonstrated that for any countable class of monotonic selection rules there is a stochastic sequence where every prefix of the sequence contains more 1 s than 0 s. (1.1) The proposal of Kolmogorov and Loveland to define stochasticity in terms of not necessarily monotonic selection rules instead of monotonic ones can be viewed as an attempt to overcome such shortcomings. However, this attempt has not been completely successfully. Shen [98] observed that the techniques described in Remark 18 can be employed to demonstrate that (1.1) remains valid if we omit the requirement that the selection rules are monotonic. Selection rules and stochastic sequences where the prevailing approach to effective randomness until the second half of the 1960s, when Martin-Löf came up with a new concept that is now known as Martin-Löf random sequence [70]. He postulated that a sequence is random if and only if the sequence satisfies all effective laws of randomness. Here a law of randomness is simply a property that holds for almost all sequences, or, equivalently, a class of uniform measure 1, and a law of randomness is effective if its complement is not just a null class but in fact is contained in the intersection of an effectively given sequence of classes where the ith class has measure less than 1/2 i. It can be shown that the concept of a Martin-Löf random sequence is more restrictive than the concept of a stochastic sequence; this is not surprising because for the latter just a single effective law of randomness, the law of large numbers, has to be satisfied for the sequence itself and for all sequences that can be selected by admissible selection rules. Shortly later, Schnorr [90] considered effective betting games and argued that such betting games yield reasonable concepts of a random sequences and 2

9 can in particularly be used to characterize Martin-Löf random sequences. In contrast to selection rules, which merely specify whether to select a bit or not, with betting games a player may bet varying amounts on the bits of a sequence. Formally, a player can be identified with a betting rule, i.e., a not necessarily total function that maps the information about the already scanned part of the unknown sequence to the next bet, where a bet is given by the place to be scanned next, a guess for the bit at this place, and a rational in the closed interval [0, 1] that is equal to the fraction of the current capital that shall be bet on this guess. Payoff is fair in the sense that at the end of each individual bet the stake is either lost or doubled. Then a sequence is said to be random with respect to a given set of admissible betting rules if we can only achieve a finite gain while betting on this sequence according to any admissible betting rule. Like for selection rules, we may consider betting rules that are monotonic, i.e., where the places of the unknown sequence are scanned in increasing order; such betting rules are called betting strategies. For an intuitive understanding of the approach to randomness via betting games, imagine a casino that offers roulette and consider the sequence of outcomes red and black that occur in the course of the game. We would certainly not call this sequence random if there were a method to determine any next bit before the corresponding drawing has actually taken place. But also if we just knew a strategy that guarantees winning an unbounded amount of money when starting with finite initial capital, this would indicate that the sequence is non-random. So we might be tempted to call a sequence non-random if there is such a strategy. The problem with this definition is that for any sequence there is a strategy that wins against this sequence, e.g., the one that works by always predicting correctly the next bit of the sequence. However, the latter is not a problem for real casinos because for them a sequence is random enough if it does not permit a winning strategy that a gambler can actually play. In general, this suggests to define randomness relative to a certain class of admissible betting strategies instead of striving for an absolute concept. In a context of effective randomness, the admissible betting strategies will be the betting strategies that are computable in a specific model of computation. In the context of recursion theory, this led for example to the consideration of computable betting strategies [8, 90, 91, 102, 106] and of the more powerful partial computable betting strategies [6] and betting rules [82]. Martin-Löf random sequences can be defined in terms of even more powerful betting strategies where the corresponding pay-off function is subcomputable, i.e., can be effectively approximated from below. In connection with the investigation of complexity classes one imposes additional resource-bounds [2, 9, 62, 63, 71] and considers for example betting strategies that can be computed in polynomial time. Part of the motivation for studying effectively stochastic and random sequences stems from the fact that random sequences often interact informatively with other concepts of interest. The material presented in what follows shows that in particular there are strong connections to effective reducibilities and to 3

10 the theory of Kolmogorov complexity, also known as algorithmic information theory. After reviewing the formal definitions of various concepts of effectively stochastic and random sequences in Chapter 2, we discuss in Chapter 3 properties of stochastic sequences. Chapter 4 is about the close connection between effective randomness and the theory of Kolmogorov complexity, also known as algorithmic information theory. Chapter 5 features the celebrated result of Gács [41] and Kučera [51, 52] that for any given sequence X there is a Martin-Löf random sequence R from which X can be decoded effectively. On first sight their result might be slightly surprising and the same remark can be made for the results of Chapter 6 on autoreducibility properties of effectively random sequences. Chapter 7 is about the separation of reducibilities by effectively random sequences; the results of this chapter give an example for the fact that effectively random sequences may possess interesting properties, hence are an alternate way to provide sequences that otherwise may only be obtained by more cumbersome explicit combinatorial constructions. Finally, in Chapter 8, we have a closer look at sequences that are stochastic with respect to selection rules that can be computed by highly restricted models of computations such as finite automata or pushdown automata. 1.2 Bibliographical notes Essentially all of the technical contributions of this text have been published previously in the form of conference or journal articles. Chapter 2 mainly reviews standard concepts and techniques in the area of effective randomness, except for the observations stated in Remarks 5 and 8; the latter were presented, together with the results of Chapter 3, to the International Conference on Automata, Languages and Programming 2002 [75] and a full version of the conference article will appear in the Journal of Symbolic Logic [77]. The material presented in Chapter 4 has been presented to the Conference on Computational Complexity 2003 [76]. The results of Chapter 5, which were obtained in collaboration with Mihailović, have been presented to the conference Mathematical Foundations of Computer Science 2001 [78]. The results presented in Chapter 6, which were obtained together with Ebert and Vollmer, are published in the SIAM Journal on Computing [38]. The latter article is based on conference articles by Ebert and Vollmer [39] and by Ebert and Merkle [37], which were presented to the conferences Mathematical Foundations of Computer Science 2000 and 2002, respectively. The fact that Martin-Löf random sequences are i.o. autoreducible and the related hat problem are due to Ebert [36]. The results of Chapter 7 were presented to the International Conference on Automata, Languages and Programming 2000 [73] and are published in the SIAM Journal on Computing [74]. The results of Chapter 8, which were obtained in collaboration with Reimann, have been presented to the conference Mathematical Foundations of Computer Science 2003 [79]. 4

11 1.3 Acknowledgements I am grateful to the members of the Heidelberg group in logic and theoretical computer science, i.e., to Klaus Ambos-Spies, Edgar Busse, Klaus Gloede, Nenad Mihailović, Paolo di Muccio, Gert H. Müller, and Jan Reimann, to the former members Levke Bentzien, Bernd Borchert, Gunther Mainhardt, André Nies, Hans-Christian Neis, Frank Stephan, Yongge Wang, and Xizhong Zheng, and to the recent visitors Denis Hirschfeldt, Tom Kent, Bakhadyr Khoussainov, Bjørn Kjos-Hanssen, Antonín Kučera, Stephen Lempp, Sasha Rubin, Ted Slaman, and Sebastiaan Terwijn. For discussion on effective randomness I would like to thank Eric Allender, Ron Book, Harry Buhrman, Cristian Calude, Todd Ebert, Lance Fortnow, Peter Gács, Peter Hertling, John Hitchcock, Marcus Hutter, Leonid Levin, Jack Lutz, Elvira Mayordomo, Dieter van Melkebeek, Philippe Moser, Olivier Powell, Boris Ryabko, Jürgen Schmidhuber, Alexander Shen, Ludwig Staiger, Martin Strauss, Leen Torenvliet, Nikolai Vereshchagin, Paul Vitányi, Heribert Vollmer, and Klaus W. Wagner. Special thanks go to Alexander Shen and Jack Lutz, who have pointed out the implications of Theorem 21 stated in Corollary 27 and Remark 29, respectively. 1.4 Notation Words and sequences Our notation is mostly standard, for unexplained terms and further details we refer to the textbooks and surveys cited in the bibliography [8, 9, 19, 58, 84, 85]. All functions are meant to be total if not explicitly attributed as being partial; in particular, a computable function is a partial computable function that is total. Words are usually over the binary alphabet {0, 1}, i.e., words are viewed as finite binary sequences. We write w for the length of a word w; the empty word is denoted by λ. Words are ordered by the usual length-lexicographical ordering and the (i + 1)st word in this ordering is denoted by s i, i.e., s 0 is the empty word λ, s 1 is the word 0, and so on. Occasionally, we identify words with natural numbers via the mapping i s i, i.e., via the isomorphism that takes the length-lexicographical ordering on {λ, 0, 1, 00,...} to the usual ordering on N. The concatenation of two words v and w is denoted by vw. For any two words u and w, the word u is a subword of w if, for appropriate words v 1 and v 2, we have w = v 1 uv 2 ; similarly, u is a prefix of w if w = uv 2 and u is a suffix of w if w = v 1 u. Prefixes and suffixes of a word w that differ from w are called proper. A set of words is prefix-free if the set does not contain a word together with a proper prefix of this word. The term sequence usually refers to an infinite binary sequence and a class is a set of sequences. A sequence S can be viewed as mapping i S(i) from N 5

12 to {0, 1}, and accordingly we have S = S(0)S(1).... The term bit i of S refers to S(i), the (i + 1)th bit of the sequence S. Unless explicitly stated otherwise, the term set refers to a subset of the natural numbers or, equivalently, to a set of words. We identify a set with a sequence S where S(i) = 1 if and only if i is in the set. Notation defined for sets is extended to the corresponding sequences and vice versa, e.g., we may speak of an oracle Turing machine that reduces one sequence to another or of the complexity of the prefixes of a set. An assignment or partial characteristic function is a function from some subset of N to {0, 1}. An assignment is finite iff its domain is finite. A word of length n is identified in the natural way with an assignment on the set {0,..., n 1}. For an assignment σ with finite domain {z 0 <... < z n 1 }, the word associated with σ is σ(z 0 )... σ(z n 1 ), i.e., the (unique) word w of length n that satisfies w(i) = σ(z i ) for i = 0,..., n 1. The restriction of an assignment σ to a set I is denoted by σ I. Hence, if viewing a sequence X as an assignment with domain N, the assignment X I has domain I and agrees there with X; in particular, restricting a sequence X to {0,..., n 1} yields a word, the prefix of X of length n. For a sequence X and an assignment σ, we write X, σ for the sequence that agrees with σ for all arguments in the domain of σ and agrees with X, otherwise. The class of all sequences is referred to as Cantor space and is denoted by {0, 1}. The class of all sequences that have a word x as common prefix is called the cylinder generated by x and is denoted by x{0, 1}. For a set W, let W {0, 1} be the union of all the cylinders x{0, 1} where the word x is in W. Recall the definition of the uniform measure (or Lebesgue measure) on Cantor space, which is the probability distribution obtained by choosing the individual bits of a sequence by independent tosses of a fair coin. We write Prob[.] for probability measures and unless explicitly stated otherwise, all probabilities refer to the uniform measure on Cantor space. Usually we write Prob[A satisfies...] instead of Prob[{A {0, 1} : A satisfies...}] in case it is understood from the context that the measure is uniform measure with respect to A. Logarithms are to base 2 and are rounded to the nearest integer that has an absolute value larger than the absolute value of the usual real-valued logarithm of the given number; e.g., we let log 3 = 2 and log 1/3 = 2. The function.,. from N N to N is the usual effective and effectively invertible pairing function [100] Kolmogorov complexity We briefly review notation and concepts related to Kolmogorov complexity; for more detailed accounts see the references [58, 31]. For any given Turing machine M, the Kolmogorov complexity C M (w) of a word w with respect to M is the length of the shortest word x such that M on input x outputs w. There are Turing machines U that yield optimal Kolmogorov complexity up to an additive constant, i.e., for any Turing machine M there is 6

13 a constant c M such that for all words w, we have C U (w) C M (w) + c M [58, Section 2.1]. We fix such an additively optimal Turing machine U as reference machine and let the Kolmogorov complexity C(w) of a word w be equal to C U (w). For a word w of length n, the uniform Kolmogorov complexity C(w; n) (or uniform complexity, for short) of w with respect to a Turing machine M is the length of the shortest word x where for all i n, the machine M outputs on input (x, i) the length i prefix of w. Again, it can be shown that for the concept of uniform complexity there are additively optimal Turing machines; we pick such a Turing machine U and let the uniform complexity C(w; n) of a word w be equal to C U (w; n) [58, 2.3.3] Reducibilities We briefly review some standard concepts of reducibilities, precise definitions and further details can for example be found in the monographs by Balcázar, Díaz, and Gabarró [19, 20], Odifreddi [84, 85] and Soare [100]. While reducibilities are usually introduced as relations between sets, they can equivalently be viewed as relations between sequences by the usual identification of sets and sequences. Recall the concept of an oracle Turing machine, i.e., a Turing machine that receives natural numbers as input, outputs binary values, and may ask during its computations queries of the form z X?, where the sequence X, the oracle, can be conceived as an additional input to the computation. Consider the computation of an oracle Turing machine M on input x and with oracle X; we write Q(M, X, x) for the set of query words occurring during the computation and we write M(X, x) for the binary output of the computation, where we say M(X, x) is undefined in case the computation does not terminate. An oracle Turing machine M is total iff M(X, x) is defined for all sequences X and all inputs x. Observe that for any total oracle Turing machine M there is a corresponding functional Γ where Γ(B) = A iff M computes the sequence A on oracle B; equivalently, the functional Γ can be viewed as a binary function from pairs of sequences and words to {0, 1} where Γ(B, x) = M(X, x). Given a reducibility r, the lower r-span of a sequence A is the class {X : X r A} of sequences that are r-reducible to A, and the lower r-span of a class C is the class of all sequences that are r-reducible to some sequence in C; corresponding concepts of upper r-span are defined in the obvious manner. Furthermore, the r-degree of a sequence A is the intersection of the lower and the upper r-span of A, i.e., the class {X : X r A and A r X}. A sequence A is Turing-reducible to a sequence B if there is an oracle Turing machine M such that M(B, x) = A(x) for all x. The definition of truth-table-reducibility is basically the same, except that in addition we require that M is total, i.e., for all oracles X and for all inputs x, the computation of M(X, x) eventually terminates. By a result due to Nerode and to Trakhtenbrot [84, Proposition III.3.2], for any {0, 1}-valued total oracle Turing machine there is an equivalent one that is again total and queries its oracle nonadaptively (i.e., M computes a list of queries that are asked simultaneously 7

14 and after receiving the answers, M is not allowed to access the oracle again). A sequence A is bounded truth-table-reducible to a sequence B if A is truth-table-reducible to B by an oracle Turing machine such that the size of the query sets Q(M, X, x) is bounded by some constant; in case this constant is just k, we say A is bounded truth-table(k)-reducible to B. A sequence A is weakly truth-table-reducible to a sequence B if A is Turing-reducible to B by an oracle Turing machine such that there is a computable function g that bounds its use, i.e., such that for all sequences X, the set Q(M, X, x) contains only numbers less than or equal to g(x). A sequence A is computably enumerable in a sequence B if there is an oracle Turing machine M such that M(B, x) = 1 in case x A and M(B, x) is undefined otherwise. For r in {btt(k), btt, tt, wtt, T, c.e.}, we say A is r-reducible to B, or A r B for short, if A is reducible to B with respect to bounded truth-table(k), to bounded truth-table, truth-table, weak truth-table, Turing, or computably enumerable reducibility, respectively. From the previous discussion it is immediate that for any k A btt(k) B A btt B A tt B A wtt B A T B A c.e. B, in fact it can be shown that all these implications are strict. We will also consider reductions of a sequence to itself. Of course, reducing a sequence to itself is easy if one does not further restrict the oracle Turing machine performing the reduction. This leads to the concepts of autoreducibility and self-reducibility. We just sketch the definitions of these concepts; for details, motivation, and further references, as well as for a discussion of the related concept of i.o autoreducibility, see Section 6.3. A sequence is Turing autoreducible or T-autoreducible, for short, if it can be reduced to itself by an oracle Turing machine that is not allowed to query the oracle at the current input, and a sequence is Turing self-reducible if it can be reduced to itself by an oracle Turing machine that may only query the oracle at places strictly less than the current input. For reducibilities other than Turing reducibility, the concepts of auto- and self-reducibility are defined in the same manner. E.g., a sequence is wttautoreducible if it is T-autoreducible by an oracle Turing machine with a computable bound on its use, and a sequence A is c.e.-self-reducible if there is an oracle Turing machine that on input x queries its oracle only at places z < x and such that M(B, x) = 1 in case x A and, otherwise, M(B, x) is undefined. Finally, recall the following polynomial time-bounded reducibilities, where the reductions are performed by oracle Turing machines such that for some k, for all input words x, and for all oracles the oracle Turing machine terminates after at most x k +k steps: Turing reducibility (p-t), truth-table reducibility (p-tt), where the queries have to be asked non-adaptively, bounded truthtable reducibility (p-btt), where for each reduction the number of queries is bounded by a constant, and, even more restrictive, p-btt(k)-reducibility, where for all reductions this constant is bounded by the natural number k. 8

15 The relation symbol p btt refers to p-btt-reducibility, and relation symbols for other reducibilities are defined in a similar fashion. Expressions such as p-treduction and p T-reduction will be used interchangeably. We represent p-btt-reductions by a pair of functions g and h computable in polynomial time where g(x) gives the set of words queried on input x and h(x) is a truth-table of a Boolean function over k variables that specifies how the answers to the queries in the set g(x) are evaluated. Here we assume, firstly, via introducing dummy variables, that the cardinality of g(x) is always exactly k and, secondly, by convention, that for i = 1,..., k, the ith argument of the Boolean function h(x) is assigned the ith query in g(x) where the queries are ordered by length-lexicographical ordering. Note that all polynomial time-bounded reducibilities mentioned above can be defined by specifying an appropriate effective sequence of total oracle Turing machines that compute the corresponding reductions. 9

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17 Chapter 2 Stochastic and random sequences Summary of Chapter 2. The definitions and basic properties of various concepts of effectively stochastic and random sequences are reviewed. 2.1 Stochastic sequences Recall from the introduction in Chapter 1 that, intuitively speaking, a selection rule defines a process that scans bits of a given sequence A. More precisely, the selection rule determines a sequence of mutually distinct places x 0, x 1,... at which A is scanned and specifies which of these places are not just scanned but are in fact selected; the bits of A at the selected places, in the order they were selected, form the sequence that is selected from A. The place x i+1 and the decision whether this place shall be selected depends solely on the previously scanned bits A(x 0 ) through A(x i ). Formally, a selection rule is a partial function that receives as input the information w = A(x 0 )... A(x i ) that has been obtained so far by scanning the unknown sequence A, and outputs a place x(w) to be scanned next and a bit b(w) that indicates whether x(w) is to be selected. Definition 1 A selection rule is a not necessarily total function s: {0, 1} N {0, 1} w (x(w), b(w)) such that whenever s(w) is defined then for all proper prefixes v of w, the value s(v) is defined and x(w) differs from x(v). Consider the situation where a selection rule s as above is applied to a sequence A. The sequence of scanned places x 0, x 1,... is defined inductively by x 0 = x(λ), x i+1 = x(a(x 0 )A(x 1 )... A(x i )). 11

18 The sequence of selected places z 0, z 1,... is the subsequence of the sequence of scanned places that is obtained by cancelling all elements where the corresponding value of b is equal to 0, i.e., x 0 is cancelled if b(λ) = 0 and x i+1 is cancelled if b(a(x 0 )A(x 1 )... A(x i )) = 0. Finally, the sequence that is selected by s from A is A(z 0 )A(z 1 ).... A selection rule s is called monotonic if for all sequences the sequence of scanned places is strictly ascending. (Equivalently, one may require x(v) < x(w) whenever both values are defined and v is a proper prefix of w.) In connection with Definition 1, consider the situation where a selection rule s, when applied to the sequence A, scans successively the places x 0, x 1,.... In case the value of s(a(x 0 )... A(x n )) is undefined for some n, the inductive definition of the sequence of scanned places gets stuck, hence the latter sequence and, accordingly, the sequence selected from A are both finite. Remark 2 The intuitive understanding of a selection rule is a function that maps the already observed pairs (x i, a i ) to a pair of a place to be scanned next and a bit that determines whether this place is to be selected. Accordingly, a selection rule with places is a function s: (N {0, 1}) N {0, 1} (x 0, a 0 ),..., (x i, a i ) (x i+1, b i+1 ) such that x i+1 differs from x 0 through x i. For such selection rules we define the concepts of sequences of scanned and of selected places in the same manner as in Definition 1. For example, given a selection rule with places s: w (x(w), b(w)) and a sequence A, the sequence x 0, x 1,... of scanned places is defined inductively by x 0 = x(λ), x i+1 = x((x 0, A(x 0 )), (x 1, A(x 1 )),..., (x i, A(x i ))). (2.1) Two selection rules (with or without places) are called equivalent if for any sequence we obtain the same sequences of scanned and of selected places, no matter which of the two selection rules we apply. Then for every selection rule there is an equivalent selection rule with places and vice versa. Furthermore, when transiting to the equivalent selection rule of the other type the properties of being partial computable and of being computable can be preserved. More precisely, any selection rule s 0 is equivalent to the selection rule with places s 1 : (x 0, a 0 ),..., (x i, a i ) s 0 (a 0... a i ). Conversely, any selection rule with places s 1 is equivalent to the selection rule s 0 : w s 1 ((x w 0, a 1 ), (x w 1, a 2 ),..., (x w i, a i )), w = a 0... a 1, where the x w j are defined inductively similar to the definition of the sequences of scanned places in (2.1), i.e., x w 0 and xw j+1 are the first components of the pairs s 1 (λ) and s 1 ((x w 0, a 1 ),..., (x w j, a j)), respectively. We leave the standard verification of these equivalences to the reader. 12

19 Definition 3 A sequence S is stochastic with respect to a given set of admissible selection rules if for every admissible selection rule the sequence of selected places z 0, z 1,... is either finite or the frequencies of 1 s in the prefixes of the selected sequence converge to 1/2, i.e., {i < n S(z i ) = 1} lim n n = 1 2. (2.2) The type of selection rule that should be admitted depends on the intended applications. For example, in Chapter 8 we discuss relations between selection rules that are computed by finite automata and normal sequences, i.e., sequences in which every word w occurs in the limit with its expected frequency 1/2 w ; similarly, when investigating complexity classes one uses selection rules that are computable within appropriate resource bounds. In a context of recursion theory, one considers computable and partial computable selection rules. In this connection, the concepts of Mises-Wald-Church stochastic and Kolmogorov-Loveland stochastic sequences are of particular interest; both concepts are defined in terms of partial computable selection rules, where the difference is that in the case of Kolmogorov-Loveland stochastic sequences the selection rules are not necessarily monotonic. Definition 4 A Kolmogorov-Loveland selection rule is a selection rule that is partial computable. A sequence is Kolmogorov-Loveland stochastic if it is stochastic with respect to Kolmogorov-Loveland selection rules. A sequence is Mises-Wald-Church stochastic if it is stochastic with respect to monotonic Kolmogorov-Loveland selection rules. A sequence is computably stochastic if it is stochastic with respect to monotonic Kolmogorov- Loveland selection rules that are total, i.e., are defined on all words. If the bits of a sequence are determined by independent tosses of a fair coin, then with probability 1 we obtain a sequence that is Kolmogorov-Loveland stochastic, hence in particular there are such sequences; for details of this probabilistic argument see Remark 18 below. By definition, being Kolmogorov-Loveland stochastic implies being Mises-Wald-Church stochastic, and the latter in turn implies being computably stochastic; we will argue in Section 2.5 that both implications are strict. From the original papers by Kolmogorov [48, 49] and by Loveland [60] where they introduce nonmonotonic selection rules and propose to investigate the corresponding stochastic sequences, it is not that clear whether these selection rules are meant to be computable or partial computable; however, elsewhere [50, 61], both authors have adopted explicitly the concept of Kolmogorov- Loveland stochastic sequence as given in Definition 4, which has become the standard one considered in the literature [8, 58, 69, 104]. In fact, the concept of Kolmogorov-Loveland stochastic sequence is robust in the sense that the concept is not changed if we replace in its definition partial computable selection rules by computable ones. This robustness and a corresponding assertion on betting rules are discussed in the following Remark 5 and in Remark 8, respectively. 13

20 While apparently these two observations went unnoticed until recently [75], it is well-known that at a similar phenomenon occurs for Martin-Löf random sequences, which can be defined equivalently in terms of sequential tests that are determined by enumerable or by computable functions [58, Definition 2.5.1]. Remark 5 The concept of Kolmogorov-Loveland stochastic sequence remains the same if we replace in its definition partial computable selection rules by computable ones. A sequence that is Kolmogorov-Loveland stochastic is in particular stochastic with respect to computable selection rules. In order to prove the reverse implication, fix any sequence S that is not Kolmogorov-Loveland stochastic. Then there is a Kolmogorov-Loveland selection rule s that selects from S a sequence of places z 0, z 1,... such that the frequencies of 1 s in the prefixes of the sequence T = S(z 0 )S(z 1 )... does not converge to 1/2. Decompose T into the subsequences T even and T odd that contain exactly the bits S(z i ) where z i is even and where z i is odd, respectively. Then at least for one of these subsequences the subsequence is infinite and the frequencies of 1 s in its prefixes do not converge to 1/2; we assume that the latter is true for T even and we omit the virtually identical considerations for T odd. We argue that there is a computable selection rule that selects the sequence T even from S, which then finished the proof. The new selection rule works basically by simulating s, however, it only scans but never selects odd numbers. Furthermore, if the simulation of s takes too many computation steps, the new selection rule simply scans the next previously unscanned odd number. When trying to render the idea outlined above more precise, it is convenient to formulate the argument in terms of selection rules with places. Recall that by Remark 2 for any selection rule there is an equivalent selection rule with places and vice versa and that the equivalent selection rules can be chosen such that the properties of being partial computable and being computable are preserved. We construct a computable selection rule with places s that works as outlined above and selects T even from S. The output pair (x i+1, b i+1 ) in s : (x 0, a 0 ),..., (x i, a i ) (x i+1, b i+1 ) is determined as follows. For a total of i steps, simulate a fixed algorithm that computes the sequence of places that are scanned by s. In case during the simulation the next place to be scanned is equal to some x j, the simulation continues by assuming that the bit at that place is a j. If the simulation continues until timeout, the output is (x, 0) where x is the least odd number different from x 0 through x i. Otherwise, i.e., if eventually a place x to be scanned comes up that differs from x 0 through x i, the simulation terminates. In this case the output is (x, 1) if x is even and would have been selected by s and the output is (x, 0), otherwise. 14

21 2.2 Random sequences Recall from the historical sketch in Section 1.1 that stochastic sequences may bear properties that are not compatible with the intuitive understanding of a random sequence. Most of the attempts to come up with a more satisfactory concept of random sequence have been formulated in terms of betting games where instead of merely selecting bits from a sequence one successively bets on the bits of a sequence, then calling a sequence random with respect to a given class of admissible betting rules if none of these rules achieves an unbounded gain on the sequence. In this section we will give a formal account of betting games and of the corresponding randomness concepts; in particular, we will introduce the concepts of partial computably and computably random sequence. In a betting game, a player successively places bets on the bits of an initially unknown sequence A. The betting proceeds in rounds i = 1, 2,.... During round i, the player receives as input the length i 1 prefix of A and then, first, decides whether to bet on bit i being 0 or 1 and, second, determines the stake that shall be bet. The stake might be any fraction between 0 and 1 of the capital accumulated so far; i.e., in particular, the player is not allowed to incur debts. Formally, a player can be identified with a betting strategy b: {0, 1} [ 1, 1] where on input w the absolute value of b(w) is the fraction of the current capital that shall be at stake and the the bet is placed on the next bit being 0 or 1 depending on whether b(w) is negative or non-negative. The player starts with positive, finite capital. At the end of each round, in case of a correct guess, the capital is increased by that round s stake and, otherwise, is decreased by the same amount. So given a betting strategy b, we can inductively compute the corresponding payoff function d b by applying the equations d b (w0) = d b (w) b(w) d b (w), d b (w1) = d b (w) + b(w) d b (w). Intuitively speaking, the payoff d b (w) is the capital the player accumulates till the end of round w by betting on a sequence that has the word w as a prefix. The payoff function d b satisfies the fairness condition d b (w) = d b(w0) + d b (w1) 2. (2.3) Conversely, any function d from words to nonnegative reals that for all words w satisfies the fairness condition (2.3) (with d b replaced by d) determines an initial capital d(λ) and a betting function b, where b(w) = d(w1) d(w0) 2 1 d(w) 15

22 in case d(w) differs from 0 and b(w) = 0, otherwise. We call a function d from words to nonnegative reals a martingale iff d(λ) > 0 and d satisfies the fairness condition (2.3) for all words w. By the discussion above, for a betting strategy b the function d b is always a martingale and, conversely, it can be shown that every martingale has the form d b for some betting strategy b. Hence betting strategies and martingales are essentially equivalent. Accordingly, we will frequently identify martingales and betting strategies via this correspondence and, if appropriate, notation introduced for betting strategies will be extended to martingales and vice versa. Definition 6 A betting strategy b succeeds on a sequence A if the corresponding payoff function d b is unbounded on the prefixes of A, i.e., if lim sup m d b (A {0,..., m}) =. A martingale d is computable if it is confined to rational values and there is a Turing machine that on input w outputs an appropriate finite representation of d(w). Observe that a martingale d with rational initial value d(λ) is computable if and only if the corresponding betting strategy is rational-valued and computable. Definition 7 A sequence is random with respect to a given set of admissible betting strategies if no admissible betting strategy succeeds on the sequence. A sequence is computably random if it is random with respect to the class of all computable betting strategies; a sequence is partial-computably random if it is random with respect to all partial computable betting strategies. Occasionally, we will also consider p-random sequences, i.e., sequences that are random with respect to betting strategies that are computable in polynomial time; p-random sequences are discussed in more detail in Section 7.2. Sequences that are random with respect to partial computable betting strategies are called partial-computably random; betting strategies of this type have been considered by Ambos-Spies [6]. Similar to selection rules, we may consider betting games where the betting has not necessarily to be done in the natural order, the corresponding nonmonotonic betting strategies will be called betting rules; hence, in particular, a betting strategy is a betting rule that is monotonic. Muchnik, Semenov, and Uspensky [82] consider partial computable betting rules; sequences that are random with respect to such betting rules are called nonmonotonic partial-computably random. In Remark 8, we argue that one obtains the same concept if one replaces in the latter definition partial computable betting rules by computable ones; this can be shown by an argument similar to the one used in Remark 5 to derive the equivalence of stochasticity with respect to computable and partial computable selection rules. Remark 8 A sequence is random with respect to computable betting rules if and only if the sequence is random with respect to partial computable betting rules. 16

23 Observe that if a betting rule succeeds on a sequence, then it also succeeds while betting just on either the odd or the even numbers, hence the bits belonging to the other half of the natural numbers are available for being scanned. By using the latter fact in a construction that is similar to the construction of the selection rule s in Remark 5, we obtain a computable betting rule that succeeds on the given sequence. 2.3 Effective measure Effective betting strategies are not only of interest in connection with the definition of effectively random sequences, but are also the basis of the theory of effective measure. Definition 9 A betting strategy succeeds on or covers a class iff the betting strategy succeeds on every sequence in the class. Recall the definition of the uniform measure (or Lebesgue measure) on Cantor space, which describes the distribution obtained by choosing the individual bits of a sequence by independent tosses of a fair coin. Ville demonstrated that a class has uniform measure 0 iff the class can be covered by some, not necessarily effective, betting strategy [9, 105]. (Remark 11 states an effective version of Ville s result, and the latter can be obtained by a noneffective version of the argument given in the remark). Ville s result justifies the following notation. A class has measure 0 with respect to a given class of betting strategies if it is covered by some betting strategy in the class. Similar to the various concepts of effectively random sequences, one obtains restricted concepts of measure 0 classes by appropriately restricting the class of admissible betting strategies. Such restricted concepts are useful when investigating classes occurring in recursion theory or complexity theory. Most of these classes are countable and hence have uniform measure 0, i.e., with respect to uniform measure all these classes have the same size. However, given a specific class C, we might try to specify a class of admissible betting strategies such that we can still cover interesting subclasses of C, but not the class C itself. For example, Lutz [62] demonstrated that computable martingales yield a reasonable measure concept for the class of computable sequences, where in particular the class of all computable sequences cannot be covered by a computable martingale; for further discussion of measure concepts for the class of computable sequences see for example Schnorr [91], Terwijn [102], and Wang [106]. 2.4 Martin-Löf random sequences The most prominent notion of effectively random sequence is the concept of Martin-Löf random sequence [70]. Its standard and more intuitive definition is in terms of effectively given covers, however, we will see in a minute that there is an equivalent, somewhat less intuitive definition in terms of betting strategies. 17

24 Let W 0, W 1,... be the standard enumeration of the computably enumerable sequences [100]. Definition 10 A class N is a Martin-Löf null class if there exists a computable function g : N N such that for all i N W g(i) {0, 1} and Prob[W g(i) {0, 1} ] 1 2 i. (2.4) A sequence is Martin-Löf random if it is not contained in any Martin-Löf null class. In the situation of Definition 10, we say that the W i form a Martin-Löf cover for the class N, i.e., a class has a Martin-Löf cover if and only if it is a Martin-Löf null class. By definition, a class N has uniform measure 0 if there is any sequence of sets V 0, V 1,... such that (2.4) is satisfied with W g(i) replaced by V i. Thus the concept of a Martin-Löf null class is indeed an effective variant of the classical concept of a class that has uniform measure 0 and, in particular, any Martin- Löf null class has uniform measure 0. By σ-additivity and since there are only countably many computable functions, also the union of all Martin-Löf null classes has uniform measure 0, hence the class of Martin-Löf random sequences has uniform measure 1. It can be shown that the union of all Martin-Löf null classes is again a Martin-Löf null class [31, Section 6.2]. Schnorr [91] showed that Martin-Löf random sequences can be equivalently defined in terms of subcomputable martingales, where a martingale d is subcomputable (sometimes also called lower semi-computable) if and only if there is a computable function d in two arguments such that for all words w, the sequence d(w, 0), d(w, 1),... is nondecreasing and converges to d(w). Remark 11 A class N is a Martin-Löf null class if and only if there is a subcomputable martingale that succeeds on every sequence in N. By letting N be equal to the singleton class {A}, this implies as a special case that a sequence A is Martin-Löf random if and only if no subcomputable martingale succeeds on A. We sketch the proof of the former assertion. Fix a subcomputable martingale d that succeeds on N; here we may assume d(λ) < 1 because otherwise, we may simply replace d by d divided by the least natural number larger than d(λ). For given i, the set U i = {w : d(w) > 2 i }. is computably enumerable in i, i.e., there is a computable function g such that U i is equal to W g(i). Furthermore, by construction N is contained in all the U i and by Remark 14 below, the class U i {0, 1} has measure of at most 1/2 i, hence the sets U i form a Martin-Löf cover for N. Conversely, assume that N is a Martin-Löf null class and let g be a computable function such that W g(0), W g(1),... is a Martin-Löf cover for N. By standard techniques, we can assume that the sets W g(i) are all prefix-free. Let d w be the martingale with initial capital 2 w that doubles along w, i.e., d w has the 18

25 value 2 u w on any prefix of u of w, the value 1 on any extension of w, and the value 0 on all other words. Then d(x) = d w (x) {w : w W g(i) for some i 0} is a subcomputable martingale that succeeds on every sequence in N. Observe that d(λ) is equal to the sum of the measures of the classes W g(i) {0, 1} and thus is indeed finite. 2.5 Relation between randomness notions Figure 2.1 summarizes the known implications between the concepts of stochastic and random sequences that we have considered so far; the shown implications are all strict, with the possible exception of A, where this is not known. X Martin-Löf random A X nonmonotonic X Kolmogorov-Loveland partial-computably random stochastisch B D X partial-computably X Mises-Wald-Church random stochastisch C E X computably random X computably stochastisch Figure 2.1: Implications between randomness notions. The implications B through E in Figure 2.1 are immediate by definition of the concepts involved; in the case of A it suffices to observe that similar to the argument in Remark 11 any sequence on which a partial computable betting rule succeeds can be covered by a Martin-Löf cover. The implications from left to right hold because stochasticity corresponds to a restricted type of randomness, i.e., randomness with respect to betting rules where the fraction of the current capital that is bet on the next bit always is either 0 or a fixed rational. The later assertion can be shown for polynomial time-bounded [10] and partial computable betting strategies [82, Theorem 7.4] by basically the same argument, which goes through for partial computable betting rules and computable betting strategies as well. In order to see that with the possible exception of A none of the implications in Figure 2.1 can be reversed, it suffices to observe that 19