V.A. Uspenskii, A.L. Semenov, and A.Kh. Shen , Can an individual sequence of zeros and ones be random? Russian Mathematical Surveys 45 (1990), pp. 121--189.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....sense should be viewed as algorithmic randomness and he proposed to call a sequence random if it is random relative to the class of computable selection functions. This gives the following first nontrivial and sound definition of randomness, where, following a suggestion by Kolmogorov (see [32]) we call randomness in the sense of von Mises stochasticity. Definition 2.1 (von Mises, Wald, Church) Let F be a countable class of selection functions. A sequence A is F stochastic if, for any selection function f in F which selects an infinite subsequence A f of A, the subsequence A f ....

....large numbers. The sequence A is computably stochastic or Church random if A is F stochastic for the class F of computable selection functions. The concept of stochasticity can be illustrated by the following alternative game theoretic characterization in terms of prediction functions (see e.g. [32]) A prediction function p is a partial function from f0; 1g to f0; 1g. This function p should be viewed as the strategy of a player P in the following game, which proceeds in infinitely many rounds and in which P attempts to predict the bits of an initially completely hidden infinite binary ....

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V. A. Uspenskii, A. L. Semenov, A. Kh. Shen. Can an individual sequence of zeros and ones be random? Russian Math. Surveys 45 (1990) 121-189.

....set of all strings ( nite words) of 0 s and 1 s, including the empty string . Call N the corresponding set of all binary sequences. A preliminary observation is that any reasonable notion of a random sequence makes sense only with respect to a given probability distribution on the set N [40]. In fact, a sequence with twice as much 0 s than 1 s would not be considered random if each digit occurs with probability p = q = 1=2 (honest coin) but it could be random if p = 1=3 and q = 2=3. That is, although originally motivated by foundational issues, the following discussion pressuposes ....

....R. Dobrushin s noted, in nity is a better approximation to Avogadro s number 6:0 10 23 , than to the number 100. 6 In this paper we only examine these. For some recent developments, see [1] Comprehensive discussions can be found in the monographs [7, 33] or the reviews [40, 35, 17]. 3 In the following we deal with Bernoulli(p; q) probability measures on N , where p q = 1. These are product measures speci ed by their action on strings, to wit: the probability of the string x 1 : x k equals p m q k m , where m is the number of 1 s and k m the number of 0 s in ....

V. A. Uspenskii, A. L. Semenov and A. Kh. Shen, Can an individual sequence of zeros and ones be random?, Russian Math. Surveys, 45, 1 (1990), 105-162.

....of randomness was proposed by von Mises [10] He called a sequence random if every subsequence obtained by an admissible selection rule satisfies the law of large numbers. A formalization of this notion, based on formal computability, was given by Church [4] in 1940. Following Kolmogorov (see [12]) we call randomness in the sense of von Mises and Church stochasticity. 522 Y. Wang For a formal definition of Church s stochasticity concept, we first formalize the notion of a selection rule. Definition 4.1. A selection function f is a partial recursive function f : # # # #. A selection ....

V. A. Uspenskii, A. L. Semenov, and A. Kh. Shen. Can an individual sequence of zeros and ones be random? Russian Math. Surveys, 45:121--189, 1990.

....B and C together form a prediction scheme in which B tries to guess the behavior of A on the set C. A is weakly (t; q; stochastic if no such scheme is better in the limit than guessing by random tosses of a fair coin. Our use of the term stochastic follows Kolmogorov s terminology [12, 28] for properties defined in terms of limiting frequencies of failure of prediction schemes. The adverb weakly distinguishes our notion from a stronger stochasticity property considered in [17] but weak stochasticity is a powerful and convenient tool. The following lemma captures the main ....

V. A. Uspenskii, A. L. Semenov, and A. Kh. Shen', Can an individual sequence of zeros and ones be random?, Russian Mathematical Surveys 45 (1990), pp. 121--189.

....1] 1: Schnorr, using technical variants of Definition 3. 18 (strong success notions involving the rate of growth of the lim sup) has shown that the weak failure of all individual Delta computable martingales on a sequence x characterizes a weak pseudorandomness condition [34,36] See also [41,42] and x6 below. In contrast, the density functions here are generalizations of the density function d of [24,Lemma 5.8] We have first used uniform systems of such density functions to define resource bounded measure, and only then used resource bounded measure to define pseudorandomness. See x6 ....

....Theorem 6.2 tells us that rec randomness is equivalent to the martingalerandomness mentioned by van Lambalgen [42,pp. 77 78] Thus if we let RAND be the set of all algorithmically random sequences of Martin Lof [28] and RANDW be the set of all weakly random sequences of Schnorr [36] see also [41,42]) then RAND RAND(rec) RANDW : If Delta is a time or space bounded complexity class, then Delta randomness is a notion of pseudorandomness that is at least as strong as (and, we conjecture, stronger than) the timeand space bounded versions of RANDW investigated by Schnorr [34,36] In any ....

V.A. Uspenskii, A.L. Semenov, and A.Kh. Shen 0 , Can an individual sequence of zeros and ones be random? Russian Mathematical Surveys 45 (1990), pp. 121--189.

....feasible deterministic algorithms, even with some nonuniform advice. This result, which appears to be very useful, is explained in this section. Properties defined in terms of limiting frequencies of failure of prediction schemes are called stochasticity properties in the terminology of Kolmogorov [KU87, USS90]. Such properties were originally proposed by von Mises [vM39] and Church [Chu40] in their efforts to define randomness. Because the prediction schemes allowed in this section are of a restricted sort, the property discussed here is a weak stochasticity property. We now make our terminology ....

V. A. Uspenskii, A. L. Semenov, and A. Kh. Shen'. Can an individual sequence of zeros and ones be random? Russian Mathematical Surveys, 45:121--189, 1990.

....von Mises to define random sequences is now known as the stochastic approach . A completely different approach to the definition of random sequences was proposed by Kolmogorov and Chaitin independently, and was further developed by Levin, Schnorr and others (see, e.g. Uspenskii, Semenov and Shen [21]) In this approach, a notion of chaoticness is used for a definition of random sequences: the entropy of a finite string x is defined to be the length of the minimal string y from which x can be generated effectively. Then an infinite sequence is chaotic if all of its initial segments have the ....

V. A. Uspenskii, A. L. Semenov, and A. Kh. Shen, Can an individual sequence of zeros and ones be random? Russian Math. Surveys 45 (1990) 121--189.

....Mises to define random sequences is now known as the stochastic approach . A completely different approach to the definition of random sequences was proposed by Kolmogorov and, independently, by Chaitin, and further developed by Levin, Schnorr and others (see, e.g. Uspenskii, Semenov and Shen [21]) In this approach, a notion of chaoticness is used for a definition of random sequences: The entropy of a finite string x is defined to be the length of the minimal string y from which x can be generated effectively. Then an infinite sequence is chaotic if all of its initial segments have the ....

....mainly concentrated on the definition of legal selection rules. For example, Church [4] suggested that legal selection rules should be some recursive processes and Kolmogorov and, independently, Loveland proposed a stronger form which is now known as KolmogorovLoveland selection rules (see e.g. [21, 26]) Di Paola [18] considered the notions of stochasticity in subrecursive hierarchies. Based on these works, Wilber [27] introduced a notion of pseudostochasticity for complexity classes, while there exists a Wilber pseudostochastic set which is not P immune. In [2] Ambos Spies et al. introduced a ....

V. A. Uspenskii, A. L. Semenov, and A. Kh. Shen. Can an individual sequence of zeros and ones be random? Russian Math. Surveys, 45:121--189, 1990.

....deterministic algorithms, even with some nonuniform advice. This result, which appears to be very useful, is explained in this section. Properties defined in terms of limiting frequencies of failure of prediction schemes are called stochasticity properties in the terminology of Kolmogorov [30, 65]. Such properties were originally proposed by von Mises [66] and Church [11] in their efforts to define randomness. Because the prediction schemes allowed in this section are of a restricted sort, the property discussed here is a weak stochasticity property. We now make our terminology precise. ....

V. A. Uspenskii, A. L. Semenov, and A. Kh. Shen', Can an individual sequence of zeros and ones be random?, Russian Mathematical Surveys 45 (1990), pp. 121--189.

....every sequence A of distinct bits of R that is chosen according to a Kolmogorov Loveland selection rule, the limiting frequency of 1 s in A is 1 2 : In the late 1980 s, Shen 0 [27] proved that the converse does not hold, thereby solving a problem that had been open for some twenty years. See [15, 29, 18] for more detailed histories of this problem and the role of stochasticity in the foundations of probability theory. Thus, the random sequences form a proper subset of the set of all Kolmogorov Loveland stochastic sequences. This note refines the method of Shen 0 [27] in order to establish a ....

....2 c : Now let n 0 2 J: Fix a prefix w v R such that d n 0 (w) 1: Then we have 2 c d n 0 (w) 2 c d(w) 2 c 1 X n=m(2c) d n (w) so n 0 m(2c) Thus J is finite. 2 The notion of Kolmogorov Loveland stochasticity was defined in [13, 14, 19, 20] detailed discussions may be found in [29, 15, 18]. A sequence is Kolmogorov Loveland stochastic if any subsequence chosen by a Kolmogorov Loveland selection rule possesses frequency stability, that is, if the proportion of 1 s in initial segments tends toward a limit of 1 2 : A Kolmogorov Loveland selection rule is a pair of partial ....

V. A. Uspenskii, A. L. Semenov, and A. Kh. Shen 0 . Can an individual sequence of zeros and ones be random? Russian Mathematical Surveys, 45:121--189, 1990.

....of randomness was proposed by von Mises [10] He called a sequence random if every subsequence obtained by an admissible selection rule satisfies the law of large numbers. A formalization of this notion, based on formal computability was given by Church [4] in 1940. Following Kolmogorov (see [12]) we call randomness in the sense of von Mises and Church stochasticity. For a formal definition of Church s stochasticity concept, we first formalize the notion of a selection rule. Definition 4.1 A selection function f is a partial recursive function f : Sigma Sigma. A selection ....

....be a p selection function. Then the selected subsequence by the selection function f satisfies the law of large numbers and there is an unbounded nondecreasing function r(n) satisfying (10) Proof. The proof of the claim is exactly the same as that for the Ville s original construction, see, e.g. [12]. In the following, we will only give the outline of the intuition. The basic idea underlying the above construction is the same as that underlying the construction in Lemma 4.8. But here there are countably many selection rules. Whence each bit of the constructed sequence is characterized by an ....

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V. A. Uspenskii, A. L. Semenov, and A. Kh. Shen. Can an individual sequence of zeros and ones be random? Russian Math. Surveys, 45:121--189, 1990.

....resource bounded randomness notions in the literature. We show that our new balanced t(n) genericity concept coincides with the resource bounded version of some other, weaker, randomness concept, namely that of Church [6] According to the classification of randomness concepts by Kolmogorov (see [16]) Church s randomness, which is based on the distribution of the 0s and 1s in effectively chosen subsequences, is a stochasticity notion, while Lutz s concept, a resource bounded version of Schnorr s randomness concept based on martingales [15] is a notion of typicalness. Our equivalence proof ....

....of randomness was proposed by von Mises [18] He called a sequence random if every subsequence obtained by an admissible selection rule satisfies the law of large numbers. A formalization of this notion, based on formal computability was given by Church [6] in 1940. Following Kolmogorov (see [16]) we call randomness in the sense of von Mises and Church stochasticity. For a formal definition of Church s stochasticity concept, we first formalize the notion of a selection rule. Definition10. A selection function f is a total recursive function f : f0; 1g f0; 1g. A selection function f ....

V. A. Uspenskii, A. L. Semenov, and A. Kh. Shen. Can an individual sequence of zeros and ones be random? Russian Math. Surveys, 45:121--189, 1990.

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V.A. Uspenskii, A.L. Semenov, and A.Kh. Shen , Can an individual sequence of zeros and ones be random? Russian Mathematical Surveys 45 (1990), pp. 121--189.

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V. A. Uspenskii, A. L. Semenov, and A. Kh. Shen'. Can an individual sequence of zeros and ones be random? Russian Math. Surveys, 45:121-189, 1990.

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