E.I. Nechiporuk, On a Boolean function, Soviet Mathematics Doklady 7:4 (1966), 999--1000. (In Russian)  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that converge to 2 [0; 1] then FREQ . De nition. Two sequences of biases are square summably equivalent, and we write , if i=0 ( i i ) 1. The next theorem is a constructive version of a classical theorem of Kakutani [17] Theorem 7.2. van Lambalgen [52, 53] Vovk [54]) Let be computable sequences of biases that converge to 2 (0; 1) RAND 0 . 6 RAND 0 = It is well known (and easy to see) that a real number is 2 computable if and only if it is the limit of a computable sequence of reals. Thus Theorems 7.1 and 7.2 tell us ....

V. G. Vovk. On a randomness criterion. Soviet Mathematics Doklady, 35:656-660, 1987.

....sequences are random and which are not. The de nition is probabilistically convincing in that it requires each random sequence to pass every algorithmically implementable statistical test of randomness. The de nition is also robust in that subsequent de nitions by Schnorr [39, 40, 41] Levin [22], Chaitin [6] Solovay [47] This work was supported in part by National Science Foundation Grants 9610461 and 9988483. and Shen [43, 44] using a variety of di erent approaches, all de ne exactly the same sequences to be random. It is noteworthy that all these approaches, like ....

....A useful characterization of random sequences is that they are those sequences that have maximal algorithmic information content. Speci cally, if K(S[0: n 1] denotes the Kolmogorov complexity (algorithmic information content) of the rst n bits of an in nite binary sequence S, then Levin [22] and Chaitin [6] have shown that S is random if and only if there is a constant c such that for all n, K(S[0: n 1] n c. Indeed Kolmogorov [19] developed what is now called C(x) the plain Kolmogorov complexity, in order to formulate such a de nition of randomness, and Martin L of, who was ....

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L. A. Levin. On the notion of a random sequence. Soviet Mathematics Doklady, 14:1413-1416, 1973. 29

....nH #n] 2# . Some of our arguments are simplified by the following constructive version of a classical theorem of Kakutani [17] Say that two bias sequences # # are square summably equivalent, and write # # , if (# i # # i ) #. Theorem 6.10. van Lambalgen [43, 44] Vovk [46]) Let # 0, and let # # be computable bias sequences with # i , # # i #] for all i N. #. Corollary 6.11. If # 0 and #] then there is an exactly computable bias sequence # # with each # # i , # i ] satisfying RAND . Proof. Assume the hypothesis. ....

V. G. Vovk. On a randomness criterion. Soviet Mathematics Doklady, 35:656--660, 1987.

.... FREQ . De nition. Two sequences of biases and 0 are square summably equivalent, and we write 2 0 , if P 1 i=0 ( i 0 i ) 1. The next theorem is a constructive version of a classical theorem of Kakutani [13] Theorem 7.2. van Lambalgen [44, 45] Vovk [46]) Let and 0 be computable sequences of biases that converge to 2 (0; 1) 1. If 2 0 , then RAND = RAND 0 . 2. If 6 2 0 , then RAND RAND 0 = It is well known (and easy to see) that a real number is 0 2 computable if and only if it ....

V. G. Vovk. On a randomness criterion. Soviet Mathematics Doklady, 35:656-660, 1987. 31

....sequences are random and which are not. The de nition is probabilistically convincing in that it requires each random sequence to pass every algorithmically implementable statistical test of randomness. The de nition is also robust in that subsequent de nitions by Schnorr [33, 34, 35] Levin [18], Chaitin [5] Solovay [40] and Shen 0 [36, 37] using a variety of di erent approaches, all de ne exactly the same sequences to be random. It is noteworthy that all these approaches, like Martin L of s, make essential use of the theory of computing. One useful characterization of random ....

....One useful characterization of random sequences is that they are those sequences that have maximal algorithmic information content. Speci cally, if K(S[0: n 1] denotes the Kolmogorov complexity (algorithmic information content) of the rst n bits of an in nite binary sequence S, then Levin [18] and Chaitin [5] have shown that S is random if and only if there is a constant c such that for all n, K(S[0: n 1] n c. Indeed Kolmogorov [15] developed what is now called C(x) the plain Kolmogorov complexity, in order to formulate such a de nition of randomness, and Martin L of, who was ....

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L. A. Levin. On the notion of a random sequence. Soviet Mathematics Doklady, 14:1413-1416, 1973.

....distributions are far too narrow for statistics: almost all probability distributions in interesting Algorithmic theory of randomness 9 statistical models are not computable. Levin also defined universal ivalues (a Bayesian analogue of Martin Lof s universal p values ) See also Levin [18] and Gacs [5] Remark 4 The algorithmic theory of randomness seems to be a natural development of Kolmogorov s complexity conception, but still it is unlikely that Kolmogorov would have approved it. Probably he would have moved in a different direction, though it is difficult to say what it ....

Leonid A Levin. Uniform tests of randomness. Soviet Mathematics Doklady, 17:337, 1976.

....the universality of Kolmogorov s definition of randomness restating it in terms of universal p values. Pi In 1971 Schnorr [22, 21] defined randomness through martingales (first introduced by Ville [26] to fix the problem with von Mises s definition of collectives) Pi In 1973 Levin [17] defined randomness wr to computable (technically, constructively closed) classes of probability distributions. This was an important step since computable probability distributions are far too narrow for statistics: almost all probability distributions in interesting Algorithmic theory of ....

Leonid A Levin. On the notion of a random sequence. Soviet Mathematics Doklady, 14:1413, 1973.

....the earlier definitions (eg, von Mises s) is that it is applicable to finite sequences and that it provides degrees of randomness; this is its crucial feature which makes practical applications possible. Later Kolmogorov s definition was developed by, among others, Martin Lof [15] 1966) Levin [11] (1973) and G acs [2] 1980) The main goal of this paper is to study computable approximations to algorithmic randomness and to apply those approximations to some benchmark datasets. The main technical tool will be Vapnik s [21] 1998) theory of Support Vector machines, but in principle it is ....

....of randomness is not the only possible: we can define uniform randomness deficiency and define, eg, the iid deficiency as the minimum of the deficiencies over all iid measures. The difference is not big when the sample space Z is compact; this follows from the following elaboration of Levin s [11] (1973) result: for any compact class P of probability measures, d L P (x) inf P2P d L P (x) 1) and d ML P (x) inf P2P d ML P (x) 2) However, in applications P is often not compact (although it is always constructively closed ) It is an important problem to study ....

Leonid A Levin. On the notion of a random sequence. Soviet Mathematics Doklady, 14:1413, 1973.

....The fact that the useful sequence K is strongly deep is no coincidence. Every sequence that is even weakly useful must be strongly deep. Bennett [5] also defines the class of weakly deep binary sequences. As noted by Bennett, this class has been investigated in other guises by Levin and V jugin [28, 31, 32, 53, 54, 55]. A sequence x 2 f0; 1g 1 is weakly deep if there do not exist a recursive time bound s : N N and an algorithmically random sequence z such that x DTIME(s) T z. Bennett [5] notes that every strongly deep sequence is weakly deep, but that there exist weakly deep sequences that are not ....

V. V. V'jugin. On Turing invariant sets. Soviet Mathematics Doklady, 17:1090--1094, 1976.

....part by National Science Foundation Grant CCR9157382, with matching funds from Rockwell International and Microware Systems Corporation. 1 Introduction Algorithmic information theory, as developed by Solomonoff [51] Kolmogorov [21, 22, 23] Chaitin [9, 10, 11, 12] Martin Lof [39, 40] Levin [26, 27, 28, 29, 30, 31, 55], Schnorr [47] G acs [15] Shen 0 [48, 49] and others, gives a satisfactory, quantitative account of the information content of individual binary strings (finite) and binary sequences (infinite) However, a given quantity of information may be organized in various ways, rendering it more or ....

L. A. Levin. Uniform tests of randomness. Soviet Mathematics Doklady, pages 337--340, 1976.

....part by National Science Foundation Grant CCR9157382, with matching funds from Rockwell International and Microware Systems Corporation. 1 Introduction Algorithmic information theory, as developed by Solomonoff [51] Kolmogorov [21, 22, 23] Chaitin [9, 10, 11, 12] Martin Lof [39, 40] Levin [26, 27, 28, 29, 30, 31, 55], Schnorr [47] G acs [15] Shen 0 [48, 49] and others, gives a satisfactory, quantitative account of the information content of individual binary strings (finite) and binary sequences (infinite) However, a given quantity of information may be organized in various ways, rendering it more or ....

.... of M 0 ; M 1 ; Delta Delta Delta ; M n Gamma1 on inputs 0; 1; n Gamma 1, respectively, will eventually determine all n bits of K [0: n Gamma 1] In contrast, consider a sequence z 2 f0; 1g 1 that is algorithmically random in the equivalent senses of Martin Lof [39] Levin [26], Schnorr [47] Chaitin [11] Solovay [52] and Shen 0 [48, 49] See section 4 below for a precise definition and basic properties of algorithmic randomness. An n bit prefix z[0: n Gamma 1] of an algorithmically random sequence z contains approximately n bits of algorithmic information ....

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L. A. Levin. On the notion of a random sequence. Soviet Mathematics Doklady, 14:1413--1416, 1973.

....fi i ) 2 1. A classical theorem of Kakutani [9] says that, if ff and fi are strongly positive sequences of biases such that ff t 2 fi, then for every set C C, X has (classical) ff measure 0 if and only if X has fi measure 0. A constructive improvement of this theorem by Vovk [28] says that, if ff and fi are strongly positive, computable sequences of biases such that ff t 2 fi, then for every set X C, X has constructive ff measure 0 if and only if X has constructive fi measure 0. The Kakutani and Vovk theorems are more general than this, but for the ....

V. G. Vovk. On a randomness criterion. Soviet Mathematics Doklady, 35:656--660, 1987.

.... complexity, also called program size complexity, was discovered independently by Solomonoff [23] Kolmogorov [13] and Chaitin [4] Self delimiting Kolmogorov complexity is a technical improvement of the original formulation that was developed independently, in slightly different forms, by Levin [18, 19], Schnorr [20] and Chaitin [5] The advantage of the self delimiting version is that it gives precise characterizations of algorithmic probability and randomness. In this paper, in order to simplify the presentation of compression depth, we very briefly develop the elements of Kolmogorov ....

L. A. Levin. On the notion of a random sequence. Soviet Mathematics Doklady, 14:1413--1416, 1973.

.... Gammatt hard language for NP is dense. The next result concerns NP completeness. The NP completeness of decision problems has two principal, well known formulations. These are the P T completeness introduced by Cook [Coo71] and the P m completeness introduced by Karp [Kar72] and Levin [Lev73]. It is widely conjectured ( LLS75, You83, LY90, Hom90] that these two notions are distinct: CvKL Conjecture. Cook versus Karp Levin ) There exists a language that is P T complete, but not P m complete, for NP. The CvKL Conjecture is very ambitious, since it implies that P 6= NP. The ....

L. A. Levin. On the notion of a random sequence. Soviet Mathematics Doklady, 14:1413--1416, 1973.

....for NP is dense. The last result that we mention in this section concerns NP completeness. The NP completeness of decision problems has two principal, well known formulations. These are the P T completeness introduced by Cook [12] and the P m completeness introduced by Karp [27] and Levin [32]. It is widely conjectured ( 31, 72, 33, 23] that these two notions are distinct: CvKL Conjecture. Cook versus Karp Levin ) There exists a language that is P T complete, but not P m complete, for NP. The CvKL Conjecture is very ambitious, since it implies that P 6= NP. The question has ....

L. A. Levin, On the notion of a random sequence, Soviet Mathematics Doklady 14 (1973), pp. 1413-- 1416.

.... = Gamma = f1; Kg and the loss functions ( fl) E ln fl = K X k=1 k ln k fl k ; fl) K X k=1 ( p k Gamma p fl k ) 2 ; fl) K X k=1 ( k Gamma fl k ) 2 fl k ; respectively (some applications of these loss functions are described in [8, 51]) Notice that the Kullback Leibler game includes the log loss game as a special case (take the degenerate s) The following result was proven in Haussler et al. [33] Example 4.3) under the assumption that K = 2. Lemma 4 The Kullback Leibler game is 1 mixable. The AA for the Kullback Leibler ....

Volodya Vovk. On a randomness criterion. Soviet Mathematics Doklady, 35:656--660, 1987.

....large numbers can be easily modified to prove that the statement of the strong law of large numbers holds for all Martin Lof random infinite sequences. Later it became clear that there is scope for non trivial results as well, even in the case of infinite sequences: see, eg, V yugin [32] or Vovk [28]. REFERENCES 12 As we already mentioned, the question of what probability is remains unanswered in Kolmogorov s complexity conception. We can follow von Mises and Reichenbach and say that, if a long binary sequence is Bernoulli, the probability of 1 is the relative frequency of 1s in that ....

Volodya Vovk. On a randomness criterion. Soviet Mathematics Doklady, 35:656--660, 1987.

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E.I. Nechiporuk, On a Boolean function, Soviet Mathematics Doklady 7:4 (1966), 999--1000. (In Russian)

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E.I. Nechiporuk, On a Boolean function, Soviet Mathematics Doklady, 7:4 (1966) 999--1000.

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V. G. Vovk. On a randomness criterion. Soviet Mathematics Doklady, 35:656--660, 1987.

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L. A. Levin. On the notion of a random sequence. Soviet Mathematics Doklady, 14:1413--1416, 1973. 40

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L. A. Levin. On the notion of a random sequence. Soviet Mathematics Doklady, 14:1413-1416, 1973.

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Leonid A. Levin. On the notion of a random sequence. Soviet Mathematics Doklady, 14:1413, 1973.

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B. Trakhtenbrot. On autoreducibility. Soviet Mathematics{Doklady, 11:814-817, 1970. 10

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B. Trakhtenbrot. On autoreducibility. Soviet Mathematics{Doklady, 11:814-817, 1970.

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