A. N. Kolmogorov. Three approaches to the quantitative definition of `information'. Problems of Information Transmission, 1:1--7, 1965. 35  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 then visiting Kolmogorov, was motivated by this idea when he de ned randomness. The quantity C(x) was also developed independently by Solomono ....

A. N. Kolmogorov. Three approaches to the quantitative de nition of `information'. Problems of Information Transmission, 1:1-7, 1965.

....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 then visiting Kolmogorov, was motivated by this idea when he de ned randomness. The quantity C(x) was also developed independently by Solomono ....

A. N. Kolmogorov. Three approaches to the quantitative denition of `information'. Problems of Information Transmission, 1:1-7, 1965.

....of the Shannon effect in ESPACE: almost every problem in ESPACE has essentially maximum circuit size complexity almost everywhere. The Kolmogorov complexity (often called the program size complexity) of binary strings and sequences was discovered independently by Solomonoff [40] Kolmogorov [18], and Chaitin [6] The extraordinary power and scope of this notion have recently been surveyed by Kolmogorov and Uspenskii [19] and Li and Vitanyi [21] In this paper we are primarily concerned with resource bounded Kolmogorov complexities, which have been investigated by Hartmanis [10] Sipser ....

....oe j n) #oe(n) c (4.4) for all x 2 f0; 1g 1 and n 2 N. As a special case of the selective Kolmogorov complexity, we have the conditional Kolmogorov complexity. This is actually a much studied special case, adapted to infinite sequences, of the conditional complexity defined by Kolmogorov [18]. Again, we are interested in resource bounded versions. Definition 4.3. Let t : N N be a resource bound and let x 2 f0; 1g 1 . a) The t time bounded conditional Kolmogorov complexity of x is the function KT t (x j Delta) KT t (x oe j Delta) where the selector oe is defined by ....

A.N. Kolmogorov, Three approaches to the quantitative definition of `information', Problems of Information Transmission 1 (1965), pp. 1--7.

....all elements of K. The meanings of efficiently and almost all are parameters of this definition that may be varied according to the context. Space bounded Kolmogorov complexity is our second measure of nonuniform complexity. Kolmogorov complexity was introduced by Solomonoff[55] Kolmogorov[31], and Chaitin[13] Resourcebounded Kolmogorov complexity has been investigated extensively [31, 20, 53, 33, 35, 7, 22, 30, 2, 3, 4, 36, 38, etc. We work with the space bounded Kolmogorov complexity of languages. Roughly speaking, for A f0; 1g , n 2 N, and a space bound t, the space bounded ....

A. N. Kolmogorov, Three approaches to the quantitative definition of `information', Problems of Information Transmission 1 (1965), pp. 1--7.

....Depth 37 7 Conclusion 40 This research was supported in 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 ....

....especially concerned with selfdelimiting Kolmogorov complexity and algorithmic randomness. The interested reader is referred to [33, 35] for more details, discussion, and proofs. Kolmogorov complexity, also called program size complexity, was discovered independently by Solomonoff [51] Kolmogorov [21], and Chaitin [9] Self delimiting Kolmogorov complexity is a technical improvement of the original formulation that was developed independently, in slightly different forms, by Levin [26, 27] Schnorr [47] and Chaitin [11] The advantage of the self delimiting version is that it gives precise ....

A. N. Kolmogorov. Three approaches to the quantitative definition of `information'. Problems of Information Transmission, 1:1--7, 1965.

....that almost every language in ESPACE is in some set X of languages if (XjESPACE) 1. In x3 below we summarize those aspects of resource bounded measure that are used in this paper. Kolmogorov complexity, discussed in several papers in this volume, was introduced by Solomonoff[Sol64] Kolmogorov[Kol65], and Chaitin[Cha66] Resourcebounded Kolmogorov complexity has been investigated extensively [Kol65, Har83, Sip83, Lev84, Lon86, BB86, Huy86, Ko86, AR88, All89, AW90, Lut90, Lut92a, etc. In this paper we work with the space bounded Kolmogorov complexity of languages. Roughly speaking, for A ....

A. N. Kolmogorov. Three approaches to the quantitative definition of `information '. Problems of Information Transmission 1:1--7, 1965.

....of a string y, denoted by x v y, if and only if jxj jyj and x = y[0: jxj Gamma 1] A string x is a proper prefix of y, denoted by x y, if and only if x v y and jxj jyj. Kolmogorov 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 ....

A. N. Kolmogorov. Three approaches to the quantitative definition of `information '. Problems of Information Transmission, 1:1--7, 1965.

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A. N. Kolmogorov. Three approaches to the quantitative definition of `information'. Problems of Information Transmission, 1:1--7, 1965. 35

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A. N. Kolmogorov. Three approaches to the quantitative definition of `information'. Problems of Information Transmission, 1:1--7, 1965.

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A. N. Kolmogorov. Three approaches to the quantitative de nition of `information'. Problems of Information Transmission, 1:1-7, 1965. 14

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A.N. Kolmogorov, Three approaches to the quantitative definition of `information', Problems of Information Transmission 1 (1965), pp. 1--7.

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