B. Ya. Ryabko. Algorithmic approach to the prediction problem. Problems of Information Transmission, 29:186-193, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....fractal sets from sets of a rather smooth geometrical nature. In the context of computability, originating from applications of Hausdorff dimension in information theory (see for example [4] a close connection between Hausdorff dimension and Kolmogorov complexity was established [6, 16, 17, 18, 20, 21], and the notion of effective dimension was introduced [14] Recently, LUTZ [13] has extended the theory of effective dimension to complexity theory by introducing resource bounded dimension. Like for classical Hausdorff dimension, LUTZ s approach yields a generalization of resource bounded ....

B. Y. Ryabko. Algorithmic approach to the prediction problem. Probl. Inf. Transm., 29(2):186--193, 1993.

....in section 3 below) is a real number dimH (X) 2 [0; 1] The Hausdor dimension is monotone, with dimH ( 0 and dimH (C) 1. Moreover, if dimH (X) dimH (C) then X is a measure 0 subset of C. Hausdor dimension thus o ers a quantitative classi cation of measure 0 sets. Moreover, Ryabko [36, 37, 38] Staiger [48, 49] and Cai and Hartmanis [3] have all proven results establishing quantitative relationships between Hausdor dimension and Kolmogorov complexity. Just as Hausdor [14] augmented Lebesgue measure with a theory of dimension, this paper augments the theory of individual random ....

....decided by tossing a 0 1 valued coin whose probability of 1 is i ) then the dimension of R is H( the binary Shannon entropy of . We defer discussion of some signi cant related work until late in the paper, where more context is available. Speci cally, results by Schnorr [40, 42] Ryabko [35, 36, 37, 38], Staiger [48, 49, 50] and Cai and Hartmanis [3] that relate martingales, supermartingales, and Kolmogorov complexity to Hausdor dimension are discussed at the end of section 6. Classical work by Besicovitch [1] Good [13] and Eggleston [9] relating limiting frequencies and Shannon entropy to ....

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B. Ya. Ryabko. Algorithmic approach to the prediction problem. Problems of Information Transmission, 29:186-193, 1993.

....e#ective dimensions to illuminate a variety of topics in algorithmic information theory and computational complexity [20, 21, 1, 7, 27, 16, 15, 11, 13, 14, 10] See [26] for a survey of some of these results. This work has also underscored and renewed the importance of earlier work by Ryabko [28, 29, 30, 31], Staiger [37, 38, 39] and Cai and Hartmanis [5] relating Kolmogorov complexity to classical Hausdor# dimension. See Section 6 of [21] for a discussion of this work. The key to all these e#ective dimensions is a simple characterization of classical Hausdor# dimension in terms of gales, which ....

B. Ya. Ryabko. Algorithmic approach to the prediction problem. Problems of Information Transmission, 29:186--193, 1993.

....strings here we shall consider the following notion of Kolmogorov complexity. Definition 2 The lower Kolmogorov complexity of an infinite string x 2 f0;1g is the value k(x) liminf K U (x=n) Utilizing Levin s universal semicomputable semimeasure (cf. ZL70] or [LV93] it was shown in [Ry93] that the exponent l (x) is bounded from above by 1 Gamma k(x) provided the gambler plays according to a computable strategy. Lemma 2 (Upper bound by Kolmogorov complexity) Let V be a computable capital function. Then (x) 1 Gamma k(x) 7) for every x 2 f0;1g . This upper bound, ....

....Proof. One inequality is Theorem 5. For the converse inequality, observe that Theorem 7 and Lemma 2 prove that g k(x) whenever g dimF , x 2 F and F f0;1g is S 2 definable. Thus, dimF sup x2F k(x) o Concluding Remark Our Theorems 7 and 8 in connection with previous results of Ryabko ([Ry86, Ry93]) and this author ( St93, St98] give evidence that there is a strong coincidence between the concepts of Kolmogorov complexity, gambling strategies and Hausdorff dimension for a class of recursive (computable) sets of infinite zeroone sequences. The results of the last section show a borderline ....

B. Ya. Ryabko, An algorithmic approach to prediction problems. Problemy Peredachi Informatsii 29 (1993), No. 2, 96--103 (in Russian). For the Arithmetical hierarchy of w-languages see e.g. [St97].

....Eggleston [5] proved that in the space of all in nite binary sequences, if we let FREQ( be the set of sequences in which 1 appears with asymptotic frequency (0 1) then the Hausdor dimension of FREQ( is precisely H( the binary entropy of . More recent investigations of Ryabko [16, 17, 18], Staiger [21, 22] and Cai and Hartmanis [3] have explored relationships between Hausdor dimension and Kolmogorov complexity (algorithmic information) Hausdor dimension was originally de ned topologically, using open covers by balls of diminishing radii [8, 6] Very recently, Lutz [14] proved ....

B. Ya. Ryabko. Algorithmic approach to the prediction problem. Problems of Information Transmission, 29:186-193, 1993.

....fractal sets from sets of a rather smooth geometrical nature. In the context of computability, originating from applications of Hausdorff dimension in information theory (see for example [4] a close connection between Hausdorff dimension and Kolmogorov complexity was established [6, 16, 17, 18, 20, 21], and the notion of effective dimension was introduced [14] Recently, LUTZ [13] has extended the theory of effective dimension to complexity theory by introducing resource bounded dimension. Like for classical Hausdorff dimension, LUTZ s approach yields a generalization of resource bounded ....

B. Y. Ryabko. Algorithmic approach to the prediction problem. Probl. Inf. Transm., 29(2):186--193, 1993.

....in section 3 below) is a real number dimH (X) 2 [0; 1] The Hausdor dimension is monotone, with dimH ( 0 and dimH (C) 1. Moreover, if dimH (X) dimH (C) then X is a measure 0 subset of C. Hausdor dimension thus o ers a quantitative classi cation of measure 0 sets. Moreover, Ryabko [30, 31, 32] Staiger [41, 42] and Cai and Hartmanis [2] have all proven results establishing quantitative relationships between Hausdor dimension and Kolmogorov complexity. 2 Just as Hausdor [11] augmented Lebesgue measure with a theory of dimension, this paper augments the theory of individual random ....

B. Ya. Ryabko. Algorithmic approach to the prediction problem. Problems of Information Transmission, 29:186-193, 1993.

....here we shall consider the following notion of Kolmogorov complexity. Definition 2 The lower Kolmogorov complexity of an infinite string x 2 f0;1g w is the value k(x) liminf n K U (x=n) n : Utilizing Levin s universal semicomputable semimeasure (cf. ZL70] or [LV93] it was shown in [Ry93] that the exponent l V (x) is bounded from above by 1 Gamma k(x) provided the gambler plays according to a computable strategy. Lemma 2 (Upper bound by Kolmogorov complexity) Let V be a computable capital function. Then l V (x) 1 Gamma k(x) 7) for every x 2 f0;1g w . This upper bound, ....

....One inequality is Theorem 5. For the converse inequality, observe that Theorem 7 and Lemma 2 prove that g k(x) whenever g dimF , x 2 F and F f0;1g w is S 2 definable. Thus, dimF sup x2F k(x) o Concluding Remark Our Theorems 7 and 8 in connection with previous results of Ryabko ([Ry86, Ry93]) and this author ( St93, St98] give evidence that there is a strong coincidence between the concepts of Kolmogorov complexity, gambling strategies and Hausdorff dimension for a class of recursive (computable) sets of infinite zeroone sequences. The results of the last section show a borderline ....

B. Ya. Ryabko, An algorithmic approach to prediction problems. Problemy Peredachi Informatsii 29 (1993), No. 2, 96--103 (in Russian). 3 For the Arithmetical hierarchy of w-languages see e.g. [St97]. How Much Can You Win 13

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B. Ya. Ryabko. Algorithmic approach to the prediction problem. Problems of Information Transmission, 29:186-193, 1993.

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B. Ya. Ryabko. Algorithmic approach to the prediction problem. Problems of Information Transmission, 29:186-193, 1993.

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B. Ya. Ryabko. Algorithmic approach to the prediction problem. Problems of Information Transmission, 29:186-193, 1993.

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B. Ya. Ryabko. Algorithmic approach to the prediction problem. Problems of Information Transmission, 29:186--193, 1993.

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B. Ya. Ryabko. Algorithmic approach to the prediction problem. Problems of Information Transmission, 29:186--193, 1993.

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B. Ya. Ryabko. Algorithmic approach to the prediction problem. Problems of Information Transmission, 29:186-193, 1993.

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B. Ya. Ryabko, An algorithmic approach to prediction problems. Problemy Peredachi Informatsii 29 (1993), No. 2, 96--103 (in Russian).

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