B. Ya. Ryabko. Noiseless coding of combinatorial sources. Problems of Information Transmission, 22:170-179, 1986.  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. Noiseless coding of combinatorial sources, Hausdorff dimension, and Kolmogorov complexity. Probl. Inf. Transm., 22:170--179, 1986.

....of constructive strong dimension and upper Kolmogorov complexity #(# ) The aim of this note is to show that, in fact, this coincidence holds for arbitrary sets of infinite strings, F . Here #(F) sup #(# ) F and #(F) sup #(# ) F . This elucidates the relations between the papers [CH, R2, R3, S1, S2] and the subject matter of [AH, L1, L2, L3] in a more precise manner than the mere remark in [L2] that Moreover, Ryabko, Staiger, and Cai and Hartmanis have all proven results establishing quantitative relationships between Hausdorff dimension and Kolmogorov complexity. As a consequence, several ....

.... dimension via s gales in [L1, AH] For a good introduction to fractal dimensions see [Fa] dim H (F) inf dim P (F) inf Theorems 5 and 7 together with the obvious inequalities dim H (F) cdim(F) and dim P (F) c(F) thus yield, on the one hand, a new proof of Ryabko s inequality [R2, Theorem 2] or [S1, Corollary 3.14] dim H (F) #(F) and (13) dim P (F) #(F) 14) on the other hand. Both inequalities show that large (in dimension) sets must contain complex infinite strings. Conditions under which equality holds in (13) are discussed in [S1, S2] On the other hand, several ....

B. Ya. Ryabko. Noiseless coding of combinatorial sources. Problems of Information Transmission, 22 (1986), 170 -- 179.

....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. Noiseless coding of combinatorial sources. Problems of Information Transmission, 22:170-179, 1986.

....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. Noiseless coding of combinatorial sources. Problems of Information Transmission, 22:170--179, 1986.

....Hausdor s 1919 de nition of dimension [9, 6] but rather on an equivalent formulation in terms of gambling strategies called gales [13] These gales (de ned precisely in section 4 below) give a convenient way of quantifying the discount rate against which a gambling strategy can succeed. Ryabko [16 18] and Staiger [23, 24] have conducted related investigations of classical Hausdor dimension in equivalent terms of the rate at which a gambling strategy can succeed in the absence of discounting. The feasible dimension dim p (X) of a set X of sequences is then de ned in terms of the maximum ....

B. Ya. Ryabko. Noiseless coding of combinatorial sources. Problems of Information Transmission, 22:170-179, 1986.

....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. Noiseless coding of combinatorial sources. Problems of Information Transmission, 22:170-179, 1986.

....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. Noiseless coding of combinatorial sources, Hausdorff dimension, and Kolmogorov complexity. Probl. Inf. Transm., 22:170--179, 1986.

....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. Noiseless coding of combinatorial sources. Problems of Information Transmission, 22:170-179, 1986.

.... one considers arbitrary subsets of A 1 , the entropy H might be interpretated as a form of the box counting dimension, see e.g. Falconer, 1990] The connection between entropy and Hausdor dimension is described in [Billingsley, 1965] and interesting results in this direction can be found in [Ryabko, 1986]. ....

Ryabko, B. Y. Noiseless coding of combinatorial sources, hausdor dimensoin, and kolmogorov complexity. Problems of Inform. Trans., 22(3): 170-179, 1986.

....which is denoted by HD(M ) We de ne the mapping : B 1 [0; 1] associating with each x 1 1 2 B 1 the real number (x 1 1 ) the binary expansion 0:x 1 x 2 . For each I B 1 , we denote as HD(I) the Hausdor dimension of set (I) 3] Then, the following facts are shown in [6] [7]: Proposition 1: 1. M( K( 2. HD(M( and 3. HD(K( In this paper, we prove the following theorem: Theorem 1: HD(K( nM( 2. Proof of Theorem First, we state two lemmas needed for the proof. Lemma 1: 6] 7] 1. A 1 A 2 ) dim (A 1 ) dim (A 2 ) 2. A) 0 ....

....set (I) 3] Then, the following facts are shown in [6] 7] Proposition 1: 1. M( K( 2. HD(M( and 3. HD(K( In this paper, we prove the following theorem: Theorem 1: HD(K( nM( 2. Proof of Theorem First, we state two lemmas needed for the proof. Lemma 1: 6] [7]) 1. A 1 A 2 ) dim (A 1 ) dim (A 2 ) 2. A) 0 ) dim (A) 1; and 3. dim ( n An ) sup n fdim (An )g. Lemma 2: 1] Let and be probability measures. Then, M fx 1 1 j lim inf n 1 log (x n 1 ) log (x n 1 ) g implies dim (M) dim (M ) 2 IEICE ....

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B.Ya. Ryabko,\Noiseless coding of combinatorial sources, Hausdor dimension, and Kolmogorov complexity," Problems of Information Transmission, vol.22, pp.170-179, 1986.

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B. Ya. Ryabko. Noiseless coding of combinatorial sources. Problems of Information Transmission, 22:170-179, 1986.

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B. Ya. Ryabko. Noiseless coding of combinatorial sources. Problems of Information Transmission, 22:170-179, 1986.

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B. Ya. Ryabko. Noiseless coding of combinatorial sources. Problems of Information Transmission, 22:170-179, 1986.

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B. Ya. Ryabko. Noiseless coding of combinatorial sources. Problems of Information Transmission, 22:170--179, 1986.

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B. Ya. Ryabko. Noiseless coding of combinatorial sources. Problems of Information Transmission, 22:170--179, 1986.

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