## The Dimensions of Individual Strings and Sequences (2003)

Venue: | INFORMATION AND COMPUTATION |

Citations: | 92 - 11 self |

### BibTeX

@ARTICLE{Lutz03thedimensions,

author = {Jack H. Lutz},

title = {The Dimensions of Individual Strings and Sequences},

journal = {INFORMATION AND COMPUTATION},

year = {2003},

volume = {187},

pages = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0, 1]. Sequences that

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Citation Context ...sk ) = Eslog ; with the proviso that 0 log 0 p = 0 so that D is continuous on [0; 1] (0; 1). It is well-known that D(sk ) 0, with equality if and only ifs= . See the text by Cover and Thomas [8] for further discussion of H() and D(sk ). Falconer [10] provides a good overview of Hausdor dimension. 6 3 Gales and Constructive Dimension In this section we dene gales and supergales and use th... |

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Citation Context ... bits of an infinite binary sequence S, then Levin [20] 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 =-=[17]-=- developed what is now called C(x), the “plain Kolmogorov complexity,” in order to formulate such a definition of randomness, and Martin-Löf, who was then visiting Kolmogorov, was motivated by this id... |

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Citation Context ... a definition of randomness, and Martin-Löf, who was then visiting Kolmogorov, was motivated by this idea when he defined randomness. (The quantity C(x) was also developed independently by Solomonoff =-=[43]-=- and Chaitin [4, 5].) Martin-Löf [27] subsequently proved that C(x) cannot be used to characterize randomness, and Levin [20] and Chaitin [6] introduced a technical modification of C(x), now called K(... |

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Citation Context ...matic achievements of the theory of computing was Martin-Lof's 1966 use of constructive measure theory to give thesrst satisfactory denition of the randomness of individual innite binary sequences [29=-=-=-]. The search for such a denition had been a major object of early twentieth-century research on the foundations of probability, but a rigorous mathematical formulation had proven so elusive that the ... |

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Citation Context ...ach random sequence to pass every algorithmically implementable statistical test of randomness. The denition is also robust in that subsequent denitions 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 0 [43, 44], using a variety of dierent approaches, all dene exactly the same seq... |

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Citation Context ...en Theorem 4.3 and Observation 4.4 tell us that dim(S) = dim(R) = : Three remarks on the proof of Theorem 4.5 should be made here. First, the proof that RAND\ 0 2 6= ; using Kreisel's Basis Lemma [21=-=, 52, 53-=-, 33] and the fact that RAND is a 0 2 set cannot directly be adapted to proving that DIM \ 0 2 6= ; because Terwijn [51] has shown that DIM is not a 0 2 set. Second, Mayordomo [31] has recently g... |

313 |
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Citation Context ...that D is continuous on [0; 1] (0; 1). It is well-known that D(sk ) 0, with equality if and only ifs= . See the text by Cover and Thomas [8] for further discussion of H() and D(sk ). Falconer [10] provides a good overview of Hausdor dimension. 6 3 Gales and Constructive Dimension In this section we dene gales and supergales and use these to dene classical and constructive Hausdor dimension... |

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Citation Context ...randomness, and Martin-Löf, who was then visiting Kolmogorov, was motivated by this idea when he defined randomness. (The quantity C(x) was also developed independently by Solomonoff [43] and Chaitin =-=[4, 5]-=-.) Martin-Löf [27] subsequently proved that C(x) cannot be used to characterize randomness, and Levin [20] and Chaitin [6] introduced a technical modification of C(x), now called K(x), the “Kolmogorov... |

211 |
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Citation Context ...and 0 2 sets of sequences. These refer to the arithmetical (i.e., eective Borel) hierarchy of sets of sequences and are not central to our development. The interested reader is referred to [34] or [3=-=2-=-] for discussion of this hierarchy. The support of a sequence S 2 C is supp(S) = fn 2 NjS[n] = 1g: The arithmetical hierarchy of sequences is dened from the arithmetical hierarchy of subsets of N usin... |

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Citation Context ...at for every f ∈ F there is a real constant α > 0 such that for all w ∈ {0,1} ∗ , g(w) ≥ αf(w). The following theorem is one of the cornerstones of algorithmic information theory. Theorem 2.1. (Levin =-=[52]-=-) 1. There is an optimal constructive subprobability measure m on {0,1} ∗ . 2. There is an optimal constructive subprobability supermeasure M on C. Throughout this paper we fix m and M as in Theorem 2... |

157 |
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Citation Context ...y convincing in that it requires each random sequence to pass every algorithmically implementable statistical test of randomness. The denition is also robust in that subsequent denitions by Schnorr [3=-=9, 40, 4-=-1], Levin [22], Chaitin [6], Solovay [47], This work was supported in part by National Science Foundation Grants 9610461 and 9988483. and Shen 0 [43, 44], using a variety of dierent approaches, all d... |

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Citation Context ... it requires each random sequence to pass every algorithmically implementable statistical test of randomness. The denition is also robust in that subsequent denitions 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 0 [43, 44], using a variety of dierent approaches, all dene exactly... |

105 | Dimension in complexity classes
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Citation Context ...ent of constructive dimension is based on supergales, which are natural generalizations of the constructive supermartingales used by Schnorr [39, 40, 41] to characterize randomness. In a recent paper =-=[2-=-7] we have shown that supergales can be used to characterize the classical Hausdor dimension, and that resource-bounded supergales can be used to dene dimension in complexity classes. In the present p... |

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Citation Context ...convincing in that it requires each random sequence to pass every algorithmically implementable statistical test of randomness. The definition is also robust in that subsequent definitions by Schnorr =-=[36, 37, 38]-=-, Levin [20], Chaitin [6], Solovay [44], ∗ This work was supported in part by National Science Foundation Grants 9610461 and 9988483.and Shen ′ [40, 41], using a variety of different approaches, all ... |

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Citation Context ... 0 2 , and 0 2 sets of sequences. These refer to the arithmetical (i.e., eective Borel) hierarchy of sets of sequences and are not central to our development. The interested reader is referred to [34=-=-=-] or [32] for discussion of this hierarchy. The support of a sequence S 2 C is supp(S) = fn 2 NjS[n] = 1g: The arithmetical hierarchy of sequences is dened from the arithmetical hierarchy of subsets o... |

71 |
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Citation Context ... w 2 f0; 1g , K(w) log 1 m(w) c 0 ; (1.3) where m is an optimal constructive subprobability measure on f0; 1g . Taken together, (1.1) and (1.2) provide a new proof of Mayordomo's recent theorem [31=-=]-=- stating that for every sequence S 2 C, dim(S) = lim inf n!1 K(S[0::n 1]) n : (1.4) Facts (1.2) and (1.4) justify the intuition that the dimension of a string or sequence is a measure of its algorithm... |

67 | Kolmogorov complexity and Hausdorff dimension
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Citation Context ... dimH(C) = 1. Moreover, if dimH(X) < dimH(C), then X is a measure 0 subset of C. Hausdorff dimension thus offers a quantitative classification of measure 0 sets. Moreover, Ryabko [33, 34, 35] Staiger =-=[46, 47]-=-, and Cai and Hartmanis [3] have all proven results establishing quantitative relationships between Hausdorff dimension and Kolmogorov complexity. Just as Hausdorff [13] augmented Lebesgue measure wit... |

63 | Random sequences
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Citation Context ... The following known result shows that constructive supermartingales can equivalently be used in place of constructive martingales in dening randomness. Theorem 3.5. (Schnorr [39, 40], van Lambalgen [52]) For every computable probability measure on C and every constructive -supermartingale d there is a constructive -martingale d 0 such that S 1 [d] S 1 [d 0 ]. If is , the uniform probability... |

49 |
Coding of combinatorial sources and Hausdorff dimension. Soviet Mathematics Doklady
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Citation Context ...H(β), the binary Shannon entropy of β. We defer discussion of some significant related work until late in the paper, where more context is available. Specifically, results by Schnorr [37, 39], Ryabko =-=[32, 33, 34, 35]-=-, Staiger [46, 47, 45], and Cai and Hartmanis [3] that relate martingales, supermartingales, and Kolmogorov complexity to Hausdorff dimension are discussed at the end of section 6. Classical work by B... |

47 |
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Citation Context ...2 slowly enough that P 1 i=0 (si 1 2 ) 2 = 1, this allowed Shen 0 to conclude that not every Kolmogorov-Loveland stochastic sequence is random, thereby solving a twenty-year-old problem of Kolmogorov =-=[18, 20]-=- and Loveland [25, 26]. Theorems 7.7 and 7.2 have the following immediate consequence concerning such sequences !s. Corollary 7.8. If !sis a computable sequence of biases that converge to 1 2 slowly e... |

45 |
The fractional dimension of a set defined by decimal properties
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(Show Context)
Citation Context ...anis [3] that relate martingales, supermartingales, and Kolmogorov complexity to Hausdorff dimension are discussed at the end of section 6. Classical work by Besicovitch [1], Good [12], and Eggleston =-=[9]-=- relating limiting frequencies and Shannon entropy to Hausdorff dimension is described briefly in section 7. 32 Preliminaries We use the set Z of integers, the set Z + of (strictly) positive integers... |

43 | A tight upper bound on Kolmogorov complexity and uniformly optimal prediction
- Staiger
- 1998
(Show Context)
Citation Context ...nd dimH (C) = 1. Moreover, if dimH (X)sdimH (C), then X is a measure 0 subset of C. Hausdor dimension thus oers a quantitative classication of measure 0 sets. Moreover, Ryabko [36, 37, 38] Staiger [48=-=, 4-=-9], 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 ... |

42 |
Gales and the constructive dimension of individual sequences
- Lutz
(Show Context)
Citation Context ...eorem 5.9. It should be noted here that Mayordomo proved Theorem 6.3 to improve the weaker result lim inf n!1 K(S[0::n 1]) n dim(S) lim sup n!1 K(S[0::n 1]) n ; which appeared in an early version [2=-=8]-=- of the present paper that lacked (among other things) section 5 and the foregoing part of section 6. As noted in the discussion at the end of this section, Mayordomo's theorem can, in turn, be used i... |

42 |
On equivalence of infinite product measures
- Kakutani
- 1948
(Show Context)
Citation Context ...uences of biases β and β ′ are square-summably equivalent, and we write β ≈ 2 β ′ , if ∑ ∞ i=0 (βi − β ′ i )2 < ∞. The next theorem is a constructive version of a classical theorem of Kakutani =-=[15]-=-. Theorem 7.2. (van Lambalgen [49, 50], Vovk [51]) Let β and β ′ be computable sequences of biases that converge to β ∈ (0,1). 1. If β ≈ 2 β ′ , then RAND β = RAND β ′. 2. If β ≈ 2 β ′ ... |

39 |
On the Length of Programs for Computing Binary Sequences
- Chaitin
- 1966
(Show Context)
Citation Context ... randomness, and Martin-Lof, who was then visiting Kolmogorov, was motivated by this idea when he dened randomness. (The quantity C(x) was also developed independently by Solomono [46] and Chaitin [4,=-= -=-5].) Martin-Lof [30] subsequently proved that C(x) cannot be used to characterize randomness, and Levin [22] and Chaitin [6] introduced a technical modication of C(x), now called K(x), the \Kolmogorov... |

39 | Incompleteness theorems for random reals - Chaitin - 1987 |

36 |
On Hausdorff and topological dimensions of the Kolmogorov complexity of the real line
- Cai, Hartmanis
- 1994
(Show Context)
Citation Context ...mH (X)sdimH (C), then X is a measure 0 subset of C. Hausdor dimension thus oers a quantitative classication 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... |

34 |
A formal theory of inductive inference
- Solomono
(Show Context)
Citation Context ...ch a denition of randomness, and Martin-Lof, who was then visiting Kolmogorov, was motivated by this idea when he dened randomness. (The quantity C(x) was also developed independently by Solomono [46] and Chaitin [4, 5].) Martin-Lof [30] subsequently proved that C(x) cannot be used to characterize randomness, and Levin [22] and Chaitin [6] introduced a technical modication of C(x), now called K(... |

31 |
On tables of random numbers
- Kolmogorov
- 1963
(Show Context)
Citation Context ...2 slowly enough that P 1 i=0 (si 1 2 ) 2 = 1, this allowed Shen 0 to conclude that not every Kolmogorov-Loveland stochastic sequence is random, thereby solving a twenty-year-old problem of Kolmogorov =-=[18, 20]-=- and Loveland [25, 26]. Theorems 7.7 and 7.2 have the following immediate consequence concerning such sequences !s. Corollary 7.8. If !sis a computable sequence of biases that converge to 1 2 slowly e... |

31 |
Dimension und ausseres mass
- Hausdorff
- 1919
(Show Context)
Citation Context ...uctive measure 0, and is nonrandom if {S} does have constructive measure 0. Neither Lebesgue measure nor constructive measure offers quantitative distinctions among measure 0 sets. In 1919, Hausdorff =-=[13]-=- augmented classical Lebesgue measure theory with a theory of dimension. This theory assigns to every subset X of a given metric space a real number dimH(X), which is now called the Hausdorff dimensio... |

26 |
Complexity oscillations in infinite binary sequences. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete
- Martin-Löf
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Citation Context ...in-Löf, who was then visiting Kolmogorov, was motivated by this idea when he defined randomness. (The quantity C(x) was also developed independently by Solomonoff [43] and Chaitin [4, 5].) Martin-Löf =-=[27]-=- subsequently proved that C(x) cannot be used to characterize randomness, and Levin [20] and Chaitin [6] introduced a technical modification of C(x), now called K(x), the “Kolmogorov complexity,” in o... |

22 |
A survey of the theory of random sequences
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- 1977
(Show Context)
Citation Context ...e dimension of R is H(), the binary Shannon entropy ofs. We defer discussion of some signicant related work until late in the paper, where more context is available. Specically, 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 sectio... |

20 |
A new interpretation of the von Mises' concept of random sequence. Zeitschr. f. math. Logik und Grundlagen d
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- 1966
(Show Context)
Citation Context ... 1 i=0 (si 1 2 ) 2 = 1, this allowed Shen 0 to conclude that not every Kolmogorov-Loveland stochastic sequence is random, thereby solving a twenty-year-old problem of Kolmogorov [18, 20] and Loveland =-=[25, 26]-=-. Theorems 7.7 and 7.2 have the following immediate consequence concerning such sequences !s. Corollary 7.8. If !sis a computable sequence of biases that converge to 1 2 slowly enough that P 1 i=0 (si... |

19 | Correspondence principles for effective dimensions. Theory of Computing Systems. To appear. Preliminary version appeared
- Hitchcock
- 2002
(Show Context)
Citation Context ...that succeeds on S. By Theorem 3.6, then, we have S ∈ S ∞ [dS] ⊆ S ∞ [d (s) ] for all S ∈ X, whence X ⊆ S ∞ [d (s) ]. Since d (s) is a constructive s-supergale, this shows that cdim(X) ≤ s. Hitchcock =-=[14]-=- has recently proven a correspondence principle for constructive dimension. This sets (a condition that is certainly ), the constructive dimension of X is precisely its classical Hausdorff dimension. ... |

18 |
On the sum of digits of real numbers represented in the dyadic system, Mathematische Annalen 110
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- 1935
(Show Context)
Citation Context ...ed in the theory of individual random sequences. This constructive dimension is used to assign every individual (innite, binary) sequence S a dimension, which is a real number dim(S) in the interval [=-=0-=-; 1]. Sequences that are random (in the sense of Martin-Lof) have dimension 1, while sequences that are decidable, 0 1 , or 0 1 have dimension 0. It is shown that for every 0 2 -computable real num... |

18 |
On relations between different algorithmic definitions of randomness
- Shen
- 1989
(Show Context)
Citation Context ... in that subsequent definitions by Schnorr [36, 37, 38], Levin [20], Chaitin [6], Solovay [44], ∗ This work was supported in part by National Science Foundation Grants 9610461 and 9988483.and Shen ′ =-=[40, 41]-=-, using a variety of different approaches, all define exactly the same sequences to be random. It is noteworthy that all these approaches, like Martin-Löf’s, make essential use of the theory of comput... |

17 |
A note on arithmetic models for consistent formulae of the predicate calculus
- Kreisel
- 1950
(Show Context)
Citation Context ...en Theorem 4.3 and Observation 4.4 tell us that dim(S) = dim(R) = : Three remarks on the proof of Theorem 4.5 should be made here. First, the proof that RAND\ 0 2 6= ; using Kreisel's Basis Lemma [21=-=, 52, 53-=-, 33] and the fact that RAND is a 0 2 set cannot directly be adapted to proving that DIM \ 0 2 6= ; because Terwijn [51] has shown that DIM is not a 0 2 set. Second, Mayordomo [31] has recently g... |

16 |
Algorithmic approach to the prediction problem
- Ryabko
- 1993
(Show Context)
Citation Context ..., with dimH (;) = 0 and dimH (C) = 1. Moreover, if dimH (X)sdimH (C), then X is a measure 0 subset of C. Hausdor dimension thus oers a quantitative classication of measure 0 sets. Moreover, Ryabko [36=-=, 37, 3-=-8] 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 Lebes... |

15 |
Three approaches to the quantitative de of information. Problems of Information Transmission
- Kolmogorov
- 1965
(Show Context)
Citation Context ...rst n bits of an innite 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 [1=-=9] de-=-veloped what is now called C(x), the \plain Kolmogorov complexity," in order to formulate such a denition of randomness, and Martin-Lof, who was then visiting Kolmogorov, was motivated by this id... |

15 |
Noiseless coding of combinatorial sources
- Ryabko
- 1986
(Show Context)
Citation Context ... with dimH(∅) = 0 and dimH(C) = 1. Moreover, if dimH(X) < dimH(C), then X is a measure 0 subset of C. Hausdorff dimension thus offers a quantitative classification of measure 0 sets. Moreover, Ryabko =-=[33, 34, 35]-=- Staiger [46, 47], and Cai and Hartmanis [3] have all proven results establishing quantitative relationships between Hausdorff dimension and Kolmogorov complexity. Just as Hausdorff [13] augmented Leb... |

15 |
The complexity and effectiveness of prediction problems
- Ryabko
- 1994
(Show Context)
Citation Context ... with dimH(∅) = 0 and dimH(C) = 1. Moreover, if dimH(X) < dimH(C), then X is a measure 0 subset of C. Hausdorff dimension thus offers a quantitative classification of measure 0 sets. Moreover, Ryabko =-=[33, 34, 35]-=- Staiger [46, 47], and Cai and Hartmanis [3] have all proven results establishing quantitative relationships between Hausdorff dimension and Kolmogorov complexity. Just as Hausdorff [13] augmented Leb... |

15 | Von Mises’ definition of random sequences reconsidered
- Lambalgen
- 1987
(Show Context)
Citation Context ... easy to see that α has an approximator (indeed, this characterizes ∆0 2-computability), and it is routine to transform an approximator of α into a nice approximator (a,b) of α. It is well known (see =-=[49, 50]-=- or [22]) that there is a sequence R ∈ RAND ∩ ∆0 2 . Let S = g (a,b)(R). Then Theorem 4.3 and Observation 4.4 tell us that dim(S) = α dim(R) = α. Three remarks on the proof of Theorem 4.5 should be ma... |

14 |
On a randomness criterion
- Vovk
- 1987
(Show Context)
Citation Context ...-summably equivalent, and we write ~s 2 ~s0 , if P 1 i=0 (sis0 i ) 2s1. The next theorem is a constructive version of a classical theorem of Kakutani [17]. Theorem 7.2. (van Lambalgen [52, 53], Vovk [=-=5-=-4]) Let ~sand ~ 0 be computable sequences of biases that converge tos2 (0; 1). 1. If ~s 2 ~ 0 , then RAND ~s= RAND ~s0 . 2. If ~s6 2 ~ 0 , then RAND ~s\ RAND ~s0 = ;. It is well-known (and easy to see... |