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24
The Dimensions of Individual Strings and Sequences
 INFORMATION AND COMPUTATION
, 2003
"... 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 ..."
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Cited by 93 (10 self)
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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
A generalization of Chaitin’s halting probability Ω and halting selfsimilar sets
 Hokkaido Math. J
, 2002
"... We generalize the concept of randomness in an infinite binary sequence in order to characterize the degree of randomness by a real number D> 0. Chaitin’s halting probability Ω is generalized to Ω D whose degree of randomness is precisely D. On the basis of this generalization, we consider the degree ..."
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Cited by 34 (12 self)
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We generalize the concept of randomness in an infinite binary sequence in order to characterize the degree of randomness by a real number D> 0. Chaitin’s halting probability Ω is generalized to Ω D whose degree of randomness is precisely D. On the basis of this generalization, we consider the degree of randomness of each point in Euclidean space through its basetwo expansion. It is then shown that the maximum value of such a degree of randomness provides the Hausdorff dimension of a selfsimilar set that is computable in a certain sense. The class of such selfsimilar sets includes familiar fractal sets such as the Cantor set, von Koch curve, and Sierpiński gasket. Knowledge of the property of Ω D allows us to show that the selfsimilar subset of [0,1] defined by the halting set of a universal algorithm has a Hausdorff dimension of one.
Relative to a random oracle, NP is not small
 In Proc. 9th Structures
, 1994
"... Resourcebounded measure as originated by Lutz is an extension of classical measure theory which provides a probabilistic means of describing the relative sizes of complexity classes. Lutz has proposed the hypothesis that NP does not have pmeasure zero, meaning loosely that NP contains a nonneglig ..."
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Cited by 18 (1 self)
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Resourcebounded measure as originated by Lutz is an extension of classical measure theory which provides a probabilistic means of describing the relative sizes of complexity classes. Lutz has proposed the hypothesis that NP does not have pmeasure zero, meaning loosely that NP contains a nonnegligible subset of exponential time. This hypothesis implies a strong separation of P from NP and is supported by a growing body of plausible consequences which are not known to follow from the weaker assertion P ̸ = NP. It is shown in this paper that relative to a random oracle, NP does not have pmeasure zero. The proof exploits the following independence property of algorithmically random sequences: if A is an algorithmically random sequence and a subsequence A0 is chosen by means of a bounded KolmogorovLoveland
A Pseudorandom Oracle Characterization of BPP
, 1993
"... . It is known from work of Bennett and Gill and AmbosSpies that the following conditions are equivalent. (i) L 2 BPP. (ii) For almost all oracles A, L 2 P A . It is shown here that the following conditions are also equivalent to (i) and (ii). (iii) The set of oracles A for which L 2 P A has psp ..."
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Cited by 13 (5 self)
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. It is known from work of Bennett and Gill and AmbosSpies that the following conditions are equivalent. (i) L 2 BPP. (ii) For almost all oracles A, L 2 P A . It is shown here that the following conditions are also equivalent to (i) and (ii). (iii) The set of oracles A for which L 2 P A has pspacemeasure 1. (iv) For every pspacerandom oracle A, L 2 P A . It follows from this characterization (and its proof) that almost every A 2 ESPACE is P T hard for BPP A . Succinctly, the main content of the proof is that pseudorandom generators exist relative to every pseudorandom oracle. 1. Introduction. The class BPP consists of those decision problems that are feasibly solvable by randomized algorithms. This class, defined by Gill [9], has been shown to admit a variety of equivalent definitions [2, 11, 32, 1, 31, 13, 30, 12, 28, 27]. A particularly elegant and useful characterization of BPP is the following. Theorem 1 (Bennett and Gill [2], AmbosSpies [1]). For a language L ` f0...
Randomness, computability, and density
 SIAM Journal of Computation
, 2002
"... 1 Introduction In this paper we are concerned with effectively generated reals in the interval (0; 1] and their relative randomness. In what follows, real and rational will mean positive real and positive rational, respectively. It will be convenient to work modulo 1, that is, identifying n + ff and ..."
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Cited by 13 (6 self)
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1 Introduction In this paper we are concerned with effectively generated reals in the interval (0; 1] and their relative randomness. In what follows, real and rational will mean positive real and positive rational, respectively. It will be convenient to work modulo 1, that is, identifying n + ff and ff for any n 2! and ff 2 (0; 1], and we do this below without further comment.
Computational depth and reducibility
 Theoretical Computer Science
, 1994
"... This paper reviews and investigates Bennett's notions of strong and weak computational depth (also called logical depth) for in nite binary sequences. Roughly, an in nite binary sequence x is de ned to be weakly useful if every element of a nonnegligible set of decidable sequences is reducible to x ..."
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Cited by 12 (2 self)
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This paper reviews and investigates Bennett's notions of strong and weak computational depth (also called logical depth) for in nite binary sequences. Roughly, an in nite binary sequence x is de ned to be weakly useful if every element of a nonnegligible set of decidable sequences is reducible to x in recursively bounded time. It is shown that every weakly useful sequence is strongly deep. This result (which generalizes Bennett's observation that the halting problem is strongly deep) implies that every high Turing degree contains strongly deep sequences. It is also shown that, in the sense of Baire category, almost
Is Complexity a Source of Incompleteness?
 IS COMPLEXITY A SOURCE OF INCOMPLETENESS
, 2004
"... ..."
The Global Power of Additional Queries to Random Oracles
"... . It is shown that, for every k 0 and every fixed algorithmically random language B, there is a language that is polynomialtime, truthtable reducible in k + 1 queries to B but not truthtable reducible in k queries in any amount of time to any algorithmically random language C. In particular, this ..."
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Cited by 8 (1 self)
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. It is shown that, for every k 0 and every fixed algorithmically random language B, there is a language that is polynomialtime, truthtable reducible in k + 1 queries to B but not truthtable reducible in k queries in any amount of time to any algorithmically random language C. In particular, this yields the separation Pktt(RAND) $ P (k+1)tt (RAND), where RAND is the set of all algorithmically random languages. 1 Introduction Will an algorithm have increased computational power when it is modified to be able to ask additional questions? One way of making this question precise is to consider it in the context of reducibilities computed by algorithms with bounds on their computational resources. In this paper, we investigate the phenomenon of increased access to oracle sets lending increased computational power for bounded truthtable reducibilities computed in polynomial time. We show that, in a strong global sense, if just one more question can be asked of sets with "maximum info...
KOLMOGOROV COMPLEXITY AND SOLOVAY FUNCTIONS
, 2009
"... Solovay [19] proved that there exists a computable upper bound f of the prefixfree Kolmogorov complexity function K such that f(x) = K(x) for infinitely many x. In this paper, we consider the class of computable functions f such that K(x) ≤ f(x)+O(1) for all x and f(x) ≤ K(x) + O(1) for infinit ..."
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Cited by 8 (3 self)
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Solovay [19] proved that there exists a computable upper bound f of the prefixfree Kolmogorov complexity function K such that f(x) = K(x) for infinitely many x. In this paper, we consider the class of computable functions f such that K(x) ≤ f(x)+O(1) for all x and f(x) ≤ K(x) + O(1) for infinitely many x, which we call Solovay functions. We show that Solovay functions present interesting connections with randomness notions such as MartinLöf randomness and Ktriviality.