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12
Algorithmic information theory
 IBM JOURNAL OF RESEARCH AND DEVELOPMENT
, 1977
"... This paper reviews algorithmic information theory, which is an attempt to apply informationtheoretic and probabilistic ideas to recursive function theory. Typical concerns in this approach are, for example, the number of bits of information required to specify an algorithm, or the probability that ..."
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Cited by 320 (19 self)
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This paper reviews algorithmic information theory, which is an attempt to apply informationtheoretic and probabilistic ideas to recursive function theory. Typical concerns in this approach are, for example, the number of bits of information required to specify an algorithm, or the probability that a program whose bits are chosen by coin flipping produces a given output. During the past few years the definitions of algorithmic information theory have been reformulated. The basic features of the new formalism are presented here and certain results of R. M. Solovay are reported.
Degrees of random sets
, 1991
"... An explicit recursiontheoretic definition of a random sequence or random set of natural numbers was given by MartinLöf in 1966. Other approaches leading to the notions of nrandomness and weak nrandomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of nrando ..."
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Cited by 46 (4 self)
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An explicit recursiontheoretic definition of a random sequence or random set of natural numbers was given by MartinLöf in 1966. Other approaches leading to the notions of nrandomness and weak nrandomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of nrandom and weakly nrandom sequences with an emphasis on the structure of their Turing degrees. After an introduction and summary, in Chapter II we present several equivalent definitions of nrandomness and weak nrandomness including a new definition in terms of a forcing relation analogous to the characterization of ngeneric sequences in terms of Cohen forcing. We also prove that, as conjectured by Kurtz, weak nrandomness is indeed strictly weaker than nrandomness. Chapter III is concerned with intrinsic properties of nrandom sequences. The main results are that an (n + 1)random sequence A satisfies the condition A (n) ≡T A⊕0 (n) (strengthening a result due originally to Sacks) and that nrandom sequences satisfy a number of strong independence properties, e.g., if A ⊕ B is nrandom then A is nrandom relative to B. It follows that any countable distributive lattice can be embedded
Informationtheoretic Limitations of Formal Systems
 JOURNAL OF THE ACM
, 1974
"... An attempt is made to apply informationtheoretic computational complexity to metamathematics. The paper studies the number of bits of instructions that must be a given to a computer for it to perform finite and infinite tasks, and also the amount of time that it takes the computer to perform these ..."
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Cited by 45 (7 self)
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An attempt is made to apply informationtheoretic computational complexity to metamathematics. The paper studies the number of bits of instructions that must be a given to a computer for it to perform finite and infinite tasks, and also the amount of time that it takes the computer to perform these tasks. This is applied to measuring the difficulty of proving a given set of theorems, in terms of the number of bits of axioms that are assumed, and the size of the proofs needed to deduce the theorems from the axioms.
Basic Elements and Problems of Probability Theory
, 1999
"... After a brief review of ontic and epistemic descriptions, and of subjective, logical and statistical interpretations of probability, we summarize the traditional axiomatization of calculus of probability in terms of Boolean algebras and its settheoretical realization in terms of Kolmogorov probabil ..."
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Cited by 6 (0 self)
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After a brief review of ontic and epistemic descriptions, and of subjective, logical and statistical interpretations of probability, we summarize the traditional axiomatization of calculus of probability in terms of Boolean algebras and its settheoretical realization in terms of Kolmogorov probability spaces. Since the axioms of mathematical probability theory say nothing about the conceptual meaning of “randomness” one considers probability as property of the generating conditions of a process so that one can relate randomness with predictability (or retrodictability). In the measuretheoretical codification of stochastic processes genuine chance processes can be defined rigorously as socalled regular processes which do not allow a longterm prediction. We stress that stochastic processes are equivalence classes of individual point functions so that they do not refer to individual processes but only to an ensemble of statistically equivalent individual processes. Less popular but conceptually more important than statistical descriptions are individual descriptions which refer to individual chaotic processes. First, we review the individual description based on the generalized harmonic analysis by Norbert Wiener. It allows the definition of individual purely chaotic processes which can be interpreted as trajectories of regular statistical stochastic processes. Another individual description refers to algorithmic procedures which connect the intrinsic randomness of a finite sequence with the complexity of the shortest program necessary to produce the sequence. Finally, we ask why there can be laws of chance. We argue that random events fulfill the laws of chance if and only if they can be reduced to (possibly hidden) deterministic events. This mathematical result may elucidate the fact that not all nonpredictable events can be grasped by the methods of mathematical probability theory.
HIGHER RANDOMNESS NOTIONS AND THEIR LOWNESS PROPERTIES
, 2007
"... Abstract. We study randomness notions given by higher recursion theory, establishing the relationships Π 1 1randomness ⊂ Π 1 1MartinLöf randomness ⊂ ∆ 1 1randomness = ∆ 1 1MartinLöf randomness. We characterize the set of reals that are low for ∆ 1 1 randomness as precisely those that are ∆ 1 ..."
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Cited by 2 (2 self)
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Abstract. We study randomness notions given by higher recursion theory, establishing the relationships Π 1 1randomness ⊂ Π 1 1MartinLöf randomness ⊂ ∆ 1 1randomness = ∆ 1 1MartinLöf randomness. We characterize the set of reals that are low for ∆ 1 1 randomness as precisely those that are ∆ 1 1traceable. We prove that there is a perfect set of such reals. 1.
Interactions of Computability and Randomness
"... We survey results relating the computability and randomness aspects of sets of natural numbers. Each aspect corresponds to several mathematical properties. Properties originally defined in very different ways are shown to coincide. For instance, lowness for MLrandomness is equivalent to Ktrivialit ..."
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Cited by 2 (0 self)
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We survey results relating the computability and randomness aspects of sets of natural numbers. Each aspect corresponds to several mathematical properties. Properties originally defined in very different ways are shown to coincide. For instance, lowness for MLrandomness is equivalent to Ktriviality. We include some interactions of randomness with computable analysis. Mathematics Subject Classification (2010). 03D15, 03D32. Keywords. Algorithmic randomness, lowness property, Ktriviality, cost function.
RANDOMNESS VIA EFFECTIVE DESCRIPTIVE SET THEORY
"... An analog of MLrandomness in the effective descriptive set theory setting is studied, where the r.e. objects are replaced by their Π1 1 counterparts. We prove the analogs of the KraftChaitin Theorem and Schnorr’s Theorem. In the new setting, while Ktrivial sets exist that are not hyperarithmetica ..."
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Cited by 1 (1 self)
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An analog of MLrandomness in the effective descriptive set theory setting is studied, where the r.e. objects are replaced by their Π1 1 counterparts. We prove the analogs of the KraftChaitin Theorem and Schnorr’s Theorem. In the new setting, while Ktrivial sets exist that are not hyperarithmetical, each low for random set is. Finally, we begin to study a very strong yet effective randomness notion: Z is Π1 1 random if Z is in no null Π1 1 class. There is a greatest Π1 1 null class, that is, a universal test for this notion.
A Concept Of Independence With Applications In Various Fields Of Mathematics
, 1980
"... We use Kohnogorov's algorithmic approach to information theory to define a concept of independence of sequences, or equivalently, the boundedness of their mutual information. This concept is applied to probability theory, intuitionistic logic, and the theory of algorithms. For each case, we study th ..."
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We use Kohnogorov's algorithmic approach to information theory to define a concept of independence of sequences, or equivalently, the boundedness of their mutual information. This concept is applied to probability theory, intuitionistic logic, and the theory of algorithms. For each case, we study the advantage of accepting the postulate that the objects studied by the theory are independent of any sequence deterinined by a nathematical property.
Centre for Discrete Mathematics and
, 2008
"... Abstract. A real x is ∆ 1 1Kurtz random (Π 1 1Kurtz random) if it in no closed null ∆ 1 1 set (Π 1 1 set). We show that there is a cone of Π 1 1Kurtz random hyperdegrees. We characterize lowness for ∆ 1 1Kurtz randomness by being ∆ 1 1dominated and ∆ 1 1semitraceable. 1. ..."
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Abstract. A real x is ∆ 1 1Kurtz random (Π 1 1Kurtz random) if it in no closed null ∆ 1 1 set (Π 1 1 set). We show that there is a cone of Π 1 1Kurtz random hyperdegrees. We characterize lowness for ∆ 1 1Kurtz randomness by being ∆ 1 1dominated and ∆ 1 1semitraceable. 1.