Recursively enumerable reals and Chaitin &Ω numbers
TLDR
It is shown that the converse implication is true: any Ω-like real in the unit interval is the halting probability of a universal self-delimiting Turing machine.About:
This article is published in Theoretical Computer Science.The article was published on 2001-03-28 and is currently open access. It has received 82 citations till now. The article focuses on the topics: Chaitin's constant & Algorithmic information theory.read more
Citations
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Gödel's Theorem: An Incomplete Guide to Its Use and Abuse
TL;DR: In this article, Franzen gives careful, non-technical explanations both of what those incompleteness theorems say and what they do not No other book aims, as his does, to address in detail the misunderstandings and abuses of the incomplethetenesstheorems that are so rife in popular discussions of their significance.
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Hierarchies of generalized kolmogorov complexities and nonenumerable universal measures computable in the limit
TL;DR: A natural hierarchy of generalizations of algorithmic probability and Kolmogorov complexity is obtained, suggesting that the "true" information content of some bitstring x is the size of the shortest nonhalting program that converges to x and nothing but x on a Turing machine that can edit its previous outputs.
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Kolmogorov Complexity and Algorithmic Randomness
TL;DR: HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not, for teaching and research institutions in France or abroad, or from public or private research centers.
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Relativizing chaitin's halting probability
TL;DR: A comparison is drawn between the jump operator from computability theory and this Omega operator, which is a natural uniform way of producing an A-random real for every A ∈ 2ω, and many other interesting properties of Omega operators.
References
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Recursively enumerable sets and degrees
TL;DR: In this paper, the relation of the structure of an R set to its degree is discussed, and the infinite injury priority method is proposed to solve the problem of scaling and splitting R sets.
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A Formal Theory of Inductive Inference. Part II
TL;DR: Four ostensibly different theoretical models of induction are presented, in which the problem dealt with is the extrapolation of a very long sequence of symbols—presumably containing all of the information to be used in the induction.
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The definition of random sequences
TL;DR: It is shown that the random elements as defined by Kolmogorov possess all conceivable statistical properties of randomness and can equivalently be considered as the elements which withstand a certain universal stochasticity test.
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On the Length of Programs for Computing Finite Binary Sequences
TL;DR: An application to the problem of defining a patternless sequence is proposed in terms of the concepts here developed to study the use of Turing machines for calculating finite binary sequences.
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Algorithmic Information Theory
TL;DR: This paper reviews algorithmic information theory, which is an attempt to apply information-theoretic and probabilistic ideas to recursive function theory.