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22
Randomness, relativization, and Turing degrees
 J. Symbolic Logic
, 2005
"... We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is nrandom if it is MartinLof random relative to . We show that a set is 2random if and only if there is a constant c such that infinitely many initial segments x of the set are cincompre ..."
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Cited by 38 (16 self)
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We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is nrandom if it is MartinLof random relative to . We show that a set is 2random if and only if there is a constant c such that infinitely many initial segments x of the set are cincompressible: C(x) c. The `only if' direction was obtained independently by Joseph Miller. This characterization can be extended to the case of timebounded Ccomplexity.
Recursively Enumerable Reals and Chaitin Ω Numbers
"... A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from b ..."
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Cited by 34 (3 self)
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A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's [23]like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability ofa universal selfdelimiting Turing machine (Chaitin's Ω number, [9]) is also a random r.e. real. Solovay showed that any Chaitin Ω number islike. In this paper we show that the converse implication is true as well: any Ωlike real in the unit interval is the halting probability of a universal selfdelimiting Turing machine.
Randomness in Computability Theory
, 2000
"... We discuss some aspects of algorithmic randomness and state some open problems in this area. The first part is devoted to the question "What is a computably random sequence?" Here we survey some of the approaches to algorithmic randomness and address some questions on these concepts. In the seco ..."
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Cited by 28 (0 self)
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We discuss some aspects of algorithmic randomness and state some open problems in this area. The first part is devoted to the question "What is a computably random sequence?" Here we survey some of the approaches to algorithmic randomness and address some questions on these concepts. In the second part we look at the Turing degrees of MartinLof random sets. Finally, in the third part we deal with relativized randomness. Here we look at oracles which do not change randomness. 1980 Mathematics Subject Classification. Primary 03D80; Secondary 03D28. 1 Introduction Formalizations of the intuitive notions of computability and randomness are among the major achievements in the foundations of mathematics in the 20th century. It is commonly accepted that various equivalent formal computability notions  like Turing computability or recursiveness  which were introduced in the 1930s and 1940s adequately capture computability in the intuitive sense. This belief is expressed in the w...
Lowness properties and approximations of the jump
 Proceedings of the Twelfth Workshop of Logic, Language, Information and Computation (WoLLIC 2005). Electronic Lecture Notes in Theoretical Computer Science 143
, 2006
"... ..."
Recursive computational depth
 Information and Computation
, 1999
"... In the 1980's, Bennett introduced computational depth as a formal measure of the amount of computational history that is evident in an object's structure. In particular, Bennett identi ed the classes of weakly deep and strongly deep sequences, and showed that the halting problem is strongly deep. Ju ..."
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Cited by 18 (2 self)
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In the 1980's, Bennett introduced computational depth as a formal measure of the amount of computational history that is evident in an object's structure. In particular, Bennett identi ed the classes of weakly deep and strongly deep sequences, and showed that the halting problem is strongly deep. Juedes, Lathrop, and Lutz subsequently extended this result by de ning the class of weakly useful sequences, and proving that every weakly useful sequence is strongly deep. The present paper investigates re nements of Bennett's notions of weak and strong depth, called recursively weak depth (introduced by Fenner, Lutz and Mayordomo) and recursively strong depth (introduced here). It is argued that these re nements naturally capture Bennett's idea that deep objects are those which \contain internal evidence of a nontrivial causal history. " The fundamental properties of recursive computational depth are developed, and it is shown that the recursively weakly (respectively, strongly) deep sequences form a proper subclass of the class of weakly (respectively, strongly) deep sequences. The abovementioned theorem of Juedes, Lathrop, and Lutz is then strengthened by proving that every weakly useful sequence is recursively strongly deep. It follows from these results that not every strongly deep sequence is weakly useful, thereby answering a question posed by Juedes.
On Schnorr and computable randomness, martingales, and machines
 Mathematical Logic Quarterly
, 2004
"... examine the randomness and triviality of reals using notions arising from martingales and prefixfree machines. ..."
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Cited by 10 (6 self)
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examine the randomness and triviality of reals using notions arising from martingales and prefixfree machines.
On the Existence and Convergence of Computable Universal Priors
 In Proc. 14th International Conf. on Algorithmic Learning Theory (ALT2003), volume 2842 of LNAI
, 2003
"... Solomonoff unified Occam's razor and Epicurus' principle of multiple explanations to one elegant, formal, universal theory of inductive inference, which initiated the field of algorithmic information theory. His central result is that the posterior of his universal semimeasure M converges rapidly to ..."
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Cited by 7 (7 self)
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Solomonoff unified Occam's razor and Epicurus' principle of multiple explanations to one elegant, formal, universal theory of inductive inference, which initiated the field of algorithmic information theory. His central result is that the posterior of his universal semimeasure M converges rapidly to the true sequence generating posterior μ, if the latter is computable. Hence, M is eligible as a universal predictor in case of unknown μ. We investigate the existence and convergence of computable universal (semi)measures for a hierarchy of computability classes: finitely computable, estimable, enumerable, and approximable. For instance, M is known...
A Separation of Two Randomness Concepts
, 1999
"... In this paper we give an afirmative answer to a conjecture by Lutz [2]. That is, we will show that there is a Schnorr random sequence which is not recrandom. Key words: Randomness, martingales 1 Introduction and notation Schnorr [3] used the martingale concept to give a uniform description of vario ..."
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Cited by 7 (0 self)
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In this paper we give an afirmative answer to a conjecture by Lutz [2]. That is, we will show that there is a Schnorr random sequence which is not recrandom. Key words: Randomness, martingales 1 Introduction and notation Schnorr [3] used the martingale concept to give a uniform description of various notions of randomness. In particular, he gave a characterization of MartinL of's randomness concept in these terms. Moreover, he criticized MartinLof's concept as being too strong and proposed a less restrictive concept as an adequate formalization of the notion of a random sequence. In addition Schnorr introduced an intermediate notion between MartinLof and Schnorr randomness which we call recrandomness. Schnorr left open the question whether recrandomness is a proper refinement of Schnorr randomness, which was conjectured to be true by Lutz [2]. We will show that recrandomness is strictly stronger than Schnorr randomness, thereby proving Lutz's conjecture. This question was also me...
The Arithmetical Complexity of Dimension and Randomness
, 2003
"... Constructive dimension and constructive strong dimension are effectivizations of the Hausdorff and packing dimensions, respectively. Each infinite binary sequence A is assigned a dimension [0, 1] and a strong dimension Dim(A) [0, 1]. ..."
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Cited by 6 (3 self)
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Constructive dimension and constructive strong dimension are effectivizations of the Hausdorff and packing dimensions, respectively. Each infinite binary sequence A is assigned a dimension [0, 1] and a strong dimension Dim(A) [0, 1].