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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.
Enumeration Reducibility, Nondeterministic Computations and Relative . . .
 RECURSION THEORY WEEK, OBERWOLFACH 1989, VOLUME 1432 OF LECTURE NOTES IN MATHEMATICS
, 1990
"... ..."
Computing a glimpse of randomness
 Experimental Mathematics
"... A Chaitin Omega number is the halting probability of a universal Chaitin (selfdelimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In ..."
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Cited by 20 (10 self)
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A Chaitin Omega number is the halting probability of a universal Chaitin (selfdelimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly noncomputable. The aim of this paper is to describe a procedure, which combines Java programming and mathematical proofs, for computing the exact values of the first 63 bits of a Chaitin Omega: 000000100000010000100000100001110111001100100111100010010011100. Full description of programs and proofs will be given elsewhere. 1
Towards Abstractions for Distributed Systems
, 2004
"... For historical, sociological and technical reasons, calculi have been the dominant theoretical paradigm in the study of functional computation. Similarly, but to a lesser degree, calculi dominate advanced mathematical accounts of concurrency. Alas, and despite its ever increasing ubiquity, an equa ..."
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Cited by 17 (5 self)
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For historical, sociological and technical reasons, calculi have been the dominant theoretical paradigm in the study of functional computation. Similarly, but to a lesser degree, calculi dominate advanced mathematical accounts of concurrency. Alas, and despite its ever increasing ubiquity, an equally convincing formal foundation for distributed computing has not been forthcoming. This thesis seeks to contribute towards ameliorating that omission. To this end, guided by the assumption that distributed computing is concurrent computing with partial failures of various kinds, we extend the asynchronous calculus with a notion of sites, the possibility of site failure, a persistence mechanism to deal with site failures, the distinction between intersite and intrasite communication, the possibility of message loss in intersite communication and a timer construct, as is often used in distributed algorithms to deal with various failure scenarios.
How can Nature help us compute
 SOFSEM 2006: Theory and Practice of Computer Science – 32nd Conference on Current Trends in Theory and Practice of Computer Science, Merin, Czech Republic, January 21–27
, 2006
"... Abstract. Ever since Alan Turing gave us a machine model of algorithmic computation, there have been questions about how widely it is applicable (some asked by Turing himself). Although the computer on our desk can be viewed in isolation as a Universal Turing Machine, there are many examples in natu ..."
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Cited by 11 (3 self)
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Abstract. Ever since Alan Turing gave us a machine model of algorithmic computation, there have been questions about how widely it is applicable (some asked by Turing himself). Although the computer on our desk can be viewed in isolation as a Universal Turing Machine, there are many examples in nature of what looks like computation, but for which there is no wellunderstood model. In many areas, we have to come to terms with emergence not being clearly algorithmic. The positive side of this is the growth of new computational paradigms based on metaphors for natural phenomena, and the devising of very informative computer simulations got from copying nature. This talk is concerned with general questions such as: • Can natural computation, in its various forms, provide us with genuinely new ways of computing? • To what extent can natural processes be captured computationally? • Is there a universal model underlying these new paradigms?
ON INTERPRETING CHAITIN’S INCOMPLETENESS THEOREM
, 1998
"... The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin’s famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number ..."
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Cited by 9 (0 self)
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The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin’s famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good measure of the strength of the theory. I exhibit certain strong counterexamples and establish conclusively that the received view is false. Moreover, I show that the limiting constants provided by the theorem do not in any way reflect the power of formalized theories, but that the values of these constants are actually determined by the chosen coding of Turing machines, and are thus quite accidental.
Exact Approximations of Omega Numbers
 International Journal of Bifurcation and Chaos 17, 1937–1954 (2007), CDMTCS report series 293. http://dx.doi.org/10.1142/S0218127407018130
"... A Chaitin Omega number is the halting probability of a universal prefixfree Turing machine. Every Omega number is simultaneously computably enumerable (the limit of a computable, increasing, converging sequence of rationals), and algorithmically random (its binary expansion is an algorithmic random ..."
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Cited by 8 (1 self)
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A Chaitin Omega number is the halting probability of a universal prefixfree Turing machine. Every Omega number is simultaneously computably enumerable (the limit of a computable, increasing, converging sequence of rationals), and algorithmically random (its binary expansion is an algorithmic random sequence), hence uncomputable. The value of an Omega number is highly machinedependent. In general, no more than finitely many scattered bits of the binary expansion of an Omega number can be exactly computed; but, in some cases, it is possible to prove that no bit can be computed. In this paper we will simplify and improve both the method and its correctness proof proposed in an earlier paper, and we will compute the exact approximations of two Omega numbers of the same prefixfree Turing machine, which is universal when used with data in base 16 or base 2: we compute 43 exact bits for the base 16 machine and 40 exact bits for the base 2 machine. 1
The emperor’s new recursiveness: The epigraph of the exponential function in two models of computability
 In Masami Ito and Teruo Imaoka, editors, Words, Languages & Combinatorics III
, 2003
"... In his book “The Emperor’s New Mind ” Roger Penrose implicitly defines some criteria which should be met by a reasonable notion of recursiveness for subsets of Euclidean space. We discuss two such notions with regard to Penrose’s criteria: one originated from computable analysis, and the one introdu ..."
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Cited by 7 (0 self)
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In his book “The Emperor’s New Mind ” Roger Penrose implicitly defines some criteria which should be met by a reasonable notion of recursiveness for subsets of Euclidean space. We discuss two such notions with regard to Penrose’s criteria: one originated from computable analysis, and the one introduced by Blum, Shub and Smale. 1
Relativizing Relativity
, 2000
"... this article; nor should they be blamed for any misconception and fallacy of the author ..."
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Cited by 4 (4 self)
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this article; nor should they be blamed for any misconception and fallacy of the author
Representations versus Numberings: On the Relationship of Two Computability Notions
 THEORETICAL COMPUTER SCIENCE
, 2001
"... This paper gives an answer to Weihrauch's question [31] whether and, if not always, when an effective map between the computable elements of two represented sets can be extended to a (partial) computable map between the represented sets. Examples are known showing that this is not possible in genera ..."
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Cited by 4 (0 self)
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This paper gives an answer to Weihrauch's question [31] whether and, if not always, when an effective map between the computable elements of two represented sets can be extended to a (partial) computable map between the represented sets. Examples are known showing that this is not possible in general. A condition is introduced and for countably based topological T 0 spaces it is shown that exactly the (partial) effective maps meeting the requirement are extendable. For total effective maps the extra condition is satisfied in the standard cases of effectively given separable metric spaces and continuous directedcomplete partial orders, in which the extendability is already known. In the first case a similar result holds also for partial effective maps, but not in the second.