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29
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
A Highly Random Number
, 2001
"... In his celebrated 1936 paper Turing defined a machine to be circular iff it performs an infinite computation outputting only finitely many symbols. We define ( as the probability that an arbitrary machine be circular and we prove that is a random number that goes beyond $2, the probability that ..."
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Cited by 15 (5 self)
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In his celebrated 1936 paper Turing defined a machine to be circular iff it performs an infinite computation outputting only finitely many symbols. We define ( as the probability that an arbitrary machine be circular and we prove that is a random number that goes beyond $2, the probability that a universal self alelimiting machine halts. The algorithmic complexity of c is strictly greater than that of $2, but similar to the algorithmic complexity of 2 , the halting probability of an oracle machine. What makes ( interesting is that it is an example of a highly random number definable without considering oracles.
Randomness and Coincidences: Reconciling Intuition and Probability Theory
, 2001
"... We argue that the apparent inconsistency between people's intuitions about chance and the normative predictions of probability theory, as expressed in judgments about randomness and coincidences, can be resolved by focussing on the evidence observations provide about the processes that generated the ..."
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Cited by 11 (3 self)
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We argue that the apparent inconsistency between people's intuitions about chance and the normative predictions of probability theory, as expressed in judgments about randomness and coincidences, can be resolved by focussing on the evidence observations provide about the processes that generated them, rather than their likelihood. This argument is supported by probabilistic modeling of sequence and number production, together with two experiments that examine people's judgments about coincidences.
BioSteps Beyond Turing
 BIOSYSTEMS
, 2004
"... Are there `biologically computing agents' capable to compute Turing uncomputable functions? It is perhaps tempting to dismiss this question with a negative answer. Quite the opposite, for the first time in the literature on molecular computing we contend that the answer is not theoretically nega ..."
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Cited by 9 (0 self)
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Are there `biologically computing agents' capable to compute Turing uncomputable functions? It is perhaps tempting to dismiss this question with a negative answer. Quite the opposite, for the first time in the literature on molecular computing we contend that the answer is not theoretically negative. Our results will be formulated in the language of membrane computing (P systems). Some mathematical results presented here are interesting in themselves. In contrast with most speedup methods which are based on nondeterminism, our results rest upon some universality results proved for deterministic P systems. These results will be used for building "accelerated P systems". In contrast with the case of Turing machines, acceleration is a part of the hardware (not a quality of the environment) and it is realised either by decreasing the size of "reactors" or by speedingup the communication channels.
Binary Lambda Calculus and Combinatory Logic.” Sep 14, 2004. http://homepages. cwi.nl/ ∼ tromp/cl/LC.pdf [64] Tadaki, K. “Upper bound by Kolmogorov complexity for the probability
 in computable POVM measurement.” Proceedings of the 5th Conference on Real Numbers and Computers, RNC5
, 2003
"... In the first part, we introduce binary representations of both lambda calculus and combinatory logic terms, and demonstrate their simplicity by providing very compact parserinterpreters for these binary languages. Along the way we also present new results on list representations, bracket abstractio ..."
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Cited by 9 (0 self)
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In the first part, we introduce binary representations of both lambda calculus and combinatory logic terms, and demonstrate their simplicity by providing very compact parserinterpreters for these binary languages. Along the way we also present new results on list representations, bracket abstraction, and fixpoint combinators. In the second part we review Algorithmic Information Theory, for which these interpreters provide a convenient vehicle. We demonstrate this with several concrete upper bounds on programsize complexity, including an elegant selfdelimiting code for binary strings. 1
Computational universes
 Chaos, Solitons & Fractals
, 2006
"... Suspicions that the world might be some sort of a machine or algorithm existing “in the mind ” of some symbolic number cruncher have lingered from antiquity. Although popular at times, the most radical forms of this idea never reached mainstream. Modern developments in physics and computer science h ..."
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Cited by 9 (5 self)
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Suspicions that the world might be some sort of a machine or algorithm existing “in the mind ” of some symbolic number cruncher have lingered from antiquity. Although popular at times, the most radical forms of this idea never reached mainstream. Modern developments in physics and computer science have lent support to the thesis, but empirical evidence is needed before it can begin to replace our contemporary world view.
Mathematical proofs at a crossroad
 Theory Is Forever, Lectures Notes in Comput. Sci. 3113
, 2004
"... Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomaticdeductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimen ..."
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Cited by 7 (7 self)
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Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomaticdeductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimental, psychological and social aspects, yesterday only marginal, but now changing radically the very essence of proof. In this paper, we try to organize this evolution, to distinguish its different steps and aspects, and to evaluate its advantages and shortcomings. Axiomaticdeductive proofs are not a posteriori work, a luxury we can marginalize nor are computerassisted proofs bad mathematics. There is hope for integration! 1
Qualitative Computing
 In Handbook of Numerical Computation. SIAM
, 2002
"... Computing is a human activity which is much more ancient than any historical record can tell, as testified by stone or bone tallies found in many prehistorical sites. It is one of the first skills to be taught to small children. However, everyone senses that there is ..."
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Cited by 7 (2 self)
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Computing is a human activity which is much more ancient than any historical record can tell, as testified by stone or bone tallies found in many prehistorical sites. It is one of the first skills to be taught to small children. However, everyone senses that there is
Quantum Information Via State Partitions and the Context Translation Principle
, 2004
"... For manyparticle systems, quantum information in base n can be defined by partitioning the set of states according to the outcomes of nary (joint) observables. Thereby, k particles can carry k nits. With regards to the randomness of single outcomes, a context translation principle is proposed. ..."
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Cited by 6 (6 self)
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For manyparticle systems, quantum information in base n can be defined by partitioning the set of states according to the outcomes of nary (joint) observables. Thereby, k particles can carry k nits. With regards to the randomness of single outcomes, a context translation principle is proposed. Quantum randomness is related to the uncontrollable degrees of freedom of the measurement interface, thereby translating a mismatch between the state prepared and the state measured.
Another Example of Higher Order Randomness
 FUNDAMENTA INFORMATICAE
, 2002
"... We consider the notion of algorithmic randomness relative to an oracle. We prove that the probability # that a program for infinite computations (a program that never halts) outputs a cofinite set is random in the second jump of the halting problem. Indeed, we prove that # is exactly as random as ..."
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Cited by 5 (2 self)
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We consider the notion of algorithmic randomness relative to an oracle. We prove that the probability # that a program for infinite computations (a program that never halts) outputs a cofinite set is random in the second jump of the halting problem. Indeed, we prove that # is exactly as random as the halting probability of a universal machine equipped with an oracle for the second jump of the halting problem, in spite of the fact that # is defined without considering oracles.