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13
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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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
Chaitin Ω Numbers, Solovay Machines, and Incompleteness
 COMPUT. SCI
, 1999
"... Computably enumerable (c.e.) reals can be coded by Chaitin machines through their halting probabilities. Tuning Solovay's construction of a Chaitin universal machine for which ZFC (if arithmetically sound) cannot determine any single bit of the binary expansion of its halting probability, we ..."
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Cited by 14 (12 self)
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Computably enumerable (c.e.) reals can be coded by Chaitin machines through their halting probabilities. Tuning Solovay's construction of a Chaitin universal machine for which ZFC (if arithmetically sound) cannot determine any single bit of the binary expansion of its halting probability, we show that every c.e. random real is the halting probability of a universal Chaitin machine for which ZFC cannot determine more than its initial block of 1 bitsas soon as you get a 0 it's all over. Finally, a constructive version of Chaitin informationtheoretic incompleteness theorem is proven.
Constructive dimension and weak truthtable degrees
 In Computation and Logic in the Real World  Third Conference of Computability in Europe. SpringerVerlag Lecture Notes in Computer Science #4497
, 2007
"... Abstract. This paper examines the constructive Hausdorff and packing dimensions of weak truthtable degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is weak truthtable equivalent to a sequence R with ..."
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Cited by 11 (3 self)
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Abstract. This paper examines the constructive Hausdorff and packing dimensions of weak truthtable degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is weak truthtable equivalent to a sequence R with dimH(R) ≥ dimH(S)/dimP(S) − ɛ, for arbitrary ɛ> 0. Furthermore, if dimP(S)> 0, then dimP(R) ≥ 1−ɛ. The reduction thus serves as a randomness extractor that increases the algorithmic randomness of S, as measured by constructive dimension. A number of applications of this result shed new light on the constructive dimensions of wtt degrees (and, by extension, Turing degrees). A lower bound of dimH(S)/dimP(S) is shown to hold for the wtt degree of any sequence S. A new proof is given of a previouslyknown zeroone law for the constructive packing dimension of wtt degrees. It is also shown that, for any regular sequence S (that is, dimH(S) = dimP(S)) such that dimH(S)> 0, the wtt degree of S has constructive Hausdorff and packing dimension equal to 1. Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor.
Constructive dimension and Turing degrees
"... This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is Turing equivalent to a sequence R with dimH(R) ≥ (dimH(S)/dimP(S)) ..."
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Cited by 8 (0 self)
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This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is Turing equivalent to a sequence R with dimH(R) ≥ (dimH(S)/dimP(S)) − ɛ, for arbitrary ɛ> 0. Furthermore, if dimP(S)> 0, then dimP(R) ≥ 1 − ɛ. The reduction thus serves as a randomness extractor that increases the algorithmic randomness of S, as measured by constructive dimension. A number of applications of this result shed new light on the constructive dimensions of Turing degrees. A lower bound of dimH(S)/dimP(S) is shown to hold for the Turing degree of any sequence S. A new proof is given of a previouslyknown zeroone law for the constructive packing dimension of Turing degrees. It is also shown that, for any regular sequence S (that is, dimH(S) = dimP(S)) such that dimH(S)> 0, the Turing degree of S has constructive Hausdorff and packing dimension equal to 1. Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor, and that bounded Turing reductions cannot extract constructive Hausdorff dimension. We also exhibit sequences on which weak truthtable and bounded Turing reductions differ in their ability to extract dimension.
Who Is Afraid of Randomness?
, 2000
"... Randomness  mark of anxiety, the cause of disarray or misfortune, the cure for boring repetitiveness, is, like it or not, one of the most powerful driving forces of life. Is it bad? Is it good? The struggle with uncertainty and risk caused by natural disasters, market downturns or terrorism is b ..."
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Randomness  mark of anxiety, the cause of disarray or misfortune, the cure for boring repetitiveness, is, like it or not, one of the most powerful driving forces of life. Is it bad? Is it good? The struggle with uncertainty and risk caused by natural disasters, market downturns or terrorism is balanced by the role played by randomness in generating diversity and innovation, in allowing complicated structures to emerge through the exploitation of serendipitous accidents. To many minds any discussion about randomness is purely academic, just another mathematical or philosophical pedantry. False! Randomness could be a matter of life or death, as in the case of Sudden Infant Death Syndrome (SIDS), a merciless childkiller. The present paper describes some difficulties regarding the mathematical modelling of randomness, contrasts siliconcomputer generated pseudorandom bits with quantumcomputer "random" bits, succinctly presents the algorithmic definition of random
Real Numbers: From Computable to Random
, 2000
"... A real is computable if it is the limit of a computable, increasing, computably converging sequence of rationals. Omitting the restriction that the sequence converges computably we arrive at the notion of computably enumerable (c.e.) real, that is, the limit of a computable, increasing, converging s ..."
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A real is computable if it is the limit of a computable, increasing, computably converging sequence of rationals. Omitting the restriction that the sequence converges computably we arrive at the notion of computably enumerable (c.e.) real, that is, the limit of a computable, increasing, converging sequence of rationals. A real is random if its binary expansion is a random sequence (equivalently, if its expansion in base b ≥ 2 is random). The aim of this paper is to review some recent results on computable, c.e. and random reals. In particular, we will present a complete characterization of the class of c.e. and random reals in terms of halting probabilities of universal Chaitin machines, and we will show that every c.e. and random real is the halting probability of some Solovay machine, that is, a universal Chaitin machine for which ZFC (if sound) cannot determine more than its initial block of 1 bits. A few open problems will be also discussed. 1 Notation and Background We will use notation that is standard in computability theory and algorithmic information theory; we will assume familiarity with Turing machine computations, computable and computably enumerable (c.e.) sets (see, for example, Soare [48] or Odifreddi [40]) and elementary algorithmic information theory (see,
Incompleteness, Complexity, Randomness and Beyond
, 2001
"... The Library is composed of an... infinite number of hexagonal galleries... [it] includes all verbal structures, all variations permitted by the twentyfive orthographical symbols, but not a single example of absolute nonsense.... These phrases, at first glance incoherent, can no doubt be justified i ..."
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The Library is composed of an... infinite number of hexagonal galleries... [it] includes all verbal structures, all variations permitted by the twentyfive orthographical symbols, but not a single example of absolute nonsense.... These phrases, at first glance incoherent, can no doubt be justified in a cryptographical or allegorical manner; such a justification is verbal and, ex hypothesi, already figures in the Library.... The certitude that some shelf in some hexagon held precious books and that these precious books were inaccessible seemed almost intolerable. A blasphemous sect suggested that... all men should juggle letters and symbols until they constructed, by an improbable gift of chance, these canonical books... but the Library is... useless, incorruptible, secret. Jorge Luis Borges, “The Library of Babel” Gödel’s Incompleteness Theorems have the same scientific status as Einstein’s principle of relativity, Heisenberg’s uncertainty principle, and Watson and Crick’s double helix model of DNA. Our aim is to discuss some new faces of the incompleteness phenomenon unveiled by an informationtheoretic approach to randomness and recent developments in quantum computing.
Entropic Measures, Markov Information Sources and Complexity
, 2001
"... conceptof entropy plays a major part in communication theory. The Shannon entropy is a measureof uncertainty with respect to a priori probability distribution. In algorithmic inf#rithmic theory the inf#3875fi3 content of a message is measured in termsof the size in bitsof the smallest programf#g co ..."
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conceptof entropy plays a major part in communication theory. The Shannon entropy is a measureof uncertainty with respect to a priori probability distribution. In algorithmic inf#rithmic theory the inf#3875fi3 content of a message is measured in termsof the size in bitsof the smallest programf#g computing that message. This paper discusses the classical entropy and entropy rate f#a discrete or continuous Markov sources, with finite or continuous alphabets, and their relations to programsize complexity and algorithmic probability. The accent is on ideas, constructions and results; no pro will be given. 1
Randomness Everywhere: Computably Enumerable Reals and Incompleteness
, 2000
"... A real is computable if it is the limit of a computable, increasing, computably converging sequence of rationals. Omitting the restriction that the sequence converges computably we arrive at the notion of computably enumerable (c.e.) real, that is, the limit of a computable, increasing, converging s ..."
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A real is computable if it is the limit of a computable, increasing, computably converging sequence of rationals. Omitting the restriction that the sequence converges computably we arrive at the notion of computably enumerable (c.e.) real, that is, the limit of a computable, increasing, converging sequence of rationals. A real is random if its binary expansion is a random sequence. The aim of these lectures is to review some recent results on computable, c.e. and random reals. In particular, we will present a complete characterization of the class of c.e. and random reals in terms of halting probabilities of universal Chaitin machines, and we will show that every c.e. and random real is the halting probability of some Solovay machine, that is, a universal Chaitin machine for which ZFC (if sound) cannot determine more than its initial block of 1 bits. A few open problems will be also discussed.