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Algorithmic Theories Of Everything
, 2000
"... The probability distribution P from which the history of our universe is sampled represents a theory of everything or TOE. We assume P is formally describable. Since most (uncountably many) distributions are not, this imposes a strong inductive bias. We show that P(x) is small for any universe x lac ..."
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Cited by 31 (15 self)
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The probability distribution P from which the history of our universe is sampled represents a theory of everything or TOE. We assume P is formally describable. Since most (uncountably many) distributions are not, this imposes a strong inductive bias. We show that P(x) is small for any universe x lacking a short description, and study the spectrum of TOEs spanned by two Ps, one reflecting the most compact constructive descriptions, the other the fastest way of computing everything. The former derives from generalizations of traditional computability, Solomonoff’s algorithmic probability, Kolmogorov complexity, and objects more random than Chaitin’s Omega, the latter from Levin’s universal search and a natural resourceoriented postulate: the cumulative prior probability of all x incomputable within time t by this optimal algorithm should be 1/t. Between both Ps we find a universal cumulatively enumerable measure that dominates traditional enumerable measures; any such CEM must assign low probability to any universe lacking a short enumerating program. We derive Pspecific consequences for evolving observers, inductive reasoning, quantum physics, philosophy, and the expected duration of our universe.
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 18 (9 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
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 show ..."
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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.
Algorithmic randomness, quantum physics, and incompleteness
 Proceedings of the Conference “Machines, Computations and Universality” (MCU’2004), number 3354 in Lecture Notes in Computer Science
, 2006
"... When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is almost certainly wrong. Arthur C. Clarke ..."
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Cited by 12 (2 self)
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When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is almost certainly wrong. Arthur C. Clarke
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
On the existence of a new family of diophantine equations for Ω
 Fundamenta Informaticae
"... Abstract. We show how to determine the kth bit of Chaitin’s algorithmically random real number, Ω, by solving k instances of the halting problem. From this we then reduce the problem of determining the kth bit of Ω to determining whether a certain Diophantine equation with two parameters, k and N, ..."
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Cited by 7 (3 self)
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Abstract. We show how to determine the kth bit of Chaitin’s algorithmically random real number, Ω, by solving k instances of the halting problem. From this we then reduce the problem of determining the kth bit of Ω to determining whether a certain Diophantine equation with two parameters, k and N, has solutions for an odd or an even number of values of N. We also demonstrate two further examples of Ω in number theory: an exponential Diophantine equation with a parameter, k, which has an odd number of solutions iff the kth bit of Ω is 1, and a polynomial of positive integer variables and a parameter, k, that takes on an odd number of positive values iff the kth bit of Ω is 1.
Every Computably Enumerable Random Real Is Provably Computably Enumerable Random
, 2009
"... We prove that every computably enumerable (c.e.) random real is provable in Peano Arithmetic (PA) to be c.e. random. A major step in the proof is to show that the theorem stating that “a real is c.e. and random iff it is the halting probability of a universal prefixfree Turing machine ” can be prov ..."
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Cited by 4 (4 self)
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We prove that every computably enumerable (c.e.) random real is provable in Peano Arithmetic (PA) to be c.e. random. A major step in the proof is to show that the theorem stating that “a real is c.e. and random iff it is the halting probability of a universal prefixfree Turing machine ” can be proven in PA. Our proof, which is simpler than the standard one, can also be used for the original theorem. Our positive result can be contrasted with the case of computable functions, where not every computable function is provably computable in PA, or even more interestingly, with the fact that almost all random finite strings are not provably random in PA. We also prove two negative results: a) there exists a universal machine whose universality cannot be proved in PA, b) there exists a universal machine U such that, based on U, PA cannot prove the randomness of its halting probability. The paper also includes a sharper form of the KraftChaitin Theorem, as well as a formal proof of this theorem written with the proof assistant Isabelle.
Truth and Light: Physical Algorithmic Randomness
, 2005
"... This thesis examines some problems related to Chaitin's Ω number. In the first section, we describe several new minimalist prefixfree machines suitable for the study of concrete algorithmic information theory; the halting probabilities of these machines are all Ω numbers. In the second part, we s ..."
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Cited by 3 (2 self)
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This thesis examines some problems related to Chaitin's Ω number. In the first section, we describe several new minimalist prefixfree machines suitable for the study of concrete algorithmic information theory; the halting probabilities of these machines are all Ω numbers. In the second part, we show that when such a sequence is the result given by a measurement of a system, the system itself can be shown to satisfy an uncertainty principle equivalent to Heisenberg's uncertainty principle. This uncertainty principle also implies Chaitin's strongest form of incompleteness. In the last part, we show that Ω can be written as an infinite product over halting programs; that there exists a "natural," or basefree formulation that does not (directly) depend on the alphabet of the universal prefixfree machine; that Tadaki's generalized halting probability is welldefined even for arbitrary univeral Turing machines and the plain complexity; and finally, that the natural generalized halting probability can be written as an infinite product over primes and has the form of a zeta function whose zeros encode halting information. We conclude with some speculation about physical systems in which partial randomness could arise, and identify many open problems.
S.: Passages of proof
 Bull. Eur. Assoc. Theor. Comput. Sci. EATCS
, 2004
"... Whether ’tis nobler in the mind to suffer The slings and arrows of outrageous fortune, Or to take arms against a sea of troubles And by opposing end them? Hamlet 3/1, by W. Shakespeare In this paper we propose a new perspective on the evolution and history of the idea of mathematical proof. Proofs w ..."
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Cited by 1 (1 self)
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Whether ’tis nobler in the mind to suffer The slings and arrows of outrageous fortune, Or to take arms against a sea of troubles And by opposing end them? Hamlet 3/1, by W. Shakespeare In this paper we propose a new perspective on the evolution and history of the idea of mathematical proof. Proofs will be studied at three levels: syntactical, semantical and pragmatical. Computerassisted proofs will be give a special attention. Finally, in a highly speculative part, we will anticipate the evolution of proofs under the assumption that the quantum computer will materialize. We will argue that there is little ‘intrinsic ’ difference between traditional and ‘unconventional ’ types of proofs. 2 Mathematical Proofs: An Evolution in Eight Stages Theory is to practice as rigour is to vigour. D. E. Knuth Reason and experiment are two ways to acquire knowledge. For a long time mathematical
Who Is Afraid of Randomness?
, 2000
"... Introduction 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 ..."
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Introduction 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 di#culties regarding the mathematical modelling of randomness, contrasts siliconcomputer generated pseudorandom bits with quantumcomputer "random" bits, succinctly presents the algorithmic definition of random