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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
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
From Heisenberg to Gödel via Chaitin
 INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS
, 2004
"... In 1927 Heisenberg discovered that the "more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa". Four years later ..."
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Cited by 11 (9 self)
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In 1927 Heisenberg discovered that the "more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa". Four years later
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
Preface Human beings have a future if they deserve to have a future!
"... 2 Chaitin coined the name AIT; this name is becoming more and more popular. vii viii Preface During its history of more than 40 years, AIT knew a significant variation in terminology. In particular, the main measures of complexity studied in AIT were called SolomonoffKolmogorovChaitin complexity, ..."
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2 Chaitin coined the name AIT; this name is becoming more and more popular. vii viii Preface During its history of more than 40 years, AIT knew a significant variation in terminology. In particular, the main measures of complexity studied in AIT were called SolomonoffKolmogorovChaitin complexity, KolmogorovChaitin complexity, Kolmogorov complexity, Chaitin complexity, algorithmic complexity, programsize complexity, etc. Solovay’s handwritten notes [22] 3, introduced and used the terms Chaitin complexity and Chaitin machine. 4 The book [21] promoted the name Kolmogorov complexity for both AIT and its main complexity. 5 The main contribution shared by AIT founding fathers in the mid 1960s was the new type of complexity—which is invariant up to an additive constant—and, with it, a new way to reason about computation. Founding fathers ’ subsequent contributions varied considerably. Solomonoff’s