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Recursively Enumerable Reals and Chaitin Ω Numbers
"... A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from b ..."
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Cited by 34 (3 self)
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A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's [23]like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability ofa universal selfdelimiting Turing machine (Chaitin's Ω number, [9]) is also a random r.e. real. Solovay showed that any Chaitin Ω number islike. In this paper we show that the converse implication is true as well: any Ωlike real in the unit interval is the halting probability of a universal selfdelimiting Turing machine.
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
Natural halting probabilities, partial randomness, and zeta functions
 Inform. and Comput
, 2006
"... We introduce the zeta number, natural halting probability and natural complexity of a Turing machine and we relate them to Chaitin’s Omega number, halting probability, and programsize complexity. A classification of Turing machines according to their zeta numbers is proposed: divergent, convergent ..."
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Cited by 17 (8 self)
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We introduce the zeta number, natural halting probability and natural complexity of a Turing machine and we relate them to Chaitin’s Omega number, halting probability, and programsize complexity. A classification of Turing machines according to their zeta numbers is proposed: divergent, convergent and tuatara. We prove the existence of universal convergent and tuatara machines. Various results on (algorithmic) randomness and partial randomness are proved. For example, we show that the zeta number of a universal tuatara machine is c.e. and random. A new type of partial randomness, asymptotic randomness, is introduced. Finally we show that in contrast to classical (algorithmic) randomness—which cannot be naturally characterised in terms of plain complexity—asymptotic randomness admits such a characterisation. 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.
Computable Approximations of Reals: An InformationTheoretic Analysis
 Fundamenta Informaticae
, 1997
"... How fast can one approximate a real by a computable sequence of rationals? We show that the answer to this question depends very much on the information content in the finite prefixes of the binary expansion of the real. Computable reals, whose binary expansions haveavery low information content, ca ..."
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Cited by 10 (3 self)
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How fast can one approximate a real by a computable sequence of rationals? We show that the answer to this question depends very much on the information content in the finite prefixes of the binary expansion of the real. Computable reals, whose binary expansions haveavery low information content, can be approximated (very fast) with a computable convergence rate. Random reals, whose binary expansions contain very much information in their prefixes, can be approximated only very slowly by computable sequences of rationals (this is the case, for example, for Chaitin's \Omega numbers) if they can be computably approximated at all. We show that one can computably approximate any computable real also very slowly, with a convergence rate slower than any computable function. However, there is still a large gap between computable reals and random reals: any computable sequence of rationals which converges (monotonically) to a random real converges slower than any computable sequence of rat...
Algorithmically Coding the Universe
 Developments in Language Theory, World Scientific
, 1994
"... All science is founded on the assumption that the physical universe is ordered. Our aim is to challenge this hypothesis using arguments from the algorithmic information theory. 1 Introduction Algorithmic information theory opens new vistas that extend far beyond the traditional boundaries of mathem ..."
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Cited by 9 (7 self)
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All science is founded on the assumption that the physical universe is ordered. Our aim is to challenge this hypothesis using arguments from the algorithmic information theory. 1 Introduction Algorithmic information theory opens new vistas that extend far beyond the traditional boundaries of mathematics and computer science. How can we describe the seemingly random processes in nature and reconcile them with the supposed order? How much can a given piece of information be compressed? These are matters of fundamental scientific importance that will be discussed below, mainly from an informal or semiformal point of view. The descriptional complexity of a sequence of bits, finite or infinite, is the length of the shortest sequence of bits defining the originally given sequence. A given sequence being random means, roughly, that its descriptional complexity equals its length. In other words the simplest way to define the sequence is to write it down. This seems to be the case for the seq...
Series Representation of LeftComputable ε–Random Reals
, 2009
"... the degree of randomness of reals (or sequences) by measuring their “degree of compression”. In what follows ε is a fixed computable real number with 0 < ε ≤ 1 and we study the ε–randomness of reals, both intrinsically and in relation to the classical notion of randomness which corresponds to ε = 1, ..."
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Cited by 4 (3 self)
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the degree of randomness of reals (or sequences) by measuring their “degree of compression”. In what follows ε is a fixed computable real number with 0 < ε ≤ 1 and we study the ε–randomness of reals, both intrinsically and in relation to the classical notion of randomness which corresponds to ε = 1, hence referred to as 1–randomness. Our main tool is the ε–universal prefixfree Turing machine, a machine that can simulate any other prefixfree machine: the length of the simulating program on the ε–universal machine is bounded up to a fixed constant by the length of the simulated program divided by ε. In case ε = 1 we get the classical notion of universal machine. Contrary to the situation in the classical theory, the difference between the prefix complexities induced by two ε–universal prefixfree Turing machines may not be bounded. We show that the halting probability of an ε–universal prefixfree Turing machine is leftcomputable and ε–random. Generalising the corresponding representability theorem of leftcomputable random reals [1, 3, 7, 10] we show that the
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.
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