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Hierarchies Of Generalized Kolmogorov Complexities And Nonenumerable Universal Measures Computable In The Limit
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
, 2000
"... The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with oneway writeonly output tape. This naturally leads to the universal enumerable SolomonoLevin measure. Here we introduce more general, nonenumerable but cumulatively enumerable m ..."
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The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with oneway writeonly output tape. This naturally leads to the universal enumerable SolomonoLevin measure. Here we introduce more general, nonenumerable but cumulatively enumerable measures (CEMs) derived from Turing machines with lexicographically nondecreasing output and random input, and even more general approximable measures and distributions computable in the limit. We obtain a natural hierarchy of generalizations of algorithmic probability and Kolmogorov complexity, suggesting that the "true" information content of some (possibly in nite) bitstring x is the size of the shortest nonhalting program that converges to x and nothing but x on a Turing machine that can edit its previous outputs. Among other things we show that there are objects computable in the limit yet more random than Chaitin's "number of wisdom" Omega, that any approximable measure of x is small for any x lacking a short description, that there is no universal approximable distribution, that there is a universal CEM, and that any nonenumerable CEM of x is small for any x lacking a short enumerating program. We briey mention consequences for universes sampled from such priors.
Hypercomputation and the Physical ChurchTuring Thesis
, 2003
"... A version of the ChurchTuring Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logicomathematical, proposition. We review the main approaches to computation beyond Tu ..."
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Cited by 21 (0 self)
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A version of the ChurchTuring Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logicomathematical, proposition. We review the main approaches to computation beyond Turing definability (`hypercomputation'): supertask, nonwellfounded, analog, quantum, and retrocausal computation. These models depend on infinite computation, explicitly or implicitly, and appear physically implausible; moreover, even if infinite computation were realizable, the Halting Problem would not be a#ected. Therefore, Thesis P is not essentially di#erent from the standard ChurchTuring Thesis.
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.
The Riemann Hypothesis
"... Abstract. In this article I describe a proof of the fact that ZFC cannot say much about a Turing machine that takes a very long time to halt (if it eventually halts). The consequences of this fact in relation to the Riemann Hypothesis are presented. ..."
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Abstract. In this article I describe a proof of the fact that ZFC cannot say much about a Turing machine that takes a very long time to halt (if it eventually halts). The consequences of this fact in relation to the Riemann Hypothesis are presented.
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.