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39
Transcending the Limits of Turing Computability
, 1998
"... Hypercomputation or superTuring computation is a “computation ” that transcends the limit imposed by Turing’s model of computability. The field still faces some basic questions, technical (can we mathematically and/or physically build a hypercomputer?), cognitive (can hypercomputers realize the AI ..."
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Cited by 18 (7 self)
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Hypercomputation or superTuring computation is a “computation ” that transcends the limit imposed by Turing’s model of computability. The field still faces some basic questions, technical (can we mathematically and/or physically build a hypercomputer?), cognitive (can hypercomputers realize the AI dream?), philosophical (is thinking more than computing?). The aim of this paper is to address the question: can we mathematically build a hypercomputer? We will discuss the solutions of the Infinite Merchant Problem, a decision problem equivalent to the Halting Problem, based on results obtained in [9, 2]. The accent will be on the new computational technique and results rather than formal proofs. 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
On the Autoreducibility of Random Sequences
, 2001
"... A binary sequence A = A(0)A(1) ... is called i.o. Turingautoreducible if A is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either A(x) or a don'tknow symbol on any given input x, and outputs A(x) for infinitely many x. If in addition ..."
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Cited by 12 (1 self)
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A binary sequence A = A(0)A(1) ... is called i.o. Turingautoreducible if A is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either A(x) or a don'tknow symbol on any given input x, and outputs A(x) for infinitely many x. If in addition the oracle Turing machine terminates on all inputs and oracles, A is called i.o. truthtableautoreducible.
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
BioSteps Beyond Turing
 BIOSYSTEMS
, 2004
"... Are there `biologically computing agents' capable to compute Turing uncomputable functions? It is perhaps tempting to dismiss this question with a negative answer. Quite the opposite, for the first time in the literature on molecular computing we contend that the answer is not theoretically nega ..."
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Cited by 9 (0 self)
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Are there `biologically computing agents' capable to compute Turing uncomputable functions? It is perhaps tempting to dismiss this question with a negative answer. Quite the opposite, for the first time in the literature on molecular computing we contend that the answer is not theoretically negative. Our results will be formulated in the language of membrane computing (P systems). Some mathematical results presented here are interesting in themselves. In contrast with most speedup methods which are based on nondeterminism, our results rest upon some universality results proved for deterministic P systems. These results will be used for building "accelerated P systems". In contrast with the case of Turing machines, acceleration is a part of the hardware (not a quality of the environment) and it is realised either by decreasing the size of "reactors" or by speedingup the communication channels.
Most programs stop quickly or never halt
 Adv. Appl. Math
"... The aim of this paper is to provide a probabilistic, but nonquantum, analysis of the Halting Problem. Our approach is to have the probability space extend over both space and time and to consider the probability that a random Nbit program has halted by a random time. We postulate an a priori compu ..."
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Cited by 9 (3 self)
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The aim of this paper is to provide a probabilistic, but nonquantum, analysis of the Halting Problem. Our approach is to have the probability space extend over both space and time and to consider the probability that a random Nbit program has halted by a random time. We postulate an a priori computable probability distribution on all possible runtimes and we prove that given an integer k> 0, we can effectively compute a time bound T such that the probability that an Nbit program will eventually halt given that it has not halted by T is smaller than 2 −k. We also show that the set of halting programs (which is computably enumerable, but not computable) can be written as a disjoint union of a computable set and a set of effectively vanishing probability. Finally, we show that “long ” runtimes are effectively rare. More formally, the set of times at which an Nbit program can stop after the time 2 N+constant has effectively zero density. 1
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
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