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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
Computing with Cells and Atoms: After Five Years
 CDMTCS Research Report Series, CDMTCS246, 2004, available at http://www.cs.auckland.ac.nz/CDMTCS/researchreports/246cris.pdf
, 2004
"... This is the text added to the Russian edition of our book Computing with Cells and Atoms (Taylor & Francis Publishers, London, 2001) to be published by Pushchino Publishing House. The translation was done by Professor Victor Vladimirovich Ivanov and Professor Robert Polozov. ..."
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Cited by 2 (0 self)
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This is the text added to the Russian edition of our book Computing with Cells and Atoms (Taylor & Francis Publishers, London, 2001) to be published by Pushchino Publishing House. The translation was done by Professor Victor Vladimirovich Ivanov and Professor Robert Polozov.
Computational Power of Infinite Quantum Parallelism
 pp.2057–2071 in International Journal of Theoretical Physics vol.44:11
, 2005
"... Recent works have independently suggested that quantum mechanics might permit procedures that fundamentally transcend the power of Turing Machines as well as of ‘standard ’ Quantum Computers. These approaches rely on and indicate that quantum mechanics seems to support some infinite variant of class ..."
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Cited by 2 (1 self)
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Recent works have independently suggested that quantum mechanics might permit procedures that fundamentally transcend the power of Turing Machines as well as of ‘standard ’ Quantum Computers. These approaches rely on and indicate that quantum mechanics seems to support some infinite variant of classical parallel computing. We compare this new one with other attempts towards hypercomputation by separating (1) its computing capabilities from (2) realizability issues. The first are shown to coincide with recursive enumerability; the second are considered in analogy to ‘existence’ in mathematical logic. KEY WORDS: Hypercomputation; quantum mechanics; recursion theory; infinite parallelism.
How to acknowledge hypercomputation?
, 2007
"... We discuss the question of how to operationally validate whether or not a “hypercomputer” performs better than the known discrete computational models. ..."
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Cited by 1 (0 self)
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We discuss the question of how to operationally validate whether or not a “hypercomputer” performs better than the known discrete computational models.
Series Preproceedings of the Workshop “Physics and Computation ” 2008
, 2008
"... In the 1940s, two different views of the brain and the computer were equally important. One was the analog technology and theory that had emerged before the war. The other was the digital technology and theory that was to become the main paradigm of computation. 1 The outcome of the contest between ..."
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In the 1940s, two different views of the brain and the computer were equally important. One was the analog technology and theory that had emerged before the war. The other was the digital technology and theory that was to become the main paradigm of computation. 1 The outcome of the contest between these two competing views derived from technological and epistemological arguments. While digital technology was improving dramatically, the technology of analog machines had already reached a significant level of development. In particular, digital technology offered a more effective way to control the precision of calculations. But the epistemological discussion was, at the time, equally relevant. For the supporters of the analog computer, the digital model — which can only process information transformed and coded in binary — wouldn’t be suitable to represent certain kinds of continuous variation that help determine brain functions. With analog machines, on the contrary, there would be few or no layers between natural objects and the work and structure of computation (cf. [4, 1]). The 1942–52 Macy Conferences in cybernetics helped to validate digital theory and logic as legitimate ways to think about the brain and the machine [4]. In particular, those conferences helped made McCullochPitts ’ digital model
Asymptotic behavior and halting probability of Turing Machines
, 2006
"... Through a straightforward Bayesian approach we show that under some general conditions a maximum running time, namely the number of discrete steps done by a computer program during its execution, can be defined such that the probability that such program will halt after that time is smaller than any ..."
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Through a straightforward Bayesian approach we show that under some general conditions a maximum running time, namely the number of discrete steps done by a computer program during its execution, can be defined such that the probability that such program will halt after that time is smaller than any arbitrary fixed value. Consistency with known results and consequences are also discussed. 1 Introductory remarks As it has been proved by Alan M. Turing in 1936 [1], if we have a program p running on an Universal Turing Machine (UTM), then we have no general, finite and deterministic algorithm which allows us to know whether and when it will halt (this is the well known halting problem). That is to say that the halting behavior of a program, with the trivial exception of the simplest ones, is not computable and predictable by a unique, general procedure. In this paper we show that, for what concerns the probability of its halt,
Asymptotic behavior and halting probability of Turing Machines ∗
, 2006
"... Through a straightforward Bayesian approach we show that under some general conditions a maximum running time, namely the number of discrete steps performed by a computer program during its execution, can be defined such that the probability that such a program will halt after that time is smaller t ..."
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Through a straightforward Bayesian approach we show that under some general conditions a maximum running time, namely the number of discrete steps performed by a computer program during its execution, can be defined such that the probability that such a program will halt after that time is smaller than any arbitrary fixed value. Consistency with known results and consequences are also discussed. 1 Introductory remarks As it has been proved by Turing in 1936 [1], if we have a program p running on an Universal Turing Machine (UTM), then we have no general, finite and deterministic algorithm which allows us to know whether and when it will halt (this is the well known halting problem). That is to say that the halting behavior of a program, with the trivial exception of the simplest ones, is not computable and predictable by a unique, general procedure. In this paper we show that, for what concerns the probability of its halt, every program running on an UTM is characterized by a peculiar asymptotic behavior in time. Similar results have been obtained by Calude et al. [2] and by Adamyan et al. [3] through a different approach, which makes use of quantum computation.
and
, 712
"... We discuss the question of how to operationally validate whether or not a “hypercomputer ” performs better than the known discrete computational models. 1 ..."
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We discuss the question of how to operationally validate whether or not a “hypercomputer ” performs better than the known discrete computational models. 1
Does Quantum Mechanics allow for Infinite Parallelism?
, 2004
"... Recent works have independently suggested that Quantum Mechanics might permit for procedures that transcend the power of Turing Machines as well as of ‘standard ’ Quantum Computers. These approaches rely on and indicate that Quantum Mechanics seems to support some infinite variant of classical paral ..."
Abstract
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Recent works have independently suggested that Quantum Mechanics might permit for procedures that transcend the power of Turing Machines as well as of ‘standard ’ Quantum Computers. These approaches rely on and indicate that Quantum Mechanics seems to support some infinite variant of classical parallel computing. We compare this new one with other attempts towards hypercomputation by separating 1) its principal computing capabilities from 2) realizability issues. The first are shown to coincide with recursive enumerability; the second are considered in analogy to ‘existence ’ in mathematical logic.