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Quantum Algorithm For Hilberts Tenth Problem
 Int.J.Theor.Phys
, 2003
"... We explore in the framework of Quantum Computation the notion of Computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert’s tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomp ..."
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Cited by 60 (10 self)
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We explore in the framework of Quantum Computation the notion of Computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert’s tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employed. If this algorithm could be physically implemented, as much as it is valid in principle—that is, if certain hamiltonian and its ground state can be physically constructed according to the proposal—quantum computability would surpass classical computability as delimited by the ChurchTuring thesis. It is thus argued that computability, and with it the limits of Mathematics, ought to be determined not solely by Mathematics itself but also by Physical Principles. 1
Computing the noncomputable
 Contemporary Physics
"... We explore in the framework of Quantum Computation the notion of computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm that exploits the quantum adiabatic which is equivalent to the Turing halting problem and known to be mathematically non ..."
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Cited by 30 (7 self)
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We explore in the framework of Quantum Computation the notion of computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm that exploits the quantum adiabatic which is equivalent to the Turing halting problem and known to be mathematically noncomputable. Generalised quantum algorithms are also considered for some other mathematical noncomputables in the same and of different noncomputability classes. The key element of all these algorithms is the measurability of both the values of physical observables and of the quantummechanical probability distributions for these values. It is argued that computability, and thus the limits of Mathematics, ought to be determined not
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
Real recursive functions and their hierarchy
, 2004
"... ... onsidered, first as a model of analog computation, and second to obtain analog characterizations of classical computational complexity classes (Unconventional Models of Computation, UMC 2002, Lecture Notes in Computer Science, Vol. 2509, Springer, Berlin, pp. 1–14). However, one of the operators ..."
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Cited by 16 (2 self)
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... onsidered, first as a model of analog computation, and second to obtain analog characterizations of classical computational complexity classes (Unconventional Models of Computation, UMC 2002, Lecture Notes in Computer Science, Vol. 2509, Springer, Berlin, pp. 1–14). However, one of the operators introduced in the seminal paper by Moore (1996), the minimalization operator, has not been considered: (a) although differential recursion (the analog counterpart of classical recurrence) is, in some extent, directly implementable in the General Purpose Analog Computer of Claude Shannon, analog minimalization is far from physical realizability, and (b) analog minimalization was borrowed from classical recursion theory and does not fit well the analytic realm of analog computation. In this paper, we show that a most natural operator captured from analysis—the operator of taking a limit—can be used properly to enhance the theory of recursion over the reals, providing good solutions to puzzling problems raised by the original model.
Elementarily computable functions over the real numbers and Rsubrecursive functions
 THEORETICAL COMPUTER SCIENCE
, 2005
"... We present an analog and machineindependent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linea ..."
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Cited by 13 (5 self)
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We present an analog and machineindependent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema. We generalize this result to all higher levels of the Grzegorczyk Hierarchy. This paper improves several previous partial characterizations and has a dual interest: • Concerning recursive analysis, our results provide machineindependent characterizations of natural classes of computable functions over the real numbers, allowing to define these classes without usual considerations on higherorder (type 2) Turing machines. • Concerning analog models, our results provide a characterization of the power of a natural class of analog models over the real numbers and provide new insights for understanding the relations between several analog computational models.
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
How can Nature help us compute
 SOFSEM 2006: Theory and Practice of Computer Science – 32nd Conference on Current Trends in Theory and Practice of Computer Science, Merin, Czech Republic, January 21–27
, 2006
"... Abstract. Ever since Alan Turing gave us a machine model of algorithmic computation, there have been questions about how widely it is applicable (some asked by Turing himself). Although the computer on our desk can be viewed in isolation as a Universal Turing Machine, there are many examples in natu ..."
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Cited by 11 (3 self)
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Abstract. Ever since Alan Turing gave us a machine model of algorithmic computation, there have been questions about how widely it is applicable (some asked by Turing himself). Although the computer on our desk can be viewed in isolation as a Universal Turing Machine, there are many examples in nature of what looks like computation, but for which there is no wellunderstood model. In many areas, we have to come to terms with emergence not being clearly algorithmic. The positive side of this is the growth of new computational paradigms based on metaphors for natural phenomena, and the devising of very informative computer simulations got from copying nature. This talk is concerned with general questions such as: • Can natural computation, in its various forms, provide us with genuinely new ways of computing? • To what extent can natural processes be captured computationally? • Is there a universal model underlying these new paradigms?
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.
Firstorder logic foundation of relativity theories
 In New Logics for the XXIst Century II, Mathematical Problems from Applied Logics, volume 5 of International Mathematical Series
, 2006
"... Abstract. Motivation and perspective for an exciting new research direction interconnecting logic, spacetime theory, relativity— including such revolutionary areas as black hole physics, relativistic computers, new cosmology—are presented in this paper. We would like to invite the logician reader to ..."
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Cited by 8 (8 self)
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Abstract. Motivation and perspective for an exciting new research direction interconnecting logic, spacetime theory, relativity— including such revolutionary areas as black hole physics, relativistic computers, new cosmology—are presented in this paper. We would like to invite the logician reader to take part in this grand enterprise of the new century. Besides general perspective and motivation, we present initial results in this direction.