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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
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.
Hypercomputability of quantum adiabatic processes: facts versus prejudices
 http://arxiv.org/quantph/0504101
, 2005
"... Abstract. We give an overview of a quantum adiabatic algorithm for Hilbert’s tenth problem, including some discussions on its fundamental aspects and the emphasis on the probabilistic correctness of its findings. For the purpose of illustration, the numerical simulation results of some simple Diopha ..."
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Cited by 12 (3 self)
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Abstract. We give an overview of a quantum adiabatic algorithm for Hilbert’s tenth problem, including some discussions on its fundamental aspects and the emphasis on the probabilistic correctness of its findings. For the purpose of illustration, the numerical simulation results of some simple Diophantine equations are presented. We also discuss some prejudicial misunderstandings as well as some plausible difficulties faced by the algorithm in its physical implementations. “To believe otherwise is merely to cling to a prejudice which only gives rise to further prejudices... ” 1
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
On the existence of a new family of diophantine equations for Ω
 Fundamenta Informaticae
"... Abstract. We show how to determine the kth bit of Chaitin’s algorithmically random real number, Ω, by solving k instances of the halting problem. From this we then reduce the problem of determining the kth bit of Ω to determining whether a certain Diophantine equation with two parameters, k and N, ..."
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Cited by 7 (3 self)
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Abstract. We show how to determine the kth bit of Chaitin’s algorithmically random real number, Ω, by solving k instances of the halting problem. From this we then reduce the problem of determining the kth bit of Ω to determining whether a certain Diophantine equation with two parameters, k and N, has solutions for an odd or an even number of values of N. We also demonstrate two further examples of Ω in number theory: an exponential Diophantine equation with a parameter, k, which has an odd number of solutions iff the kth bit of Ω is 1, and a polynomial of positive integer variables and a parameter, k, that takes on an odd number of positive values iff the kth bit of Ω is 1.
On the identification of the ground state based on occupation probabilities: An investigation of Smith’s apparent counterexample
 Journal of Applied Mathematics and Computation
, 2005
"... Abstract. We study a set of truncated matrices, given by Smith [8], in connection to an identification criterion for the ground state in our proposed quantum adiabatic algorithm for Hilbert’s tenth problem. We identify the origin of the trouble for this truncated example and show that for a suitable ..."
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Cited by 7 (0 self)
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Abstract. We study a set of truncated matrices, given by Smith [8], in connection to an identification criterion for the ground state in our proposed quantum adiabatic algorithm for Hilbert’s tenth problem. We identify the origin of the trouble for this truncated example and show that for a suitable choice of some parameter it can always be removed. We also argue that it is only an artefact of the truncation of the underlying Hilbert spaces, through showing its sensitivity to different boundary conditions available for such a truncation. It is maintained that the criterion, in general, should be applicable provided certain conditions are satisfied. We also point out that, apart from this one, other criteria serving the same identification purpose may also be available. In a proposal of a quantum adiabatic algorithm for Hilbert’s tenth problem [5], we employ an adiabatic process with a timedependent Hamiltonian (1) H(t) = (1 − t/T)HI + (t/T)HP. Here t is time and this Hamiltonian metamorphoses from HI when t = 0 to HP when t = T. The final Hamiltonian HP encodes the Diophantine equation in consideration, while the initial HI is universal and independent of the Diophantine equation, except only on its number of variables K. The process is captured by the Schrödinger equation (2) ∂tψ(t)〉
The church–turing thesis: Consensus and opposition
 Logical Approaches to Computational Barriers: Second Conference on Computability in Europe, CiE 2006
, 2006
"... ..."
Using biased coins as oracles
, 2004
"... Abstract. While it is well known that a Turing machine equipped with the ability to flip a fair coin cannot compute more that a standard Turing machine, we show that this is not true for a biased coin. Indeed, any oracle set X may be coded as a probability pX such that if a Turing machine is given a ..."
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Cited by 4 (3 self)
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Abstract. While it is well known that a Turing machine equipped with the ability to flip a fair coin cannot compute more that a standard Turing machine, we show that this is not true for a biased coin. Indeed, any oracle set X may be coded as a probability pX such that if a Turing machine is given a coin which lands heads with probability pX it can compute any function recursive in X with arbitrarily high probability. We also show how the assumption of a nonrecursive bias can be weakened by using a sequence of increasingly accurate recursive biases or by choosing the bias at random from a distribution with a nonrecursive mean. We conclude by briefly mentioning some implications regarding the physical realisability of such methods.
Physicallyrelativized ChurchTuring Hypotheses. Applied Mathematics and Computation 215, 4
 in the School of Mathematics at the University of Leeds, U.K. © 2012 ACM 00010782/12/03 $10.00 march 2012  vol. 55  no. 3  communications of the acm 83
"... Abstract. We turn ‘the ’ ChurchTuring Hypothesis from an ambiguous source of sensational speculations into a (collection of) sound and welldefined scientific problem(s): Examining recent controversies, and causes for misunderstanding, concerning the state of the ChurchTuring Hypothesis (CTH), sug ..."
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Cited by 4 (0 self)
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Abstract. We turn ‘the ’ ChurchTuring Hypothesis from an ambiguous source of sensational speculations into a (collection of) sound and welldefined scientific problem(s): Examining recent controversies, and causes for misunderstanding, concerning the state of the ChurchTuring Hypothesis (CTH), suggests to study the CTH relative to an arbitrary but specific physical theory—rather than vaguely referring to “nature ” in general. To this end we combine (and compare) physical structuralism with (models of computation in) complexity theory. The benefit of this formal framework is illustrated by reporting on some previous, and giving one new, example result(s) of computability