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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 62 (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 Turing ..."
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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.
Algorithmic randomness, quantum physics, and incompleteness
 PROCEEDINGS OF THE CONFERENCE “MACHINES, COMPUTATIONS AND UNIVERSALITY” (MCU’2004), LECTURES NOTES IN COMPUT. SCI. 3354
, 2004
"... ..."
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
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 9 (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 ChurchTuring thesis: Consensus and opposition
 Logical Approaches to Computational Barriers: Second Conference on Computability in Europe, CiE 2006
, 2006
"... Many years ago, I wrote [7]: It is truly remarkable (Gödel...speaks of a kind of miracle) that it has proved possible to give a precise mathematical characterization of the class of processes that can be carried out by purely machanical means. It is in fact the possibility of such a characterization ..."
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Cited by 8 (0 self)
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Many years ago, I wrote [7]: It is truly remarkable (Gödel...speaks of a kind of miracle) that it has proved possible to give a precise mathematical characterization of the class of processes that can be carried out by purely machanical means. It is in fact the possibility of such a characterization that underlies the ubiquitous applicability of digital computers. In addition it has made it possible to prove the algorithmic unsolvability of important problems, has provided a key tool in mathematical logic, has made available an array of fundamental models in theoretical computer science, and has been the basis of a rich new branch of mathemtics. A few years later I wrote [8]: Thesubject...isAlanTuring’sdiscoveryoftheuniversal(orallpurpose) digitalcomputerasamathematicalabstraction....Wewill tryto show how this very abstract work helped to lead Turing and John von Neumann to the modern concept of the electronic computer.
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 existence of truly autonomic computing systems and the link with quantum computing, arXiv: cs.LO/0411094
"... A theoretical model of truly autonomic computing systems (ACS), with infinitely many constraints, is proposed. An argument similar to Turing’s for the unsolvability of the halting problem, which is permitted in classical logic, shows that such systems cannot exist. Turing’s argument fails in the rec ..."
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Cited by 3 (2 self)
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A theoretical model of truly autonomic computing systems (ACS), with infinitely many constraints, is proposed. An argument similar to Turing’s for the unsolvability of the halting problem, which is permitted in classical logic, shows that such systems cannot exist. Turing’s argument fails in the recently proposed nonAristotelian finitary logic (NAFL), which permits the existence of ACS. NAFL also justifies quantum superposition and entanglement, which are essential ingredients of quantum algorithms, and resolves the EinsteinPodolskyRosen (EPR) paradox in favour of quantum mechanics and nonlocality. NAFL requires that the autonomic manager (AM) must be conceptually and architecturally distinct from the managed element, in order for the ACS to exist as a nonselfreferential entity. Such a scenario is possible if the AM uses quantum algorithms and is protected from all problems by (unbreakable) quantum encryption, while the managed element remains classical. NAFL supports such a link between autonomic and quantum computing, with the AM existing as a metamathematical entity. NAFL also allows quantum algorithms to access truly random elements and thereby supports nonstandard models of quantum (hyper) computation that permit infinite parallelism. 1.
2005 A possible hypercomputational quantum algorithm Quantum
 SPIE) Proc. SPIE 5815 219–26
, 2005
"... The term ‘hypermachine ’ denotes any data processing device (theoretical or that can be implemented) capable of carrying out tasks that cannot be performed by a Turing machine. We present a possible quantum algorithm for a classically noncomputable decision problem, Hilbert’s tenth problem; more sp ..."
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Cited by 3 (3 self)
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The term ‘hypermachine ’ denotes any data processing device (theoretical or that can be implemented) capable of carrying out tasks that cannot be performed by a Turing machine. We present a possible quantum algorithm for a classically noncomputable decision problem, Hilbert’s tenth problem; more specifically, we present a possible hypercomputation model based on quantum computation. Our algorithm is inspired by the one proposed by Tien D. Kieu, but we have selected the infinite square well instead of the (onedimensional) simple harmonic oscillator as the underlying physical system. Our model exploits the quantum adiabatic process and the characteristics of the representation of the dynamical Lie algebra su(1, 1) associated to the infinite square well.
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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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