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The quantum algorithm of Kieu does not solve the Hilbert’s tenth problem
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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
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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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)〉
Reply to “the quantum algorithm of Kieu does not solve the Hilbert’s tenth problem”. Archive preprint http://arxiv.org/abs/quantph/0111020
, 2001
"... The arguments employed in quantph/0111009, to claim that the quantum algorithm in quantph/0110136 does not work, are so general that were they true then the adiabatic theorem itself would have been wrong. As a matter of fact, those arguments are only valid for the sudden approximation, not the adi ..."
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The arguments employed in quantph/0111009, to claim that the quantum algorithm in quantph/0110136 does not work, are so general that were they true then the adiabatic theorem itself would have been wrong. As a matter of fact, those arguments are only valid for the sudden approximation, not the adiabatic process. The author of [1] carefully distinguishes between the general groundstate oracle from the algorithm which explicitly employs the adiabatic evolution, both proposed for the Hilbert’s tenth problem in [2]. Then it is concluded that this latter quantum algorithm is untenable. However, the arguments employed to reach this conclusion is so general. They are apparently applicable not only to the quantum algorithm but also to any adiabatic process. Were they true then the adiabatic theorem would have been wrong. In the below we examine the crucial steps in the arguments and point out their shortcoming. We follow the notations of [1] and just pick up at the crucial inequality (the unnumbered, last inequality of the paper) ‖g(T) 〉 − g0(T)〉 ‖ ≤ T ‖HP gI〉‖. (1) where g(T) 〉 is the end state arrived at some time T in a supposedly adiabatic process which starts with the initial state gI 〉 and ends with the hamiltonian HP. The state g0(T) 〉 is constructed so that it only differs from the initial state gI 〉 by a Tdependent phase factor. Then from the fact that lim xmin→ ∞ ‖HP gI〉 ‖ = lim xmin→ ∞ 〈xmingI〉  = 0 (2) (where xmin 〉 is the soughtafter state, contained in HP), it was concluded that the left hand side of (1) can be vanishingly small and thus that g(T) 〉 can never be closed to xmin〉
A Formalization of the ChurchTuring Thesis for StateTransition Models
"... Abstract. Our goal is to formalize the ChurchTuring Thesis for a very large class of computational models. Specifically, the notion of an “effective model of computation ” over an arbitrary countable domain is axiomatized. This is accomplished by modifying Gurevich’s “Abstract State Machine ” postu ..."
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Abstract. Our goal is to formalize the ChurchTuring Thesis for a very large class of computational models. Specifically, the notion of an “effective model of computation ” over an arbitrary countable domain is axiomatized. This is accomplished by modifying Gurevich’s “Abstract State Machine ” postulates for statetransition systems. A proof is provided that all models satisfying our axioms, regardless of underlying data structure—and including all standard statetransition models—are equivalent to (up to isomorphism), or weaker than, Turing machines. To allow the comparison of arbitrary models operating over arbitrary domains, we employ a quasiordering on computational models, based on their extensionality. LCMs can do anything that could be described as “rule of thumb ” or “purely mechanical”.... This is sufficiently well established that it is now agreed amongst logicians that “calculable by means of an LCM” is the correct accurate rendering of such phrases. 1
A possible hypercomputational quantum algorithm
 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 spe ..."
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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.
Towards a theory of intelligence
 Theoretical Computer Science
"... In 1950, Turing suggested that intelligent behavior might require “a departure from the completely disciplined behaviour involved in computation”, but nothing that a digital computer could not do. In this paper, I want to explore Turing’s suggestion by asking what it is, beyond computation, that int ..."
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In 1950, Turing suggested that intelligent behavior might require “a departure from the completely disciplined behaviour involved in computation”, but nothing that a digital computer could not do. In this paper, I want to explore Turing’s suggestion by asking what it is, beyond computation, that intelligence might require, why it might require it and what knowing the answers to the first two questions might do to help us understand artificial and natural intelligence.
MATHEMATICAL COMPUTABILITY QUESTIONS FOR SOME CLASSES OF LINEAR AND NONLINEAR DIFFERENTIAL EQUATIONS ORIGINATED FROM HILBERT’S TENTH PROBLEM
, 2005
"... Abstract. Inspired by Quantum Mechanics, we reformulate Hilbert’s tenth problem in the domain of integer arithmetics into problems involving either a set of infinitelycoupled nonlinear differential equations or a class of linear Schrödinger equations with some appropriate timedependent Hamiltonian ..."
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Abstract. Inspired by Quantum Mechanics, we reformulate Hilbert’s tenth problem in the domain of integer arithmetics into problems involving either a set of infinitelycoupled nonlinear differential equations or a class of linear Schrödinger equations with some appropriate timedependent Hamiltonians. We then raise the questions whether these two classes of differential equations are computable or not in some computation models of computable analysis. These are nontrivial and important questions given that: (i) not all computation models of computable analysis are equivalent, unlike the case with classical recursion theory; (ii) and not all models necessarily and inevitably reduce computability of real functions to discrete computations on Turing machines. However unlikely the positive answers to our computability questions, their existence should deserve special attention and be satisfactorily settled since such positive answers may also have interesting logical consequence back in the classical recursion theory for the ChurchTuring thesis.