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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 61 (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 33 (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
TWIN PARADOX AND THE LOGICAL FOUNDATION OF RELATIVITY THEORY
, 2005
"... Abstract. We study the foundation of spacetime theory in the framework of firstorder logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for spacetime theory (or relativity). First we recall a ..."
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Cited by 25 (17 self)
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Abstract. We study the foundation of spacetime theory in the framework of firstorder logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for spacetime theory (or relativity). First we recall a simple and streamlined FOLaxiomatization Specrel of special relativity from the literature. Specrel is complete with respect to questions about inertial motion. Then we ask ourselves whether we can prove the usual relativistic properties of accelerated motion (e.g., clocks in acceleration) in Specrel. As it turns out, this is practically equivalent to asking whether Specrel is strong enough to “handle ” (or treat) accelerated observers. We show that there is a mathematical principle called induction (IND) coming from real analysis which needs to be added to Specrel in order to handle situations involving relativistic acceleration. We present an extended version AccRel of Specrel which is strong enough to handle accelerated motion, in particular, accelerated observers. Among others, we show that the Twin Paradox becomes provable in AccRel, but it is not provable without IND. Key words: twin paradox, relativity theory, accelerated observers, firstorder logic, axiomatization, foundation of relativity theory 1.
Logic of spacetime and relativity theory
, 2006
"... 2.1 Motivation for special relativistic kinematics in place of Newtonian kinematics......................... 4 ..."
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Cited by 23 (12 self)
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2.1 Motivation for special relativistic kinematics in place of Newtonian kinematics......................... 4
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 22 (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.
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
Elementarily computable functions over the real numbers and subrecursive functions
 Theoretical Computer Science, 348(23):130 – 147, 2005. Automata, Languages and Programming: Algorithms and Complexity (ICALPA
, 2004
"... Abstract 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 composit ..."
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Cited by 17 (6 self)
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Abstract 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 dene 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. 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 15 (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
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 14 (14 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.