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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
Finite Automata Models of Quantized Systems: Conceptual Status and Outlook
 Developments in Language Theory. Proceedings of the 6th International Conference, DLT 2002
, 2002
"... Since Edward Moore, finite automata theory has been inspired by physics, in particular by quantum complementarity. We review automaton complementarity, reversible automata and the connections to generalized urn models. Recent developments in quantum information theory may have appropriate formal ..."
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Cited by 3 (3 self)
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Since Edward Moore, finite automata theory has been inspired by physics, in particular by quantum complementarity. We review automaton complementarity, reversible automata and the connections to generalized urn models. Recent developments in quantum information theory may have appropriate formalizations in the automaton context.
Computational Power of Infinite Quantum Parallelism
 pp.2057–2071 in International Journal of Theoretical Physics vol.44:11
, 2005
"... Recent works have independently suggested that quantum mechanics might permit procedures that fundamentally transcend the power of Turing Machines as well as of ‘standard ’ Quantum Computers. These approaches rely on and indicate that quantum mechanics seems to support some infinite variant of class ..."
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Cited by 2 (1 self)
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Recent works have independently suggested that quantum mechanics might permit procedures that fundamentally transcend the power of Turing Machines as well as of ‘standard ’ Quantum Computers. These approaches rely on and indicate that quantum mechanics seems to support some infinite variant of classical parallel computing. We compare this new one with other attempts towards hypercomputation by separating (1) its computing capabilities from (2) realizability issues. The first are shown to coincide with recursive enumerability; the second are considered in analogy to ‘existence’ in mathematical logic. KEY WORDS: Hypercomputation; quantum mechanics; recursion theory; infinite parallelism.
BioSystems 77 (2004) 175–194 Biosteps beyond Turing
, 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 negativ ..."
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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. Consequently, two acceleration postulates of biological inspiration are introduced; each of them poses specific questions to biology. Finally, in a more speculative part of the paper, we will deal with Turing noncomputability activity of the brain and possible forms of (extraterrestrial) intelligence.
Turing machines can be efficiently simulated by the General Purpose Analog Computer
"... Abstract. The ChurchTuring thesis states that any sufficiently powerful computational model which captures the notion of algorithm is computationally equivalent to the Turing machine. This equivalence usually holds both at a computability level and at a computational complexity level modulo polynom ..."
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Abstract. The ChurchTuring thesis states that any sufficiently powerful computational model which captures the notion of algorithm is computationally equivalent to the Turing machine. This equivalence usually holds both at a computability level and at a computational complexity level modulo polynomial reductions. However, the situation is less clear in what concerns models of computation using real numbers, and no analog of the ChurchTuring thesis exists for this case. Recently it was shown that some models of computation with real numbers were equivalent from a computability perspective. In particular it was shown that Shannon’s General Purpose Analog Computer (GPAC) is equivalent to Computable Analysis. However, little is known about what happens at a computational complexity level. In this paper we shed some light on the connections between this two models, from a computational complexity level, by showing that, modulo polynomial reductions, computations of Turing machines can be simulated by GPACs, without the need of using more (space) resources than those used in the original Turing computation, as long as we are talking about bounded computations. In other words, computations done by the GPAC are as spaceefficient as computations done in the context of Computable Analysis. 1