Algorithmic information theory
Algorithmic information theory
Hilbert's tenth problem
SIAM Journal on Computing
Quantum computation and quantum information
Quantum computation and quantum information
Minds and Machines
On the existence of a new family of Diophantine equations for Ω
Fundamenta Informaticae
Zeno machines and hypercomputation
Theoretical Computer Science
An explicit solution to Post's Problem over the reals
Journal of Complexity
Computational Complexity in Non-Turing Models of Computation
Electronic Notes in Theoretical Computer Science (ENTCS)
On the Possibilities of Hypercomputing Supertasks
Minds and Machines
An explicit solution to post's problem over the reals
FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
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Despite the recursive non-computability of Hilbert's tenth problem, we outline and argue for a quantum algorithm that is based on the Quantum Adiabatic Theorem. It is explained how this algorithm can solve Hilbert's tenth problem. The algorithm is then considered in the context of several "no-go" arguments against such hypercomputation. Logical arguments are usually based on Cantor's diagonal technique used for proving non-computability of the Turing halting problem, which is related to Hilbert's tenth problem. Physical arguments are related to the limited computability of a class of quantum computation based on qubits and dimensionally finite quantum logical gates.