A version of the Church-Turing 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 logico-mathematical, proposition. We review the main approaches to computation beyond Turing definability (`hypercomputation'): supertask, non-well-founded, 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 Church-Turing Thesis.

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

We give an update on a quantum adiabatic algorithm for the Turing noncomputable Hilbert’s tenth problem, and briefly go over some relevant issues and misleading objections to the algorithm. Keywords: Quantum adiabatic computation; Turing computability; Hilbert’s tenth problem 1.

The arguments employed in quant-ph/0111009, to claim that the quantum algorithm in quant-ph/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 ground-state 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 T-dependent phase factor. Then from the fact that lim xmin→ ∞ ‖HP |gI〉 ‖ = lim xmin→ ∞ |〈xmin|gI〉 | = 0 (2) (where |xmin 〉 is the sought-after 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〉

We overview different approaches to the study of hypercomputation and other investigations on the plausibility of the physical Church–Turing thesis. We propose five thesis to classify investigation in this area. Sly does it. Tiptoe catspaws. Slide and creep.

[26] argues that the diagonal argument of the number theorist Cantor can be used to elucidate issues that arose in the socialist calculation debate of the 1930s. In particular he contends that the diagonal argument buttresses the claims of the Austrian economists regarding the impossibility of rational planning. We challenge Murphy’s argument, both at the number theoretic level and from the standpoint of economic realism.