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. 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.

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Church-Turing Thesis as saying something about the scope and limitations of physical computing machines. Although this was not the intention of Church or Turing, the Physical Church Turing thesis is interesting in its own right. Consider, for example, Wolfram’s formulation: One can expect in fact that universal computers are as powerful in their computational capabilities as any physically realizable system can be, that they can simulate any physical system...Nophysically implementable procedure could then shortcut a computationally irreducible process. (Wolfram 1985) Wolfram’s thesis consists of two parts: (a) Any physical system can be simulated (to any degree of approximation) by a universal Turing machine (b) Complexity bounds on Turing machine simulations have physical significance. For example, suppose that the computation of the minimum energy of some system of n particles takes at least exponentially (in n) many steps. Then the relaxation time of the actual physical system to its minimum energy state will also take exponential time. An even more extreme formulation of (more or less) the same thesis is due to Aharonov (1998): A probabilistic Turing machine can simulate any reasonable physical device in polynomial cost. She calls this The Modern Church Thesis. Aharonov refers here to probabilistic Turing machines that use random numbers in addition to the usual deterministic table of steps. It seems that such machines are capable to perform certain tasks faster than fully deterministic machines. The most famous randomized algorithm of that kind concerns the decision whether a given natural number is prime. A probabilistic algorithm that decides primality in a number of

Accelerating Turing machines are abstract devices that have the same computational structure as Turing machines, but can perform super-tasks. I argue that performing super-tasks alone does not buy more computational power, and that accelerating Turing machines do not solve the halting problem. To show this, I analyze the reasoning that leads to Thomson's paradox, point out that the paradox rests on a conflation of different perspectives of accelerating processes, and conclude that the same conflation underlies the claim that accelerating Turing machines can solve the halting problem.

The notes I handle no better than many pianists. But the pauses between the notes-- ah, that is where the art resides. "-- Artur Schabel `Antidisestablishmentarianism ' is the longest word. But what is longest possible word? And what is the shortest possible word? Reflection on these questions have prompted me write the longest essay that has ever been written. This is it. So, sit back. Nothing will ever be longer because this one contains infinitely many sentences. Word length = ∞?This may seem impossible. Only finitely many symbols can be inscribed on a page. Even if I wrote smaller and smaller, I would eventually run out of inscribable surfaces. I cannot autograph an atom. Even if I had unlimited time, I would run out of space. I appear condemned to produce only finitely many sentences.

We discuss the question of how to operationally validate whether or not a “hypercomputer ” performs better than the known discrete computational models. 1.

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The intuition guiding the de…nition of computation has shifted over time, a process that is re‡ected in the changing formulations of the Church-Turing thesis. The theory of computation began with logic and gradually moved to the capacity of …nite automata. Consequently, modern computer models rely on general physical principles, with quantum computers representing the extreme case. The paper discusses this development, and the challenges to the Church-Turing thesis in its physical form, in particular, Kieu’s quantum computer and relativistic hyper-computation. Finally, the robustness of the boundary between polynomial and exponential time complexity is considered in connection with quantum computers and quantum information theory. Key words: Church-Turing thesis, hyper-computation, quantum computers 1 The Church-Turing thesis and the meaning of ‘computable function’ The common formulation of the Church-Turing thesis runs as follows: Every computable function is computable by a Turing machine

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