We describe a possible physical device that computes a function that cannot be computed by a Turing machine. The device is physical in the sense that it is compatible with General Relativity. We discuss some objections, focusing on those which deny that the device is either a computer or computes a function that is not Turing computable. Finally, we argue that the existence of the device does not refute the Church–Turing thesis, but nevertheless may be a counterexample to Gandy’s thesis.

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

Abstract. One of the biggest challenges of science today is to outline connections between the subjective world of human experience, as studied by psychology, and the objective world of measurable brain events, as studied by neuroscience. In this paper a series of approximations to neural dynamics is outlined, leading to a phenomenological theory of mind based on concepts directly related to human cognition. Behaviorism is based on an engineering approach, treating the mind as a control system for the organism. This corresponds to an approximation of the recurrent neural dynamics (brain states) by finite state automata (behavioral states). Another approximations to neural dynamics is described, leading to a Platonic-like model of mind based on psychological spaces. Objects and events in these spaces correspond to quasi-stable states of brain dynamics and may be interpreted from psychological point of view. Platonic model bridges the gap between the neurophysiological brain events and higher cognitive functions realized by the mind. Categorization experiments with human subjects are presented as a challenge for mind-brain theories. Wider implications of this model as a basis for cognitive science are discussed and possible extensions outlined. 1 1.1

Taking the view that computation is after all physical, we argue that physics, particularly quantum physics, could help extend the notion of computability. Here, we list the important and unique features of quantum mechanics and then outline a quantum mechanical “algorithm” for one of the insoluble problems of mathematics, the Hilbert’s tenth and equivalently the Turing halting problem. The key element of this algorithm is the computability and measurability of both the values of physical observables and of the quantum-mechanical probability distributions for these values. The fact is that quantum computers can prove theorems by methods that neither a human brain nor any other Turing-computational arbiter will ever be able to reproduce. What if a quantum algorithm delivered a theorem that it was infeasible to prove classically. No such algorithm is yet known, but nor is anything known to rule out such a possibility, and this raises a question of principle: should we still accept such a theorem as undoubtedly proved? We believe that the rational answer ot this question is yes, for our confidence in quantum proofs rests upon the same foundation as our confidence in classical proofs: our acceptance of the physical laws underlying the computing operations. D. Deustch, A. Ekert and R. Lupacchini [1] 1