We describe an emerging field, that of nonclassical computability and nonclassical computing machinery. According to the nonclassicist, the set of well-defined computations is not exhausted by the computations that can be carried out by a Turing machine. We provide an overview of the field and a philosophical defence of its foundations.

Abstract not found

We propose certain non-Turing models of computation, but our intent is not to advocate models that surpass the power of Turing Machines (TMs), but to defend the need for models with orthogonal notions of power. We review the nature of models and argue that they are relative to a domain of application and are ill-suited to use outside that domain. Hence we review the presuppositions and context of the TM model and show that it is unsuited to natural computation (computation occurring in or inspired by nature). Therefore we must consider an expanded definition of computation that includes alternative (especially analog) models as well as the TM. Finally we present an alternative model, of continuous computation, more suited to natural computation. We conclude with remarks on the expressivity of formal mathematics. Key words: analog computation, analog computer, biocomputation, computability, computation on reals, continuous computation, formal system, hypercomputation,

Does Nature permit the implementation of behaviours that cannot be simulated computationally? We consider the meaning of physical computationality in some detail, and present arguments in favour of physical hypercomputation: for example, modern scientific method does not allow the specification of any experiment capable of refuting hypercomputation. We consider the implications of relativistic algorithms capable of solving the (Turing) Halting Problem. We also reject as a fallacy the argument that hypercomputation has no relevance because non-computable values are indistinguishable from sufficiently close computable approximations. In addition to

Accelerated Turing machines are Turing machines that perform tasks commonly regarded as impossible, such as computing the halting function. The existence of these notional machines has obvious implications concerning the theoretical limits of computability. 2 1. Introduction Neither Turing nor Post, in their descriptions of the devices we now call Turing machines, made much mention of time (Turing 1936, Post 1936). 1 They listed the primitive operations that their devices perform - read a square of the tape, write a single symbol on a square of the tape (first deleting any symbol already present), move one square to the right, and so forth - but they made no mention of the duration of each primitive operation. The crucial concept is that of whether or not the machine halts after a finite number of operations. Temporal considerations are not relevant to the functioning of the devices as described, nor - so we are clearly supposed to believe - to the soundness of the proofs that Turi...

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.

to a practical application. A dozen years later the first stored-program electronic digital computers began to spring into existence. All were modelled on the universal Turing machine. Today's digital computers also are in essence universal Turing machines. 2. Is There a Known Upper Bound to Computability? Many textbooks on the fundamentals of computer science offer examples of informationprocessing tasks that are, it is claimed, absolutely uncomputable, in the sense that no machine can be specified to carry out these tasks. For example, it is said that no machine can repond to any given (finite) string of binary digits in accordance with the following rules: 3 (1) Answer '1' if the string is a program that will cause a universal Turing machine on whose tape it is inscribed to execute only a finite number of operations (such programs are called 'terminating'). (2) Answer '0' if the string is not a terminating program; i.e. if the st

Abstract. In this paper we will try to understand how oracles and advice functions, which are mathematical abstractions in the theory of computability and complexity, can be seen as physical measurements in Classical Physics. First, we consider how physical measurements are a natural external source of information to an algorithmic computation. We argue that oracles and advice functions can help us to understand how the structure of space and time has information content that can be processed by Turing machines (after Cooper and Odifreddi [10] and Copeland and Proudfoot [11, 12]). We show that non-uniform complexity is an adequate framework for classifying feasible computations by Turing machines interacting with an oracle in Nature. By classifying the information content of such an oracle using Kolmogorov complexity, we obtain a hierarchical structure for advice classes. 1

Church thesis and its variants say roughly that all reasonable models of computation do not have more power than Turing Machines. In a contrapositive way, they say that any model with super-Turing power must have something unreasonable. Our aim is to discuss how much theoretical computer science can quantify this, by considering several classes of continuous time dynamical systems, and by studying how much they can be proved Turing or super-Turing. 1

Peter Wegner’s definition of computability differs markedly from the classical term as established by Church, Kleene, Markov, Post, Turing et al. Wegner identifies interaction as the main feature of today’s systems which is lacking in the classical treatment of computability. We compare the different approaches and argue whether or not Wegner’s criticism is appropriate. Taking into account the major arguments from the literature, we show that Church’s thesis still holds. 1.