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We discuss combining physical experiments with machine computations and introduce a form of analogue-digital (AD) Turing machine. We examine in detail a case study where an experimental procedure based on Newtonian kinematics is combined with a class of Turing machines. Three forms of AD machine are studied, in which physical parameters can be set exactly and approximately. Using non-uniform complexity theory, and some probability, we prove theorems that show that these machines can compute more than classical Turing machines.

We pose the following question: If a physical experiment were to be completely controlled by an algorithm, what effect would the algorithm have on the physical measurements made possible by the experiment?In a programme to study the nature of computation possible by physical systems, and by algorithms coupled with physical systems, we have begun to analyse (i) the algorithmic nature of experimental procedures, and (ii) the idea of using a physical experiment as an oracle to Turing Machines. To answer the question, we will extend our theory of experimental oracles in order to use Turing machines to model the experimental procedures that govern the conduct of physical experiments. First, we specify an experiment that measures mass via collisions in Newtonian Dynamics; we examine its properties in preparation for its use as an oracle. We start to classify the computational power of polynomial time Turing machines with this experimental oracle using non-uniform complexity classes. Second, we show that modelling an experimenter and experimental procedure algorithmically imposes a limit on what can be measured with equipment. Indeed, the theorems suggest a new form of uncertainty principle for our knowledge of physical quantities measured in simple physical experiments. We argue that the results established here are representative of a huge class of experiments.

Earlier, to explore the idea of combining physical experiments with algorithms, we introduced a new form of analogue-digital (AD) Turing machine. We examined in detail a case study where an experimental procedure, based on Newtonian kinematics, is used as an oracle with classes of Turing machines. The physical cost of oracle calls was counted and three forms of AD queries were studied, in which physical parameters can be set exactly and approximately. Here, in this sequel, we complete the classification of the computational power of these AD Turing machines and determine precisely what they can compute, using non-uniform complexity classes and probabilities.

We study a restricted version of Shannon's general purpose analog computer in which we only allow the machine to solve linear differential equations. We show that if this computer is allowed to sense inequalities in a differentiable way, then it can compute exactly the elementary functions, the smallest known recursive class closed under time and space complexity. Furthermore, we show that if the machine has access to a function f ðxÞ with a suitable growth as x goes to infinity, then it can compute functions on any given level of the Grzegorczyk hierarchy. More precisely, we show that the model contains exactly the nth level of the Grzegorczyk hierarchy if it is allowed to solve n À 3 non-linear differential equations of a certain kind. Therefore, we claim that, at least in this region of the complexity hierarchy, there is a close connection between analog complexity classes, the dynamical systems that compute them, and classical sets of subrecursive functions. # 2002 Elsevier Science (USA)

In the last years, recursive functions over the reals (Theoret. Comput. Sci. 162 (1996) 23) have been considered, first as a model of analog computation, and second to obtain analog characterizations of classical computational complexity classes (Unconventional Models of Computation, UMC 2002, minimalization operator, has not been considered: (a) although differential recursion (the analog counterpart of classical recurrence) is, in some extent, directly implementable in the General Purpose Analog Computer of Claude Shannon, analog minimalization is far from physical realizability, and (b) analog minimalization was borrowed from classical recursion theory and does not fit well the analytic realm of analog computation. In this paper, we show that a most natural operator captured from analysis-the operator of taking a limit-can be used properly to enhance the theory of recursion over the reals, providing good solutions to puzzling problems raised by the original model.

Shannon's general purpose analog computer (GPAC) is an elegant model of analog computation in continuous time. In this paper, we consider whether the set G of GPAC-computable functions is closed under iteration, that is, whether for any function f (x) # G there is a function F(x, t) # G such that F(x, t)= f t (x) for nonnegative integers t. We show that G is not closed under iteration, but a simple extension of it is. In particular, if we relax the definition of the GPAC slightly to include unique solutions to boundary value problems, or equivalently if we allow functions x k %(x) that sense inequalities in a differentiable way, the resulting class, which we call G+% k , is closed under iteration. Furthermore, G+% k includes all primitive recursive functions and has the additional closure property that if T(x) is in G+% k , then any function of x computable by a Turing machine in T(x) time is also. Academic Press

We developed earlier a theory of combining algorithms with physical systems, on the basis of using physical experiments as oracles to algorithms. Although our concepts and methods are general, each physical oracle requires its own analysis, on the basis of some fragment of physical theory that specifies the equipment and its behaviour. For specific examples of physical systems (mechanical, optical, electrical), the computational power has been characterized using non-uniform complexity classes. The powers of the known examples vary according to assumptions on precision and timing but seem to lead to the same complexity classes, namely P/ log and BPP// log . In this study, we develop sets of axioms for the interface between physical equipment and algorithms that allow us to prove general characterizations, in terms of P/ log and BPP// log , for large classes of physical oracles, in a uniform way. Sufficient conditions on physical equipment are given that ensure a physical system satisfies the axioms.

Link to this article: http://journals.cambridge.org/abstract_S0960129511000557How to cite this article: EDWIN J. BEGGS, JOSÉ FÉLIX COSTA and JOHN V. TUCKER (2012). The impact of models of a physical oracle on computational power.Using physical experiments as oracles for algorithms, we can characterise the computational power of classes of physical systems. Here we show that two different physical models of the apparatus for a single experiment can have different computational power. The experiment is the scatter machine experiment (SME), which was first presented in Beggs and Tucker (2007b). Our first physical model contained a wedge with a sharp vertex that made the experiment non-deterministic with constant runtime. We showed that Turing machines with polynomial time and an oracle based on a sharp wedge computed the non-uniform complexity class P /poly. Here we reconsider the experiment with a refined physical model where the sharp vertex of the wedge is replaced by any suitable smooth curve with vertex at the same point. These smooth models of the experimental apparatus are deterministic. We show that no matter what shape is chosen for the apparatus:(i) the time of detection of the scattered particles increases at least exponentially with the size of the query; and (ii) Turing machines with polynomial time and an oracle based on a smooth wedge compute the non-uniform complexity class P /log * $ P /poly.We discuss evidence that many experiments that measure quantities have exponential runtimes and a computational power of P /log * . † The authors would like to thank EPSRC for their support under grant EP/C525361/1. The research of José Félix Costa is also supported by Fundação para a Ciência e Tecnologia, Financiamento Base 2010-ISFL-1-209.

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