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Hypercomputation and the Physical ChurchTuring Thesis
, 2003
"... A version of the ChurchTuring 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 logicomathematical, proposition. We review the main approaches to computation beyond Tu ..."
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Cited by 21 (0 self)
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A version of the ChurchTuring 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 logicomathematical, proposition. We review the main approaches to computation beyond Turing definability (`hypercomputation'): supertask, nonwellfounded, 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 ChurchTuring Thesis.
Can Newtonian systems, bounded in space, time, mass and energy compute all functions?
"... In the theoretical analysis of the physical basis of computation there is a great deal of confusion and controversy (e.g., on the existence of hypercomputers). First, we present a methodology for making a theoretical analysis of computation by physical systems. We focus on the construction and anal ..."
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In the theoretical analysis of the physical basis of computation there is a great deal of confusion and controversy (e.g., on the existence of hypercomputers). First, we present a methodology for making a theoretical analysis of computation by physical systems. We focus on the construction and analysis of simple examples that are models of simple subtheories of physical theories. Then we illustrate the methodology, by presenting a simple example for Newtonian Kinematics, and a critique that leads to a substantial extension of the methodology. The example proves that for any set A of natural numbers there exists a 3dimensional Newtonian kinematic system MA, with an infinite family of particles Pn whose total mass is bounded, and whose observable behaviour can decide whether or not n ∈ A for all n ∈ N in constant time. In particular, the example implies that simple Newtonian kinematic systems that are bounded in space, time, mass and energy can compute all possible sets and functions on discrete data. The system is a form of marble run and is a model of a small fragment of Newtonian Kinematics. Next, we use the example to extend the methodology. The marble run shows that a formal theory for computation by physical systems needs strong conditions on the notion of experimental procedure and, specifically, on methods for the construction of equipment. We propose to extend the methodology by defining languages to express experimental procedures and the construction of equipment. We conjecture that the functions computed by experimental computation in Newtonian Kinematics are “equivalent” to those computed by algorithms, i.e. the partial computable functions.
How much can analog and hybrid systems be proved (super)Turing
 Applied Mathematics and Computation
, 2006
"... 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 superTuring power must have something unreasonable. Our aim is to discuss how much theoretical computer science can ..."
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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 superTuring 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 superTuring. 1
Elements of a More Comprehensive Theory of Computing
 BioSystems
, 1999
"... Problems implementing DNA computers stem from the physical nature of molecules and their reactions. The present theory of computation requires assumptions that, at best, are extremely crude approximations of the physical chemistry. Here I consider the hypothesis that discarding those assumptions in ..."
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Cited by 3 (2 self)
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Problems implementing DNA computers stem from the physical nature of molecules and their reactions. The present theory of computation requires assumptions that, at best, are extremely crude approximations of the physical chemistry. Here I consider the hypothesis that discarding those assumptions in favor of more physically realistic descriptions would produce a more comprehensive theory of computing, yielding both theoretical insights and help in designing better molecular computers. I describe the discordances between the theories of physical biochemistry and computation, indicate some elements of a more comprehensive theory, and discuss some of the challenges the construction of a unified theory faces. keywords: molecular computing DNA computing physical biochemistry theory of computing 1 Hypothesis Molecular computing is justifiably exciting, not least for the alluring prospect of biologicallyinspired machines nicely handling NPcomplete problems. However, current molecular ...
Five views of hypercomputation
"... 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. ..."
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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.
After the Turing Machine
, 1999
"... Problems in implementing DNA and other types of molecular computers stem from the inherent physical nature of molecules and their reactions. The current theory of computation makes assumptions that at best are very crude approximations of the physical chemistry. A theory which took into account the ..."
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Problems in implementing DNA and other types of molecular computers stem from the inherent physical nature of molecules and their reactions. The current theory of computation makes assumptions that at best are very crude approximations of the physical chemistry. A theory which took into account the physical chemistry would likely be very different from what we have now and should help in designing more optimal molecular computing systems. In this paper I describe briefly the discordance between these assumptions and the physical chemistry; indicate some of the properties a more physically realistic model might have; and sketch some of the possibilities for a computing system which exploited the physical chemistry. We are developing a model for networks of biochemical reactions and molecules incorporating treatments of both continuous and discrete aspects of biochemical systems. The formal system of the model provides an example of what a CPC might be if the problems of encodi...
Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains
"... Abstract. In this paper we consider the computational complexity of solving initialvalue problems defined with analytic ordinary differential equations (ODEs) over unbounded domains of R n and C n, under the Computable Analysis setting. We show that the solution can be computed in polynomial time o ..."
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Abstract. In this paper we consider the computational complexity of solving initialvalue problems defined with analytic ordinary differential equations (ODEs) over unbounded domains of R n and C n, under the Computable Analysis setting. We show that the solution can be computed in polynomial time over its maximal interval of definition, provided it satisfies a very generous bound on its growth, and that the function admits an analytic extension to the complex plane. 1