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29
Computations via experiments with kinematic systems
, 2004
"... Consider the idea of computing functions using experiments with kinematic systems. We prove that for any set A of natural numbers there exists a 2dimensional kinematic system BA with a single particle P whose observable behaviour decides n ∈ A for all n ∈ N. The system is a bagatelle and can be des ..."
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Consider the idea of computing functions using experiments with kinematic systems. We prove that for any set A of natural numbers there exists a 2dimensional kinematic system BA with a single particle P whose observable behaviour decides n ∈ A for all n ∈ N. The system is a bagatelle and can be designed to operate under (a) Newtonian mechanics or (b) Relativistic mechanics. The theorem proves that valid models of mechanical systems can compute all possible functions on discrete data. The proofs show how any information (coded by some A) can be embedded in the structure of a simple kinematic system and retrieved by simple observations of its behaviour. We reflect on this undesirable situation and argue that mechanics must be extended to include a formal theory for performing experiments, which includes the construction of systems. We conjecture that in such an extended mechanics the functions computed by experiments are precisely those computed by algorithms. We set these theorems and ideas in the context of the literature on the general problem “Is physical behaviour computable? ” and state some open problems.
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.
Reducibility of Domain Representations and CantorWeihrauch Domain Representations
, 2006
"... We introduce a notion of reducibility of representations of topological spaces and study some basic properties of this notion for domain representations. A representation reduces to another if its representing map factors through the other representation. Reductions form a preorder on representatio ..."
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Cited by 8 (4 self)
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We introduce a notion of reducibility of representations of topological spaces and study some basic properties of this notion for domain representations. A representation reduces to another if its representing map factors through the other representation. Reductions form a preorder on representations. A spectrum is a class of representations divided by the equivalence relation induced by reductions. We establish some basic properties of spectra, such as, nontriviality. Equivalent representations represent the same set of functions on the represented space. Within a class of representations, a representation is universal if all representations in the class reduce to it. We show that notions of admissibility, considered both for domains and within Weihrauch’s TTE, are universality concepts in the appropriate spectra. Viewing TTE representations as domain representations, the reduction notion here is a natural generalisation of the one from TTE. To illustrate the framework, we consider some domain representations of real numbers and show that the usual interval domain representation, which is universal among dense representations, does not reduce to various Cantor domain representations. On the other hand, however, we show that a substructure of the interval domain more suitable for efficient computation of operations is equivalent to the usual interval domain with respect to reducibility. 1.
Oracles and Advice as Measurements
"... 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 ..."
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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 nonuniform 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
Computable total functions on metric algebras, universal algebraic specifications and dynamical systems
 THE JOURNAL OF LOGIC AND ALGEBRAIC PROGRAMMING
, 2005
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How to compare the power of computational models
 In Computability in Europe 2005: New Computational Paradigms
, 2005
"... Abstract. We argue that there is currently no satisfactory general framework for comparing the extensional computational power of arbitrary computational models operating over arbitrary domains. We propose a conceptual framework for comparison, by linking computational models to hypothetical physica ..."
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Abstract. We argue that there is currently no satisfactory general framework for comparing the extensional computational power of arbitrary computational models operating over arbitrary domains. We propose a conceptual framework for comparison, by linking computational models to hypothetical physical devices. Accordingly, we deduce a mathematical notion of relative computational power, allowing the comparison of arbitrary models over arbitrary domains. In addition, we claim that the method commonly used in the literature for “strictly more powerful” is problematic, as it allows for a model to be more powerful than itself. On the positive side, we prove that Turing machines and the recursive functions are “complete ” models, in the sense that they are not susceptible to this anomaly, justifying the standard means of showing that a model is “hypercomputational.” 1
A network model of analogue computation over metric algebras
 Torenvliet (Eds.), Computability in Europe, 2005, Springer Lecture Notes in Computer Science
, 2005
"... Abstract. We define a general concept of a network of analogue modules connected by channels, processing data from a metric space A, and operating with respect to a global continuous clock T. The inputs and outputs of the network are continuous streams u: T → A, and the inputoutput behaviour of the ..."
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Abstract. We define a general concept of a network of analogue modules connected by channels, processing data from a metric space A, and operating with respect to a global continuous clock T. The inputs and outputs of the network are continuous streams u: T → A, and the inputoutput behaviour of the network with system parameters from A is modelled by a function Φ: C[T,A] p ×A r →C[T,A] q (p, q> 0,r ≥ 0), where C[T,A] is the set of all continuous streams equipped with the compactopen topology. We give an equational specification of the network, and a semantics which involves solving a fixed point equation over C[T,A] using a contraction principle. We analyse a case study involving a mechanical system. Finally, we introduce a custommade concrete computation theory over C[T,A] and show that if the modules are concretely computable then so is the function Φ. 1
Continuity of Operators on Continuous and Discrete Time Streams
"... its diversity and scale and, especially, in its conceptual and technical depth. His achievement stands out and defies the simple conventions of contemporary judgment and praise. As a scientist and intellectual, and a longstanding friend and collaborator of ours, he is unique. For this birthday celeb ..."
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Cited by 1 (1 self)
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its diversity and scale and, especially, in its conceptual and technical depth. His achievement stands out and defies the simple conventions of contemporary judgment and praise. As a scientist and intellectual, and a longstanding friend and collaborator of ours, he is unique. For this birthday celebration, we offer congratulations and encouragement, and record our admiration and gratitude. We consider the semantics of networks processing streams of data from a complete metric space. We consider two types of data streams: those based on continuous time (used in networks of physical components and analog devices), and those based on discrete time (used in concurrent algorithms). The networks are both governed by global clocks and together model a huge range of systems. Previously, we have investigated these two types of networks separately. Here we combine their study in a unified theory of stream transformers, given as fixed points of equations. We begin to develop this theory by using the standard mathematical techniques of topology to prove certain computationally desirable properties of these semantic functions, notably continuity, which is significant for models of a physical system, according to Hadamard’s principle. Key words and phrases: analog computing, analog networks, compact open topology, continuous stream operations, continuous time streams, discrete time streams, fixed points, Hadamard’s principle, synchronous concurrent algorithms, topological algebras. 2 1