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A survey on continuous time computations
 New Computational Paradigms
"... Abstract. We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing resu ..."
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Abstract. We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature. 1
Abstract Embedding infinitely parallel computation in Newtonian kinematics
"... First, we reflect on computing sets and functions using measurements from experiments with a class of physical systems. We call this experimental computation. We outline a programme to analyse theoretically experimental computation in which a central problem is: Given a physical theory T, explore an ..."
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First, we reflect on computing sets and functions using measurements from experiments with a class of physical systems. We call this experimental computation. We outline a programme to analyse theoretically experimental computation in which a central problem is: Given a physical theory T, explore and classify the computational models that can be embedded in, and abstracted from, the physical systems specified by the physical theory T. We consider the embedding of arbitrary sets, functions, programs, and computers into designs for systems that can be specified in subtheories or fragments T of Newtonian kinematics in order to explore some of the physical assumptions of T that allows its systems to qualify as hypercomputers, i.e. physical models that compute sets and functions that cannot be computed in classical computability theory. In designing systems we work strictly within the chosen theory T and do not concern ourself with whether or not T is valid of the world today. We are interested in exploring the subtheory from a computational point of view and especially in restrictions on the assumptions of T that allow us to return from hypercomputation to classical computation. Secondly, we give a construction of an infinitely parallel machine that can decide all the arithmetical sets of natural numbers. We embed this hypercomputer as system in 3dimensions obeying the laws of a fragment of Newtonian kinematics. In particular, the example shows that communication allowable in Newtonian kinematics is especially powerful. We conclude with further reflections and open problems.
Embedding infinitely parallel computation in Newtonian kinematics
"... First, we reflect on computing sets and functions using measurements from experiments with a class of physical systems. We call this experimental computation. We outline a programme to analyse theoretically experimental computation in which a central problem is: Given a physical theory T, explore an ..."
Abstract
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First, we reflect on computing sets and functions using measurements from experiments with a class of physical systems. We call this experimental computation. We outline a programme to analyse theoretically experimental computation in which a central problem is: Given a physical theory T, explore and classify the computational models that can be embedded in, and abstracted from, the physical systems specified by the physical theory T. We consider the embedding arbitrary sets, functions, programs, and computers into designs for systems that can be specified in subtheories or fragments T of Newtonian kinematics in order to explore some of the physical assumptions of T that allowed its systems to qualify as hypercomputers, i.e., physical models that compute sets and functions that cannot be computed in classical computability theory. In designing systems we work strictly within the chosen theory T and do not concern ourself with whether or not T is valid of the world today. We are interested in exploring the subtheory from a computational point of view and especially in restrictions on the assumptions of T that allow us to return from hypercomputation to classical computation. Secondly, we give a construction of an infinitely parallel machine that can decide all the arithmetical sets of natural numbers. We embed this hypercomputer as system in 3dimensions obeying the laws of a fragment of Newtonian kinematics. In particular, the example shows that communication allowable in Newtonian kinematics is especially powerful. We conclude with further reflections and open problems.
12345efghi UNIVERSITY OF WALES SWANSEA REPORT SERIES
"... Newtonian mechanics and infinitely parallel computation by ..."
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Introduction
"... Motivation We decided to have two chapters about our motivation. Thus, Chapter 1 presents dynamical systems, with a mathematical point of view, rather classical, with wellknown examples from literature. ..."
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Motivation We decided to have two chapters about our motivation. Thus, Chapter 1 presents dynamical systems, with a mathematical point of view, rather classical, with wellknown examples from literature.
A Survey on Continuous Time Computations (Chapter of the book “New Computational Paradigms”, To appear, Springer)
"... Abstract. We provide an overview of computation theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, sum ..."
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Abstract. We provide an overview of computation theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature. 1