Results 1  10
of
22
Abstract versus concrete computation on metric partial algebras
 ACM Transactions on Computational Logic
, 2004
"... Data types containing infinite data, such as the real numbers, functions, bit streams and waveforms, are modelled by topological manysorted algebras. In the theory of computation on topological algebras there is a considerable gap between socalled abstract and concrete models of computation. We pr ..."
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Cited by 30 (19 self)
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Data types containing infinite data, such as the real numbers, functions, bit streams and waveforms, are modelled by topological manysorted algebras. In the theory of computation on topological algebras there is a considerable gap between socalled abstract and concrete models of computation. We prove theorems that bridge the gap in the case of metric algebras with partial operations. With an abstract model of computation on an algebra, the computations are invariant under isomorphisms and do not depend on any representation of the algebra. Examples of such models are the ‘while ’ programming language and the BCSS model. With a concrete model of computation, the computations depend on the choice of a representation of the algebra and are not invariant under isomorphisms. Usually, the representations are made from the set N of natural numbers, and computability is reduced to classical computability on N. Examples of such models are computability via effective metric spaces, effective domain representations, and type two enumerability. The theory of abstract models is stable: there are many models of computation, and
Science, Computational Science and Computer Science: At a Crossroads
 Comm. ACM
, 1993
"... We describe computational science as an interdisciplinary approach to doing science on computers. Our purpose is to introduce computational science as a legitimate interest of computer scientists. We present a foundation for computational science based on the need to incorporate computation at the s ..."
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Cited by 25 (2 self)
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We describe computational science as an interdisciplinary approach to doing science on computers. Our purpose is to introduce computational science as a legitimate interest of computer scientists. We present a foundation for computational science based on the need to incorporate computation at the scientific level; i.e., computational aspects must be considered when a model is formulated. We next present some obstacles to computer scientists' participation in computational science, including a cultural bias in computer science that inhibits participation. Finally, we look at some areas of conventional computer science and indicate areas of mutual interest between computational science and computer science. Keywords: education, computational science. 1 What is Computational Science ? In December, 1991, the U. S. Congress passed the High Performance Computing and Communications Act, commonly known as the HPCC . This act focuses on several aspects of computing technology, but two have...
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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Cited by 14 (5 self)
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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.
Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a ..."
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Cited by 12 (5 self)
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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.
Number Computability and Domain Theory
 Information and Computation
, 1996
"... We present the different constructive definitions of real number that can be found in the literature. Using domain theory we analyse the notion of computability that is substantiated by these definitions and we give a definition of computability for real numbers and for functions acting on them. Thi ..."
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Cited by 8 (0 self)
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We present the different constructive definitions of real number that can be found in the literature. Using domain theory we analyse the notion of computability that is substantiated by these definitions and we give a definition of computability for real numbers and for functions acting on them. This definition of computability turns out to be equivalent to other definitions given in the literature using different methods. Domain theory is a useful tool to study higher order computability on real numbers. An interesting connection between Scotttopology and the standard topologies on the real line and on the space of continuous functions on reals is stated. An important result in this paper is the proof that every computable functional on real numbers is continuous w.r.t. the compact open topology on the function space. 1
Π 0 1 sets and models of WKL0
"... We show that any two Medvedev complete Π 0 1 subsets of 2 ω are recursively homeomorphic. We obtain a Π 0 1 set Q ′ of countable coded ωmodels of WKL0 with a strong homogeneity property. We show that if G is a generic element of Q ′ , then the ωmodel of WKL0 coded by G satisfies ∀X∀Y (if X is de ..."
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Cited by 8 (5 self)
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We show that any two Medvedev complete Π 0 1 subsets of 2 ω are recursively homeomorphic. We obtain a Π 0 1 set Q ′ of countable coded ωmodels of WKL0 with a strong homogeneity property. We show that if G is a generic element of Q ′ , then the ωmodel of WKL0 coded by G satisfies ∀X∀Y (if X is definable from Y, then X is Turing reducible to Y). We use a result of Kučera to refute some plausible conjectures concerning ωmodels of WKL0. We generalize our results to nonωmodels of WKL0. We discuss the significance of our results for foundations of mathematics.
Effective Computability of Solutions of Differential Inclusions The Ten Thousand Monkeys Approach
"... Abstract: In this paper we consider the computability of the solution of the initialvalue problem for differential inclusions with semicontinuous righthand side. We present algorithms for the computation of the solution using the “ten thousand monkeys” approach, in which we generate all possible so ..."
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Cited by 5 (2 self)
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Abstract: In this paper we consider the computability of the solution of the initialvalue problem for differential inclusions with semicontinuous righthand side. We present algorithms for the computation of the solution using the “ten thousand monkeys” approach, in which we generate all possible solution tubes, and then check which are valid. In this way, we show that the solution of an uppersemicontinuous differential inclusion is uppersemicomputable, and the solution of a differential inclusion defined by a onesided locally Lipschitz function is lowersemicomputable computable. We show that the solution of a locally Lipschitz differential equation is computable even if the function is not effectively locally Lipschitz. We also recover a result of Ruohonen, in which it is shown that if the solution is unique, then it is computable, even if the righthand side is not locally Lipschitz. We also prove that the maximal interval of existence for the solution must be effectively enumerable open, and give an example of a computable locally Lipschitz function which is not effectively locally Lipschitz.
A Constructive Theory of PointSet Nearness
 in Proceedings of Topology in Computer Science: Constructivity; Asymmetry and Partiality; Digitization, Seminar in Dagstuhl, Germany, 4–9 June 2000; Springer Lecture Notes in Computer Science
, 2001
"... An axiomatic constructive development of the theory of nearness and apartness of a point and a set is introduced as a setting for topology. ..."
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Cited by 2 (1 self)
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An axiomatic constructive development of the theory of nearness and apartness of a point and a set is introduced as a setting for topology.
On the Stability of Fast Polynomial Arithmetic
"... Operations on univariate dense polynomials—multiplication, division with remainder, multipoint evaluation—constitute central primitives entering as buildup blocks into many higher applications and algorithms. Fast Fourier Transform permits to accelerate them from naive quadratic to running time O(n ..."
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Cited by 2 (0 self)
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Operations on univariate dense polynomials—multiplication, division with remainder, multipoint evaluation—constitute central primitives entering as buildup blocks into many higher applications and algorithms. Fast Fourier Transform permits to accelerate them from naive quadratic to running time O(n · polylogn), that is softly linear in the degree n of the input. This is routinely employed in complexity theoretic considerations and, over integers and finite fields, in practical number theoretic calculations. The present work explores the benefit of fast polynomial arithmetic over the field of real numbers where the precision of approximation becomes crucial. To this end, we study the computability of the above operations in the sense of Recursive Analysis as an effective refinement of continuity. This theoretical worstcase stability analysis is then complemented by an empirical evaluation: We use GMP and the iRRAM to find the precision required for the intermediate calculations in order to achieve a desired output accuracy. 1
Constructive Mathematics and Quantum Physics
, 1999
"... This paper is dedicated to the memory of Prof. Gottfried T. Ru ttimann ..."
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Cited by 1 (0 self)
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This paper is dedicated to the memory of Prof. Gottfried T. Ru ttimann