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Abstract Geometrical Computation and the Linear Blum, Shub and Smale Model

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Computation and Logic in the Real World (CiE 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4497))

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Abstract

Abstract geometrical computation naturally arises as a continuous counterpart of cellular automata. It relies on signals (dimensionless points) traveling at constant speed in a continuous space in continuous time. When signals collide, they are replaced by new signals according to some collision rules. This simple dynamics relies on real numbers with exact precision and is already known to be able to carry out any (discrete) Turing-computation. The Blum, Shub and Small (BSS) model is famous for computing over ℝ (considered here as a ℝ unlimited register machine) by performing algebraic computations.

We prove that signal machines (set of signals and corresponding rules) and the infinite-dimension linear (multiplications are only by constants) BSS machines can simulate one another.

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Authors and Affiliations

  1. Laboratoire d’Informatique Fondamentale d’Orléans, Université d’Orléans, B.P. 6759, F-45067 ORLÉANS Cedex 2, France

    Jérôme Durand-Lose

Authors
  1. Jérôme Durand-Lose
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Editors and Affiliations

  1. Dept. of Pure Mathematics, University of Leeds, LS2 9JT, Leeds, UK

    S. Barry Cooper

  2. Institue for Logic, Language and Computation (ILLC), Universiteit van Amsterdam, Plantage Muidergracht 24, 1018, TV Amsterdam, The Netherlands

    Benedikt Löwe

  3. Dipartimento di Scienze Matematiche ed Informatiche “R. Magari”, University of Siena, 53100, Siena, Italy

    Andrea Sorbi

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Durand-Lose, J. (2007). Abstract Geometrical Computation and the Linear Blum, Shub and Smale Model. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds) Computation and Logic in the Real World. CiE 2007. Lecture Notes in Computer Science, vol 4497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73001-9_25

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  • DOI: https://doi.org/10.1007/978-3-540-73001-9_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73000-2

  • Online ISBN: 978-3-540-73001-9

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