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Geometrical Accumulations and Computably Enumerable Real Numbers

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Book cover Unconventional Computation (UC 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6714))

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Abstract

Abstract geometrical computation involves drawing colored line segments (traces of signals) according to rules: signals with similar color are parallel and when they intersect, they are replaced according to their colors. Time and space are continuous and accumulations can be devised to unlimitedly accelerate a computation and provide, in a finite duration, exact analog values as limits.

In the present paper, we show that starting with rational numbers for coordinates and speeds, the time of any accumulation is a c.e. (computably enumerable) real number and moreover, there is a signal machine and an initial configuration that accumulates at any c.e. time. Similarly, we show that the spatial positions of accumulations are exactly the d-c.e. (difference of computably enumerable) numbers. Moreover, there is a signal machine that can accumulate at any c.e. time or d-c.e. position.

http://www.univ-orleans.fr/lifo/Members/Jerome.Durand-Lose ,

Jerome.Durand-Lose@univ-orleans.fr

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Authors and Affiliations

  1. LIFO, Université d’Orléans, B.P. 6759, F-45067, ORLÉANS Cedex 2

    Jérôme Durand-Lose

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  1. Jérôme Durand-Lose
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Editors and Affiliations

  1. Department of Computer Science, Science Centre, The University of Auckland, 38 Princes Street, 1142, Auckland, New Zealand

    Cristian S. Calude

  2. Department of Mathematics, University of Turku, 20014, Turku, Finland

    Jarkko Kari

  3. Department of Information Technologies, Åbo Akademi University, ICT Building, Joukahaisenkatu 3-5 A, 20520, Turku, Finland

    Ion Petre

  4. Leiden Institute of Center Advanced Computer Science (LIACS), Leiden University, Niels Bohrweg 1, 2333, Leiden, CA, The Netherlands

    Grzegorz Rozenberg

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Durand-Lose, J. (2011). Geometrical Accumulations and Computably Enumerable Real Numbers. In: Calude, C.S., Kari, J., Petre, I., Rozenberg, G. (eds) Unconventional Computation. UC 2011. Lecture Notes in Computer Science, vol 6714. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21341-0_15

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  • DOI: https://doi.org/10.1007/978-3-642-21341-0_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-21340-3

  • Online ISBN: 978-3-642-21341-0

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