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Irrationality Is Needed to Compute with Signal Machines with Only Three Speeds

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The Nature of Computation. Logic, Algorithms, Applications (CiE 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7921))

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Abstract

Space-time diagrams of signal machines on finite configurations are composed of interconnected line segments in the Euclidean plane. As the system runs, a network emerges. If segments extend only in one or two directions, the dynamics is finite and simplistic. With four directions, it is known that fractal generation, accumulation and any Turing computation are possible.

This communication deals with the three directions/speeds case. If there is no irrational ratio (between initial distances between signals or between speeds) then the network follows a mesh preventing accumulation and forcing a cyclic behavior. With an irrational ratio (here, the Golden ratio) between initial distances, it becomes possible to provoke an accumulation that generates infinitely many interacting signals in a bounded portion of the Euclidean plane. This behavior is then controlled and used in order to simulate a Turing machine and generate a 25-state 3-speed Turing-universal signal machine.

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Authors and Affiliations

  1. Laboratoire d’Informatique Fondamentale d’Orléans (LIFO), Université d’Orléans, B.P. 6759, 45067, ORLÉANS Cedex 2, France

    Jérôme Durand-Lose

Authors
  1. Jérôme Durand-Lose
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Editors and Affiliations

  1. Dipartimento die Informatica, Sistemistica e Comunicazione, Università degli Studi di Milano-Bicocca, Piazza dell’Ateneo Nuovo 1, 20126, Milan, Italy

    Paola Bonizzoni

  2. Institute for Theoretical Computer Science, Mathematics and Operations Research, Faculty of Computer Science, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577, Neubiberg, Germany

    Vasco Brattka

  3. Institute for Logic, Language and Computation, University of Amsterdam, Postbus 94242, 1090 GE, Amsterdam, The Netherlands

    Benedikt Löwe

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Durand-Lose, J. (2013). Irrationality Is Needed to Compute with Signal Machines with Only Three Speeds. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds) The Nature of Computation. Logic, Algorithms, Applications. CiE 2013. Lecture Notes in Computer Science, vol 7921. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39053-1_12

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  • DOI: https://doi.org/10.1007/978-3-642-39053-1_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-39052-4

  • Online ISBN: 978-3-642-39053-1

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