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Computing in Perfect Euclidean Frameworks

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Advances in Unconventional Computing

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 22))

Abstract

This chapter presents what kind of computation can be carried out using an Euclidean space—as input, memory, output...—with dedicated primitives. Various understandings of computing are encountered in such a setting allowing classical (Turing, discrete) computations as well as, for some, hyper and analog computations thanks to the continuity of space. The encountered time scales are discrete or hybrid (continuous evolution between discrete transitions). The first half of the chapter presents three models of computation based on geometric concepts—namely: ruler and compass, local constrains and emergence of polyhedra and piece-wise constant derivative. The other half concentrates on signal machines: line segments are extended; when they meet, they are replaced by others. Not only are these machines capable of classical computation but moreover, using the continuous nature of space and time they can also perform hyper-computation and analog computation. It is possible to build fractals and to go one step further on to use their partial generation to solve, e.g., quantified SAT in “constant space and time”.

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Notes

  1. 1.

    \(\varSigma _{0}\) is the recursive sets, \(\varSigma _{1}\) is recursively enumerable sets, e.g. the Halting problem.

  2. 2.

    Extension of the arithmetical hierarchy to ordinal indices.

  3. 3.

    The author is not satisfied enough with its code to put it on the internet but send it on request.

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Authors and Affiliations

  1. Laboratoire d’Informatique Fondamentale d’Orléans, Université d’Orléans, Batiment IIIA Rue Léonard de Vinci B.P. 6759 F-45067 Orléans Cedex 2, Orléans, France

    Jérôme Durand-Lose

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  1. Jérôme Durand-Lose
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Correspondence to Jérôme Durand-Lose .

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  1. Unconventional Computing Centre, University of the West of England Unconventional Computing Centre, Bristol, United Kingdom

    Andrew Adamatzky

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Durand-Lose, J. (2017). Computing in Perfect Euclidean Frameworks. In: Adamatzky, A. (eds) Advances in Unconventional Computing. Emergence, Complexity and Computation, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-33924-5_6

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  • DOI: https://doi.org/10.1007/978-3-319-33924-5_6

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