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Conference on Computability in Europe

CiE 2019: Computing with Foresight and Industry pp 13–24Cite as

Surreal Blum-Shub-Smale Machines

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11558))

Abstract

Blum-Shub-Smale machines are a classical model of computability over the real line. In [9], Koepke and Seyfferth generalised Blum-Shub-Smale machines to a transfinite model of computability by allowing them to run for a transfinite amount of time. The model of Koepke and Seyfferth is asymmetric in the following sense: while their machines can run for a transfinite number of steps, they use real numbers rather than their transfinite analogues. In this paper we will use the surreal numbers in order to define a generalisation of Blum-Shub-Smale machines in which both time and register content are transfinite.

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Notes

  1. 1.

    A stronger version of infinite time Blum-Shub-Smale machines could be obtained by allowing infinite time Blum-Shub-Smale machines to use rational functions with real coefficients, but this was not done in [11].

  2. 2.

    By this we mean the notion of limit coming from the order topology over \(\mathrm {No}\).

  3. 3.

    A totally ordered field is a field together with a total order \(\le \) such that for all x, y, and z, we have that if \(x\le y\), then \(x+z \le y+z\), and if x and y are positive, then \(x \cdot y\) is positive. The cofinality of an ordered field is the least cardinal \(\lambda \) such that there is a sequence of length \(\lambda \) cofinal in the field.

  4. 4.

    By abuse of notation we write \(\wp (\mathrm {No})\) for the class of subsets of \(\mathrm {No}\).

  5. 5.

    In this sentence \(1+\alpha \) should be read as the ordinal addition so that for \(\alpha \ge \omega \) we have \(1+\alpha =\alpha \).

  6. 6.

    Once again the operations in \((\omega \times S)+H\) must be interpreted as ordinal operations.

  7. 7.

    The class function \(\delta _{\mathrm {No}}\) is not literally an extension of \(\delta _{\mathbb {Q}_\kappa }\) just because in [4] we assumed \(\mathrm {dom}(\delta _{\mathbb {Q}_\kappa })\subset 2^\kappa \) rather than \(\mathrm {dom}(\delta _{\mathbb {Q}_\kappa })\subset 2^{<\kappa }\). This does not make much of a difference in our algorithms as far as we have a marker for the end of the code of the sign sequence (i.e., the last two bits in the definition of \(\delta _{\mathrm {No}}\)).

  8. 8.

    Note that, if K is a set \(H^{n,m}_{K}\) is also a set.

References

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Authors and Affiliations

  1. Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146, Hamburg, Germany

    Lorenzo Galeotti

  2. Institute for Logic, Language and Computation, Universiteit van Amsterdam, Postbus 94242, 1090 GE, Amsterdam, The Netherlands

    Lorenzo Galeotti

  3. Amsterdam University College, Postbus 94160, 1090 GD, Amsterdam, The Netherlands

    Lorenzo Galeotti

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  1. Lorenzo Galeotti
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Correspondence to Lorenzo Galeotti .

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Editors and Affiliations

  1. Christian Albrechts University of Kiel, Kiel, Germany

    Florin Manea

  2. Durham University, Durham, UK

    Barnaby Martin

  3. Durham University, Durham, UK

    Daniël Paulusma

  4. Università degli Studi di Milano, Milan, Italy

    Giuseppe Primiero

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Galeotti, L. (2019). Surreal Blum-Shub-Smale Machines. In: Manea, F., Martin, B., Paulusma, D., Primiero, G. (eds) Computing with Foresight and Industry. CiE 2019. Lecture Notes in Computer Science(), vol 11558. Springer, Cham. https://doi.org/10.1007/978-3-030-22996-2_2

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  • DOI: https://doi.org/10.1007/978-3-030-22996-2_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-22995-5

  • Online ISBN: 978-3-030-22996-2

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