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Discrete Transfinite Computation

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Turing’s Revolution

Abstract

We describe various computational models based initially, but not exclusively, on that of the Turing machine, that are generalized to allow for transfinitely many computational steps. Variants of such machines are considered that have longer tapes than the standard model, or that work on ordinals rather than numbers. We outline the connections between such models and the older theories of recursion in higher types, generalized recursion theory, and recursion on ordinals such as α-recursion. We conclude that, in particular, polynomial time computation on ω-strings is well modelled by several convergent conceptions.

AMS Classifications: 03D10, 03D65, 03E45, 03D30

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Author information

Authors and Affiliations

  1. School of Mathematics, University of Bristol, Bristol, BS8 1TW, England

    P. D. Welch

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  1. P. D. Welch
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Correspondence to P. D. Welch .

Editor information

Editors and Affiliations

  1. Department of Humanities, Social and Political Sciences, ETH Zurich, Zurich, Switzerland

    Giovanni Sommaruga

  2. Institute of Computer Science and Applied Mathematics, University of Bern, Bern, Switzerland

    Thomas Strahm

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Welch, P.D. (2015). Discrete Transfinite Computation. In: Sommaruga, G., Strahm, T. (eds) Turing’s Revolution. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-22156-4_6

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