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Some Observations on Infinitary Complexity

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Sailing Routes in the World of Computation (CiE 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10936))

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Abstract

Continuing the study of complexity theory of Koepke’s Ordinal Turing Machines (OTMs) that was done in [CLR], we prove the following results:

  1. 1.

    An analogue of Ladner’s theorem for OTMs holds: That is, there are languages \(\mathcal {L}\) which are NP\(^{\infty }\), but neither P\(^{\infty }\) nor NP\(^{\infty }\)-complete. This answers an open question of [CLR].

  2. 2.

    The speedup theorem for Turing machines, which allows us to bring down the computation time and space usage of a Turing machine program down by an aribtrary positive factor under relatively mild side conditions by expanding the working alphabet does not hold for OTMs.

  3. 3.

    We show that, for \(\alpha <\beta \) such that \(\alpha \) is the halting time of some OTM-program, there are decision problems that are OTM-decidable in time bounded by \(|w|^{\beta }\cdot \gamma \) for some \(\gamma \in \text {On}\), but not in time bounded by \(|w|^{\alpha }\cdot \gamma \) for any \(\gamma \in \text {On}\).

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Notes

  1. 1.

    Multitape OTMs can be simulated by single-tape OTMs quickly enough so that this does not make a difference for Theorem 2, our main result. However, the proofs of Theorem 3 and Corollary 2 both make crucial use of the availability of several tapes.

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Acknowledgements

We thank Philipp Schlicht for checking our proof of Theorem 2. We also thank our three anonymous referees for a considerable amount of constructive comments on the exposition.

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Authors and Affiliations

  1. Fachbereich Mathematik und Statistik, Universität Konstanz, Konstanz, Germany

    Merlin Carl

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  1. Merlin Carl
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Correspondence to Merlin Carl .

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Editors and Affiliations

  1. Kiel University, Kiel, Germany

    Florin Manea

  2. Queens College, CUNY, Queens, New York, USA

    Russell G. Miller

  3. Kiel University, Kiel, Germany

    Dirk Nowotka

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Carl, M. (2018). Some Observations on Infinitary Complexity. In: Manea, F., Miller, R., Nowotka, D. (eds) Sailing Routes in the World of Computation. CiE 2018. Lecture Notes in Computer Science(), vol 10936. Springer, Cham. https://doi.org/10.1007/978-3-319-94418-0_12

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  • DOI: https://doi.org/10.1007/978-3-319-94418-0_12

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