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Polynomial time ultrapowers and the consistency of circuit lower bounds

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Abstract

A polynomial time ultrapower is a structure given by the set of polynomial time computable functions modulo some ultrafilter. They model the universal theory \(\forall \mathsf {PV}\) of all polynomial time functions. Generalizing a theorem of Hirschfeld (Israel J Math 20(2):111–126, 1975), we show that every countable model of \(\forall \mathsf {PV}\) is isomorphic to an existentially closed substructure of a polynomial time ultrapower. Moreover, one can take a substructure of a special form, namely a limit polynomial time ultrapower in the classical sense of Keisler (in: Bergelson, V., Blass, A., Di Nasso, M., Jin, R. (eds.) Ultrafilters across mathematics, contemporary mathematics vol 530, pp 163–179. AMS, New York, 1963). Using a polynomial time ultrapower over a nonstandard Herbrand saturated model of \(\forall \mathsf {PV}\) we show that \(\forall \mathsf {PV}\) is consistent with a formal statement of a polynomial size circuit lower bound for a polynomial time computable function. This improves upon a recent result of Krajíček and Oliveira (Logical methods in computer science 13 (1:4), 2017).

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Notes

  1. All relevant technical notions will be defined precisely later.

  2. For \(\ell ,k\in \mathbb {N}\), the theory \(\forall \mathsf {PV}\cup \{\exists n LB [g](c\cdot n^\ell ,n)\mid c\in \mathbb {N}\}\) does not seem to imply \(\forall z\exists n LB [g](|z|\cdot n^k, n)\). For all we know, there could be a nonstandard model of the former theory that contains only standard n witnessing the lower bounds, while every nonstandard model of \(\forall z\exists n LB [g](|z|\cdot n^k, n)\) witnesses the lower bound with some nonstandard n.

  3. This is a slight abuse of standard terminology according to which a structure is existentially closed if it is existentially closed as a substructure of any of its extensions. We shall not use this terminology.

  4. In contrast to the setting of [17] and of this paper, [11, 27, 31, 37] consider ultrafilters over restricted Boolean algebras, namely those consisting of sets with characteristic function in F.

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Acknowledgements

We want to thank the anonymous referee for detailed comments and suggestions. We further thank Ján Pich for many helpful conversations about the topic of the current paper. Jan Bydžovský is currently partially supported by the Austrian Science Fund (FWF) under Project P31063. Moritz Müller is currently supported by the European Research Council (ERC) under the European Unions Horizon 2020 research programme (Grant Agreement ERC-2014-CoG 648276 AUTAR); the main part of the current work has been done while supported by the Austrian Science Fund (FWF) under Project P28699.

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  1. Institute of Discrete Mathematics and Geometry, Technische Universität Wien, Vienna, Austria

    Jan Bydžovský

  2. Department of Computer Science, Universitat Politècnica de Catalunya, Barcelona, Spain

    Moritz Müller

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  1. Jan Bydžovský
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Correspondence to Jan Bydžovský.

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Based on Jan Bydžovský’s Master Thesis [3] written under the supervision of Moritz Müller.

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Bydžovský, J., Müller, M. Polynomial time ultrapowers and the consistency of circuit lower bounds. Arch. Math. Logic 59, 127–147 (2020). https://doi.org/10.1007/s00153-019-00681-y

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