Polynomial time ultrapowers and the consistency of circuit lower bounds

  • Jan BydžovskýEmail author
  • Moritz Müller
Original Study


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).


Restricted ultrapowers Bounded arithmetic Circuit lower bounds 

Mathematics Subject Classification

03C20 03C98 03B70 68Q17 



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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Copyright information

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Authors and Affiliations

  1. 1.Institute of Discrete Mathematics and GeometryTechnische Universität WienViennaAustria
  2. 2.Department of Computer ScienceUniversitat Politècnica de CatalunyaBarcelonaSpain

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