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Polynomial time ultrapowers and the consistency of circuit lower bounds

  • Jan BydžovskýEmail author
  • Moritz Müller
Original Study
  • 10 Downloads

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).

Keywords

Restricted ultrapowers Bounded arithmetic Circuit lower bounds 

Mathematics Subject Classification

03C20 03C98 03B70 68Q17 

Notes

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.

References

  1. 1.
    Avigad, J.: Saturated models of universal theories. Ann. Pure Appl. Logic 118(3), 219–234 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Buss, S.R.: Bounded arithmetic and propositional proof complexity. In: H. Schwichtenberg (ed.) Logic of Computation, pp. 67–122. Springer, Berlin (1997)Google Scholar
  3. 3.
    Bydžovský, J.: Powers of models in weak arithmetics. MSc. Thesis, University of Vienna, 2018. http://dmg.tuwien.ac.at/bydzovsky/mthesis.pdf
  4. 4.
    Cook, S.A.: Feasibly constructive proofs and the propositional calculus. In: Proceedings of the Seventh Annual ACM Symposium on Theory of Computing (STOC), pp. 83–97. ACM, New York (1975)Google Scholar
  5. 5.
    Cook, S.A., Krajíček, J.: Consequences of the provability of NP \(\subseteq \) P/poly. J. Symb. Logic 72(4), 1353–1371 (2007)CrossRefGoogle Scholar
  6. 6.
    Cobham, A.: The instrinsic computational difficulty of functions. In: Bar Hillel, Y. (ed.) Proceedings of the 1964 International Congress for Logic, Methodology, and the Philosophy of Science, pp. 24–30. North-Holland Publising Co., Amsterdam (1965)Google Scholar
  7. 7.
    Davis, M.: Hilbert’s tenth problem is unsolvable. Am. Math. Mon. 80(3), 233–269 (1973)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Frayne, T.E., Morel, A.C., Scott, D.S.: Reduced direct products. Fundam. Math. 51(3), 195–228 (1962)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Garlík, M.: Construction of models of bounded arithmetic by restricted reduced powers. Arch. Math. Logic 55, 625–648 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hájek, P., Pudlák, P.: Metamathematics of first order arithmetic. Springer/ASL Perspectives in Logic (1993)Google Scholar
  11. 11.
    Hirschfeld, J.: Models of arithmetic and recursive functions. Israel J. Math. 20(2), 111–126 (1975)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Jeřábek, E.: Approximate counting in bounded arithmetic. J. Symb. Logic 72(3), 959–993 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Keisler, H.J.: On the class of limit ultrapowers of a relational system. Not. Am. Math. Soc. 7, 878–879 (1960)Google Scholar
  14. 14.
    Keisler, H.J.: Limit ultrapowers. Trans. AMS 107, 382–408 (1963)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Keisler, H.J.: The ultraproduct construction. In: Bergelson, V., Blass, A., Di Nasso, M., Jin, R. (eds.) Ultrafilters Across Mathematics, Contemporary Mathematics 530, pp. 163-179. AMS, New York (2010)Google Scholar
  16. 16.
    Kochen, S.B.: Ultraproducts in the theory of models. Ann. Math. 74(2), 221–261 (1961)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kochen, S.B., Kripke, S.A.: Non-standard models of Peano arithmetic. L’Enseignement Mathématique 28, 211–231 (1982)Google Scholar
  18. 18.
    Krajíček, J.: Bounded Arithmetic, Propositional Logic, and Complexity Theory Encyclopedia of Mathematics and Its Applications, vol. 60. Cambridge University Press, Cambridge (1995)Google Scholar
  19. 19.
    Krajíček, J.: Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic. J. Symb. Logic 62(2), 457–486 (1997)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Krajíček, J.: Extensions of models of PV. In: Makowsky, J. A., Ravve, E.V. (eds.) Logic Colloquium’95, ASL/Springer Series Lecture Notes in Logic 11, pp. 104–114 (1998)Google Scholar
  21. 21.
    Krajíček, J.: Forcing with random variables and proof complexity. London Mathematical Society Lecture Note Series 382, Cambridge University Press, Cambridge (2011)Google Scholar
  22. 22.
    Krajíček, J., Oliveira, I.C.: Unprovability of circuit upper bounds in Cook’s theory PV. Logical Methods in Computer Science 13 (1:4) (2017)Google Scholar
  23. 23.
    Krajíček, J., Pudlák, P., Takeuti, G.: Bounded arithmetic and polynomial hierarchy. Ann. Pure Appl. Logic 52, 143–154 (1991)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Łos, J.: Quelques remarques, théorèmes et problèmes sur les classes définissables d’algèbres. Mathematical Interpretations of Formal Systems, North Holland, pp. 98–113 (1955)Google Scholar
  25. 25.
    Mac Dowell, R., Specker, E.: Modelle der Arithmetik. In: Infinitistic Methods. Proceedings of the Symposium on Foundations of Mathematics 1959, pp. 257–263. Pergamon Press, Warsaw (1961)Google Scholar
  26. 26.
    Maly, J., Müller, M.: A remark on pseudo proof systems and hard instances of the satisfiability problem. Math. Logic Q 64(6), 418–428 (2018)MathSciNetCrossRefGoogle Scholar
  27. 27.
    McLaughlin, T.G.: Sub-arithmetical ultrapowers: a survey. Ann. Pure Appl. Logic 49(2), 143–191 (1990)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Müller, M., Pich, J.: Feasibly constructive proofs of succinct weak circuit lower bounds. Preprint at Electronic Colloqium of Computational Complexity, Technical Report TR17-144 (2017)Google Scholar
  29. 29.
    Paris, J., Harrington, L.: A mathematical incompleteness in Peano arithmetic. Handbook of Mathematical Logic, North Holland, pp. 1133–1142 (1977)Google Scholar
  30. 30.
    Pudlák, P.: Logical Foundations of Mathematics and Computational Complexity, a Gentle Introduction. Springer, Berlin (2013)Google Scholar
  31. 31.
    Pudlák, P.: Randomness, pseudorandomness and models of arithmetic. In: Cégielski, P., Cornaros, Ch. (eds.) New Studies in Weak Arithmetics, pp. 199–216. CSLI Publications, Stanford (2013)Google Scholar
  32. 32.
    Razborov, A.A.: On provably disjoint NP-pairs. Basic Research in Computer Science BRICS RS-94-36 (1994)Google Scholar
  33. 33.
    Razborov, A.A.: Bounded arithmetic and lower bounds in Boolean complexity. Feasible Math. II, 344–386 (1995)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Razborov, A.A.: Unprovability of lower bounds on the circuit size in certain fragments of bounded arithmetic. Izv. Rus. Acad. Sci. 59, 201–224 (1995)MathSciNetGoogle Scholar
  35. 35.
    Razborov, A.A., Rudich, S.: Natural proofs. J. Comput. Syst. Sci. 55(1), 24–35 (1997)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Santhanam, R., Williams, R.: On uniformity and circuit lower bounds. Comput. Complex. 23(2), 177–205 (2014)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Scott, D.: On constructing models of arithmetic. In: Infinitistic Methods. Proceedings of the Symposium on Foundations of Mathematics 1959, pp. 235–255. Pergamon Press, Warsaw (1961)Google Scholar
  38. 38.
    Shelah, S.: Every two elementarily equivalent models have isomorphic ultrapowers. Israel J. Math. 10, 224–233 (1971)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Skolem, T.: Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen. Fundam. Math. 23(1), 150–161 (1934)CrossRefGoogle Scholar
  40. 40.
    Zambella, D.: Notes on polynomially bounded arithmetic. J. Symb. Logic 61(3), 942–966 (1996)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

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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