Skip to main content
Springer Nature Link
Log in

Propositional Proofs in Frege and Extended Frege Systems (Abstract)

  • Conference paper
  • First Online:
Computer Science -- Theory and Applications (CSR 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9139))

Included in the following conference series:

  • 516 Accesses

Abstract

We discuss recent results on the propositional proof complexity of Frege proof systems, including some recently discovered quasipolynomial size proofs for the pigeonhole principle and the Kneser-Lovász theorem. These are closely related to formalizability in bounded arithmetic.

S. Buss—Supported in part by NSF grants CCF-121351 and DMS-1101228, and a Simons Foundation Fellowship 306202.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    See Jeřábek [18] for an alternative formulation of extended Frege systems based directly on Boolean circuits.

References

  1. Aisenberg, J., Bonet, M.L., Buss, S.: Quasi-polynomial size Frege proofs of Frankl’s theorem on the trace of finite sets. J. Symbolic Log. (to appear)

    Google Scholar 

  2. Aisenberg, J., Bonet, M.L., Buss, S., Crãciun, A., Istrate, G.: Short proofs of the Kneser-Lovász principle. In: Proceedings of 42th International Colloquium on Automata, Languages, and Programming (ICALP 2015) (2015) (to appear)

    Google Scholar 

  3. Avigad, J.: Plausibly hard combinatorial tautologies. In: Beame, P., Buss, S.R. (eds.) Proof Complexity and Feasible Arithmetics, pp. 1–12. American Mathematical Society, Rutgers (1997)

    Google Scholar 

  4. Beame, P.: Proof complexity. In: Rudich, S., Wigderson, A. (eds.) Computational Complexity Theory. IAS/Park City Mathematical Series, vol. 10, pp. 199–246. American Mathematical Society, Princeton (2004)

    Google Scholar 

  5. Beame, P., Pitassi, T.: Propositional proof complexity: past, present and future. In: Paun, G., Rozenberg, G., Salomaa, A. (eds.) Current Trends in Theoretical Computer Science Entering the 21st Century, pp. 42–70. World Scientific Publishing Co. Ltd, Singapore (2001). Earlier version appeared in Computational Complexity Column, Bulletin of the EATCS (2000)

    Google Scholar 

  6. Beckmann, A., Buss, S.R.: Improved witnessing and local improvement principles for second-order bounded arithmetic. ACM Trans. Comput. Logic 15(1), 35 (2014)

    Article  MathSciNet  Google Scholar 

  7. Bonet, M.L., Buss, S.R., Pitassi, T.: Are there hard examples for Frege systems? In: Clote, P., Remmel, J. (eds.) Feasible Mathematics II, pp. 30–56. Birkhäuser, Boston (1995)

    Chapter  Google Scholar 

  8. Buss, S.: Quasipolynomial size proofs of the propositional pigeonhole principle. Theor. Comput. Sci. 576, 77–84 (2015)

    Article  MathSciNet  Google Scholar 

  9. Buss, S.R.: Bounded Arithmetic. Ph.D. thesis, Bibliopolis, Princeton University (revision of 1985) (1986)

    Google Scholar 

  10. Buss, S.R.: Polynomial size proofs of the propositional pigeonhole principle. J. Symbolic Logic 52, 916–927 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  11. Buss, S.R.: Propositional consistency proofs. Ann. Pure Appl. Logic 52, 3–29 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  12. Buss, S.R.: Propositional proof complexity: an introduction. In: Berger, U., Schwichtenberg, H. (eds.) Computational Logic, pp. 127–178. Springer-Verlag, Berlin (1999)

    Chapter  Google Scholar 

  13. Buss, S.R.: Towards NP-P via proof complexity and proof search. Ann. Pure Appl. Logic 163(9), 1163–1182 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cook, S.A., Reckhow, R.A.: On the lengths of proofs in the propositional calculus, preliminary version. In: Proceedings of the Sixth Annual ACM Symposium on the Theory of Computing, pp. 135–148 (1974)

    Google Scholar 

  15. Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. J. Symbolic Logic 44, 36–50 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hrubeš, P., Tzameret, I.: Short proofs for determinant identities. SIAM J. Comput. 44(2), 340–383 (2015)

    Article  MathSciNet  Google Scholar 

  17. Istrate, G., Crãciun, A.: Proof complexity and the Kneser-Lovász theorem. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 138–153. Springer, Heidelberg (2014)

    Google Scholar 

  18. Jeřábek, E.: Dual weak pigeonhole principle, boolean complexity, and derandomization. Ann. Pure Appl. Logic 124, 1–37 (2004)

    Article  Google Scholar 

  19. Kołodziejczyk, L.A., Nguyen, P., Thapen, N.: The provably total NP search problems of weak second-order bounded arithmetic. Ann. Pure Appl. Logic 162(2), 419–446 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Krajíček, J.: Bounded Arithmetic: Propositional Calculus and Complexity Theory. Cambridge University Press, New York (1995)

    Book  MATH  Google Scholar 

  21. Lovász, L.: Kneser’s conjecture, chromatic number, and homotopy. J. Comb. Theor. A 25(3), 319–324 (1978)

    Article  MATH  Google Scholar 

  22. Matoušek, J.: A combinatorial proof of Kneser’s conjecture. Combinatorica 24(1), 163–170 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Pudlák, P.: Twelve problems in proof complexity. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds.) Computer Science – Theory and Applications. LNCS, vol. 5010, pp. 13–27. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  24. Reckhow, R.A.: On the lengths of proofs in the propositional calculus. Ph.D. thesis, Department of Computer Science, University of Toronto, Technical report #87 (1976)

    Google Scholar 

  25. Segerlind, N.: The complexity of propositional proofs. Bull. Symbolic Logic 13(4), 417–481 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Tsejtin, G.S.: On the complexity of derivation in propositional logic. Studies in Constructive Mathematics and Mathematical Logic 2, pp. 115–125 (1968). Reprinted in J. Siekmann and G. Wrightson, Automation of Reasoning, vol. 2, Springer-Verlag, pp. 466–483 (1983)

    Google Scholar 

Download references

Acknowledgments

We thank Lev Beklemishev and Vladimir Podolskii for helpful comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, San Diego, La Jolla, CA, 92130-0112, USA

    Sam Buss

Authors
  1. Sam Buss
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sam Buss .

Editor information

Editors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, Lomonosov Moscow State University and National Research University Higher School of Economics, Moscow, Russia

    Lev D. Beklemishev

  2. Moscow Institute of Physics and Technology, Moscow, Russia and Kazan (Volga Region) Federal University, Kazan, Kabardino-Balkar.Rep, Russia

    Daniil V. Musatov

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Buss, S. (2015). Propositional Proofs in Frege and Extended Frege Systems (Abstract). In: Beklemishev, L., Musatov, D. (eds) Computer Science -- Theory and Applications. CSR 2015. Lecture Notes in Computer Science(), vol 9139. Springer, Cham. https://doi.org/10.1007/978-3-319-20297-6_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-20297-6_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20296-9

  • Online ISBN: 978-3-319-20297-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation