Lower bounds for prepositional proofs and independence results in bounded arithmetic

  • Alexander A. Razborov
Invited Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1099)


We begin with a highly informal discussion of the role played by Bounded Arithmetic and propositional proof systems in the reasoning about the world of feasible computations. Then we survey some known lower bounds on the complexity of proofs in various propositional proof systems, paying special attention to recent attempts on reducing such bounds to some purely complexity results or assumptions. As one of the main motivations for this research we discuss provability of extremely important propositional formulae that express hardness of explicit Boolean functions with respect to various non-uniform computational models.


Boolean Function Proof System Proof Theory Propositional Formula Interpolation Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Ajtai. The complexity of the pigeonhole principle. In Proceedings of the 29th IEEE Symposium on Foundations of Computer Science, pages 346–355, 1988.Google Scholar
  2. 2.
    M. Ajtai. Parity and the pigeonhole principle. In S. R. Buss and P. J. Scott, editors, Feasible Mathematics, pages 1–24. Birkhauser, 1990.Google Scholar
  3. 3.
    M. Ajtai. The independence of the modulo p counting principle. In Proceedings of the 26th ACM STOC, pages 402–411, 1994.Google Scholar
  4. 4.
    N. Alon and R. Boppana. The monotone circuit complexity of Boolean functions. Combinatorica, 7(1):1–22, 1987.Google Scholar
  5. 5.
    P. Beame, R. Impagliazzo, J. Krajíček, T. Pitassi, and P. Pudlák. Lower bounds on Hilbert's Nullstellensatz and propositional proofs. To appear in Proc. of the London Math. Soc., 1994.Google Scholar
  6. 6.
    P. Beame and T. Pitassi. Exponential separation between the matching principles and the pigeonhole principle. Submitted to Annals of Pure and Applied Logic, 1993.Google Scholar
  7. 7.
    S. Bellantoni, T. Pitassi, and A. Urquhart. Approximation of small depth Frege proofs. SIAM Journal on Computing, 21(6):1161–1179, 1992.CrossRefGoogle Scholar
  8. 8.
    A. Blake. Canonical expressions in Boolean algebra. PhD thesis, University of Chicago, 1937.Google Scholar
  9. 9.
    M. Bonet, T. Pitassi, and R. Raz. Lower bounds for cutting planes proofs with small coefficients. In Proceedings of the 27th ACM STOC, pages 575–584, 1995.Google Scholar
  10. 10.
    S. R. Buss. Bounded Arithmetic. Bibliopolis, Napoli, 1986.Google Scholar
  11. 11.
    S. Buss, R. Impagliazzo, J. Krajíček, P. Pudlák, A. Razborov, and J. Sgall. Proof complexity in algebraic systems and bounded depth Frege systems with modular counting. Submitted to Computational Complexity, 1996.Google Scholar
  12. 12.
    V. Chvátal and E. Szemerédi. Many hard examples for resolution. Journal of the ACM, 35(4):759–768, 1988.CrossRefGoogle Scholar
  13. 13.
    S. A. Cook. Feasibly constructive proofs and the propositional calculus. In Proceedings of the 7th Annual ACM Symposium on the Theory of Computing, pages 83–97, 1975.Google Scholar
  14. 14.
    S. A. Cook and A. R. Reckhow. The relative efficiency of propositional proof systems. Journal of Symbolic Logic, 44(1):36–50, 1979.Google Scholar
  15. 15.
    W. Cook, C. R. Coullard, and G. Turán. On the complexity of cutting plane proofs. Discrete Applied Mathematics, 18:25–38, 1987.CrossRefGoogle Scholar
  16. 16.
    M. Davis and H. Putnam. A computing procedure for quantification theory. Journal of the ACM, 7(3):210–215, 1960.CrossRefGoogle Scholar
  17. 17.
    P. Hájek and P. Pudlák. Metamathematics of First-Order Arithmetic. Springer-Verlag, 1993.Google Scholar
  18. 18.
    A. Haken. The intractability or fesolution. Theoretical Computer Science, 39:297–308, 1985.CrossRefGoogle Scholar
  19. 19.
    A. Haken. Counting bottlenecks to show monotone P ≠ NP. In Proceedings of the 36th IEEE FOCS, 1995.Google Scholar
  20. 20.
    J. Håstad. Computational limitations on Small Depth Circuits. PhD thesis, Massachusetts Institute of Technology, 1986.Google Scholar
  21. 21.
    J. Krajíček. Lower bounds to the size of constant-depth propositional proofs. Journal of Symbolic Logic, 59(1):73–86, 1994.Google Scholar
  22. 22.
    J. Krajíček. Bounded arithmetic, propositional logic and complexity theory. Cambridge University Press, 1994.Google Scholar
  23. 23.
    J. Krajíček. Interpolation theorems, lower bounds for proof systems and independence results for bounded arithmetic. To appear in Journal of Symbolic Logic, 1994.Google Scholar
  24. 24.
    J. Krajíček and P. Pudlák. Some consequences of cryptographical conjectures for S 21 and EF. To appear in the Proceedings of the meeting Logic and Computational Complexity, Ed. D. Leivant, 1995.Google Scholar
  25. 25.
    J. Krajíček, P. Pudlák, and A. R. Woods. Exponential lower bounds to the size of bounded depth frege proofs of the pigeonhole principle. Random Structures and Algorithms, 7(1):15–39, 1995.Google Scholar
  26. 26.
    T. Pitassi, P. Beame, and R. Impagliazzo. Exponential lower bounds for the pigeonhole principle. Computational Complexity, 3:97–140, 1993.CrossRefGoogle Scholar
  27. 27.
    P. Pudlák. Lower bounds for resolution and cutting planes proofs and monotone computations. Submitted to Journal of Symbolic Logic, 1995.Google Scholar
  28. 28.
    А. А. Разборов. Нижн ие оценки размера схе м ограниченной глуби ны в полном базисе, сод ержащем функцию логи ческого сложения. Ма тем. Зам., 41(4):598–607, 1987. A. A. Razborov, Lower bounds on the size of bounded-depth networks over a complete basis with logical addition, Mathem. Notes of the Academy of Sci. of the USSR, 41(4):333–338, 1987.Google Scholar
  29. 29.
    A. Razborov. An equivalence between second order bounded domain bounded arithmetic and first order bounded arithmetic. In P. Clote and J. Krajíček, editors, Arithmetic, Proof Theory and Computational Complexity, pages 247–277. Oxford University Press, 1992.Google Scholar
  30. 30.
    A. Razborov. On small depth threshold circuits. In Proceedings of the SWAT 92, Lecture Notes in Computer Science, 621, pages 42–52, New York/Berlin, 1992. Springer-Verlag.Google Scholar
  31. 31.
    A. Razborov. Bounded Arithmetic and lower bounds in Boolean complexity. In P. Clote and J. Remmel, editors, Feasible Mathematics II. Progress in Computer Science and Applied Logic, vol. 13, pages 344–386. Birkhaüser, 1995.Google Scholar
  32. 32.
    A. Razborov. Unprovability of lower bounds on circuit size in certain fragments of Bounded Arithmetic. Из в. АН СССР, сер. матем. (Izvestiya of the RAN), 59(1):201–222, 1995. See also Izvestiya: Mathematics 59:1, 205–227.Google Scholar
  33. 33.
    A. Razborov. On provably disjoint NP-pairs. Technical Report RS-94-36, Basic Research in Computer Science Center, Aarhus, Denmark, 1994.Google Scholar
  34. 34.
    A. Razborov and S. Rudich. Natural proofs. To appear in Journal of Computer and System Sciences (for the preliminary version see Proceedings of the 26th ACM Symposium on Theory of Computing, pp. 204–213), 1994.Google Scholar
  35. 35.
    R. A. Reckhow. On the lengths of proofs in the propositional calculus. Technical Report 87, University of Toronto, 1976.Google Scholar
  36. 36.
    S. Riis. Independence in Bounded Arithmetic. PhD thesis, Oxford University, 1993.Google Scholar
  37. 37.
    S. Riis. Count(q) does not imply Count(p). Technical Report RS-94-21, Basic Research in Computer Science Center, Aarhus, Denmark, 1994.Google Scholar
  38. 38.
    J. A. Robinson. A machine-oriented logic based on the resolution principle. Journal of the ACM, 12(1):23–41, 1965.CrossRefGoogle Scholar
  39. 39.
    A. L. Selman. Complexity issues in cryptography. Proceedings of Symposia in Applied Mathematics, 38:92–107, 1989.Google Scholar
  40. 40.
    R. Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proceedings of the 19th ACM Symposium on Theory of Computing, pages 77–82, 1987.Google Scholar
  41. 41.
    G. Takeuti. RSUV isomorphisms. In P. Clote and J. Krajíček, editors, Arithmetic, Proof Theory and Computational Complexity, pages 364–386. Oxford University Press, 1992.Google Scholar
  42. 42.
    Г. С. Цейтин. О сложност и вывода в исчислении высказываний, In А. О. Сли сенко, editor, Исследования по конструктивной ма тематике и математич еской логике, II; Записк инаучных семинаров Л ОМИ, m. 8, pages 234–259. Наука, Ленинг рад, 1968. Engl. translation: G. C. Tseitin, On the complexity of derivations in propositional calculus, in: Studies in mathematics and mathematical logic, Part II, ed. A. O. Slissenko, pp. 115–125.Google Scholar
  43. 43.
    A. Urquhart. Hard examples for resolution. Journal of the ACM, 34(1):209–219, 1987.CrossRefGoogle Scholar
  44. 44.
    A. Urquhart. The complexity of propositional proofs. Bulletin of Symbolic Logic, 1:425–467, 1995.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Alexander A. Razborov
    • 1
  1. 1.Steklov Mathematical InstituteMoscowRussia

Personalised recommendations