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Twelve Problems in Proof Complexity

  • Pavel Pudlák
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5010)

Abstract

Proof complexity is a research area that studies the concept of complexity from the point of view of logic. Although it is very much connected with computational complexity, the goals are different. In proof complexity we are studying the question how difficult is to prove a theorem? There are various ways how one can measure the “complexity” of a theorem. We may ask what is the length of the shortest proof of the theorem in a given formal system. Thus the complexity is the size of proofs. This corresponds to questions in computational complexity about the size of circuits, the number of steps of Turing machines etc. needed to compute a given function. But we may also ask how strong theory is needed to prove the theorem. This also has a counterpart in computational complexity—the questions about the smallest complexity class to which a given set or function belongs.

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References

  1. 1.
    Ajtai, M.: The complexity of the pigeonhole principle. In: Proc. IEEE 29th Annual Symp. on Foundation of Computer Science, pp. 346–355 (1988)Google Scholar
  2. 2.
    Alekhnovich, M., Ben-Sasson, E., Razborov, A.A., Wigderson, A.: Pseudorandom Generators in Propositional Proof Complexity. SIAM Journal on Computing 34(1), 67–88 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Beame, P.: A switching lemma primer. Technical Report UW-CSE-95-07-01, Department of Computer Science and Engineering, University of Washington (November 1994)Google Scholar
  4. 4.
    Beame, P., Pitassi, T., Segerlind, N.: Lower Bounds for Lovász-Schrijver Systems and Beyond Follow from Multiparty Communication Complexity. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1176–1188. Springer, Heidelberg (2005)Google Scholar
  5. 5.
    Bonet, M., Pitassi, T., Raz, R.: No feasible interpolation for TC 0-Frege proofs. In: Proc. 38-th FOCS, pp. 254–263 (1997)Google Scholar
  6. 6.
    Buss, S.R.: Bounded Arithmetic. Bibliopolis (1986)Google Scholar
  7. 7.
    Buss, S., Impagliazzo, R., Krajíček, J., Pudlák, P., Razborov, A.A., Sgall, J.: Proof complexity in algebraic systems and constant depth Frege systems with modular counting. Computational Complexity 6, 256–298 (1996(/1997)Google Scholar
  8. 8.
    Buss, S., Pudlák, P.: On the computational content of intuitionistic propositional proofs. Annals of Pure and Applied Logic 109, 49–64 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Chattopadhyay, A., Ada, A.: Multiparty Communication Complexity of Disjointness. arXiv e-print (arXiv:0801.3624)Google Scholar
  10. 10.
    Clegg, M., Edmonds, J., Impagliazzo, R.: Using the Groebner basis algorithm to find proofs of unsatisfiability. In: Proc. 28-th ACM STOC, pp. 174–183 (1996)Google Scholar
  11. 11.
    Clote, P., Krajíček, J.: Open Problems. In: Clote, P., Krajíček, J. (eds.) Arithmetic, Proof Theory and Computational Complexity, pp. 1–19. Oxford Press (1993)Google Scholar
  12. 12.
    Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. Journ. Symbolic Logic 44, 25–38 (1987)MathSciNetGoogle Scholar
  13. 13.
    Dalla Chiara, M.L., Giuntini, R.: Quantum Logics. In: Gabbay, Guenthner (eds.) Handbook of Philosophical Logic, pp. 129–228. Kluwer Academic Publishers, Dordrecht (2002)Google Scholar
  14. 14.
    Dash, S.: An Exponential Lower Bound on the Length of Some Classes of Branch-and-Cut Proofs. In: Cook, W.J., Schulz, A.S. (eds.) IPCO 2002. LNCS, vol. 2337, pp. 145–160. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. 15.
    Egly, U., Tompits, H.: On different proof-search strategies for orthologic. Stud. Log. 73, 131–152 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Hrubeš, P.: A lower bound for intuitionistic logic. Ann. Pure Appl. Logic 146(1), 72–90 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Hrubeš, P.: Lower bounds for modal logics. Journ. Symbolic Logic (to appear)Google Scholar
  18. 18.
    Kojevnikov, A., Itsykson, D.: Lower Bounds of Static Lovász-Schrijver Calculus Proofs for Tseitin Tautologies. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 323–334. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    Krajíček, J.: Lower Bounds to the Size of Constant-Depth Propositional Proofs. J. of Symbolic Logic 59(1), 73–86 (1994)CrossRefzbMATHGoogle Scholar
  20. 20.
    Krajıček, J.: Bounded Arithmetic, Propositional Logic, and Complexity Theory. In: Encyclopedia of Mathematics and its Applications 60, Cambridge Univ. Press, Cambridge (1995)Google Scholar
  21. 21.
    Krajíček, J.: A fundamental problem of mathematical logic. Collegium Logicum, Annals of Kurt Gödel Society 2, 56–64 (1996)Google Scholar
  22. 22.
    Krajıček, J.: Lower bounds for a proof system with an exponential speed-up over constant-depth Frege systems and over polynomial calculus. In: Privara, I., Ružička, P. (eds.) MFCS 1997. LNCS, vol. 1295, pp. 85–90. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  23. 23.
    Krajíček, J.: On methods for proving lower bounds in propositional logic. In: Dalla Chiara, M.L., et al. (eds.) Logic and Scientific Methods, pp. 69–83. Kluwer Acad. Publ., DordrechtGoogle Scholar
  24. 24.
    Krajíček, J.: On the weak pigeonhole principle. Fundamenta Mathematicae 170(1-3), 123–140 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Krajíček, J.: Proof complexity. In: Laptev, A. (ed.) European congress of mathematics (ECM), Stockholm, Sweden, June 27–July 2, 2004, pp. 221–231. European Mathematical Society (2005)Google Scholar
  26. 26.
    Krajíček, J.: A proof complexity generator. In: Glymour, C., Wang, W., Westerstahl, D. (eds.) Proc. 13th Int. Congress of Logic, Methodology and Philosophy of Science, Beijing. ser. Studies in Logic and the Foundations of Mathematics. King’s College Publications, London (to appear, 2007)Google Scholar
  27. 27.
    Krajíček, J., Pudlák, P., Takeuti, G.: Bounded Arithmetic and the Polynomial Hierarchy. Annals of Pure and Applied Logic 52, 143–153 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Krajíček, J., Pudlák, P., Woods, A.: Exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle. Random Structures and Algorithms 7(1), 15–39 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Krajíček, J., Pudlák, P.: Some consequences of cryptographical conjectures for \(S^1_2\) and EF. Information and Computation 140, 82–94 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Krajíček, J., Impagliazzo, R.: A note on conservativity relations among bounded arithmetic theories. Mathematical Logic Quarterly 48(3), 375–377 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Krajíček, J., Skelley, A., Thapen, N.: NP search problems in low fragments of bounded arithmetic. J. of Symbolic Logic 72(2), 649–672 (2007)CrossRefzbMATHGoogle Scholar
  32. 32.
    Lee, T., Schraibman, A.: Disjointness is hard in the multi-party number on the forehead model. arXiv e-print (arXiv:0712.4279)Google Scholar
  33. 33.
    Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optimization 1(2), 166–190 (1991)CrossRefzbMATHGoogle Scholar
  34. 34.
    Pitassi, T., Beame, P., Impagliazzo, R.: Exponential Lower Bounds for the Pigeonhole Principle. Computational Complexity 3, 97–140 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    Pudlák, P.: Lower bounds for resolution and cutting planes proofs and monotone computations. J. Symbolic Logic 62(3), 981–998 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Pudlák, P.: The lengths of proofs. In: Handbook of Proof Theory, pp. 547–637. Elsevier, Amsterdam (1998)CrossRefGoogle Scholar
  37. 37.
    Pudlák, P.: On reducibility and symmetry of disjoint NP-pairs. Theor. Comput. Science 295, 323–339 (2003)CrossRefzbMATHGoogle Scholar
  38. 38.
    Pudlák, P.: Consistency and games—in search of new combinatorial principles. In: Helsinki, Stoltenberg-Hansen, V., Vaananen, J. (eds.) Proc. Logic Colloquium 2003. Assoc. for Symbolic Logic, pp. 244–281 (2006)Google Scholar
  39. 39.
    Pudlák, P., Sgall, J.: Algebraic models of computation and interpolation for algebraic proof systems. In: Beame, P.W., Buss, S.R. (eds.) Proof Complexity and Feasible Arithmetics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 39, pp. 279–295Google Scholar
  40. 40.
    Razborov, A.A.: Lower bounds on the size of bounded-depth networks over a complete basis with logical addition (Russian). Matematicheskie Zametki 41(4), 598–607 (1987); English translation in: Mathematical Notes of the Academy of Sci. of the USSR 41(4), 333–338 (1987) Google Scholar
  41. 41.
    Razborov, A.A.: On provably disjoint NP-pairs. BRICS Report Series RS-94-36 (1994), http://www.brics.dk/RS/94/36/index.html
  42. 42.
    Razborov, A.A.: Unprovability of lower bounds on the circuit size in certain fragments of Bounded Arithmetic. Izvestiya of the R.A.N. 59(1), 201–222 (1995); see also Izvestiya: Mathematics 59(1), 205–227Google Scholar
  43. 43.
    Razborov, A.A.: Bounded Arithmetic and Lower Bounds in Boolean Complexity. In: Feasible Mathematics II, pp. 344–386. Birkhäuser Verlag (1995)Google Scholar
  44. 44.
    Razborov, A.A.: Lower bound for the polynomial calculus. Computational Complexity 7(4), 291–324 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  45. 45.
    Razborov, A.A.: Pseudorandom Generators Hard for k-DNF Resolution and Polynomial Calculus Resolution (2002-2003), http://www.mi.ras.ru/~razborov/resk.ps
  46. 46.
    Skelley, A., Thapen, N.: The provably total search problems of Bounded Arithmetic (preprint, 2007)Google Scholar
  47. 47.
    Smolensky, R.: Algebraic Methods in the Theory of Lower Bounds for Boolean Circuit Complexity. In: STOC, pp. 77–82 (1987)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Pavel Pudlák
    • 1
  1. 1.Mathematical InstitutePrague

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