The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs

  • Albert Atserias
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7962)

Abstract

In recent years there has been some progress in our understanding of the proof-search problem for very low-depth proof systems, e.g. proof systems that manipulate formulas of very low complexity such as clauses (i.e. resolution), DNF-formulas (i.e. R(k) systems), or polynomial inequalities (i.e. semi-algebraic proof systems). In this talk I will overview this progress. I will start with bounded-width resolution, whose specialized proof-search algorithm is as easy as uninteresting, but whose proof-search problem is unintentionally solved by certain versions of conflict-driven clause-learning algorithms with restarts. I will continue with R(k) systems, whose proof-search problem turned out to hide the complexity of certain two-player games of interest in the area of systems synthesis and verification. And I will close with bounded-degree semi-algebraic proof systems, whose proof-search problem turned out to hide the complexity of systems of linear equations over finite fields, among other problems.

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References

  1. 1.
    Alekhnovich, M., Razborov, A.A.: Resolution Is Not Automatizable Unless W[P] Is Tractable. SIAM J. Comput. 38(4), 1347–1363 (2008)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Atserias, A., Bonet, M.L.: On the Automatizability of Resolution and Related Propositional Proof Systems. Information and Computation 189(2), 182–201 (2004)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Atserias, A., Fichte, J.K., Thurley, M.: Clause-Learning Algorithms with Many Restarts and Bounded-Width Resolution. Journal of Artificial Intelligence Research 40, 353–373 (2011)MathSciNetMATHGoogle Scholar
  4. 4.
    Atserias, A., Maneva, E.: Mean-payoff games and propositional proofs. Information and Computation 209(4), 664–691 (2011)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Barak, B., Brandão, F., Harrow, A., Kelner, J., Zhou, Y.: Hypercontractivity, Sum-of-Squares Proofs, and their Applications. In: Proc. of 44th ACM Symposium on Theory of Computing (STOC), pp. 307–326 (2012)Google Scholar
  6. 6.
    Beame, P., Karp, R., Pitassi, T., Saks, M.: The efficiency of resolution and Davis-Putnam procedures. SIAM J. Comput. 31(4), 1048–1075 (2002)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Beame, P., Kautz, H., Sabharwal, A.: Towards Understanding and Harnessing the Potential of Clause Learning. Journal of Artificial Intelligence Research 22, 319–351 (2004)MathSciNetMATHGoogle Scholar
  8. 8.
    Beame, P., Pitassi, T.: Simplified and improved Resolution lower bounds. In: Proc. of the 27th IEEE Foundations of Computer Science (FOCS), pp. 274–282 (1996)Google Scholar
  9. 9.
    Beckmann, A., Pudlák, P., Thapen, N.: Parity games and propositional proofs (April 2013) (in preparation )Google Scholar
  10. 10.
    Ben-Sasson, E.: Hard examples for the bounded depth Frege proof system. Computational Complexity 11, 109–136 (2002)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Ben-Sasson, E., Impagliazzo, R., Wigderson, A.: Near Optimal Separation of Tree-Like and General Resolution. Combinatorica 24(4), 585–604 (2003)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ben-Sasson, E., Wigderson, A.: Short proofs are narrow–resolution made simple. Journal of the ACM 48(2), 149–169 (2001)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Bochnak, J., Coste, M., Roy, M.-F.: Real algebraic geometry. Springer (1999)Google Scholar
  14. 14.
    Bonet, M.L., Domingo, C., Gavaldà, R., Maciel, A., Pitassi, T.: Non-automatizability of bounded-depth Frege proofs. Computational Complexity 13, 47–68 (2004)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Bonet, M.L., Esteban, J.L., Galesi, N., Johansen, J.: On the Relative Complexity of Resolution Refinements and Cutting Planes Proof Systems. SIAM J. Comput. 30(5), 1462–1484 (2000)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Bonet, M.L., Galesi, N.: Optimality of Size-Width Tradeoffs for Resolution. Computational Complexity 10(4), 261–276 (2001)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Bonet, M.L., Pitassi, T., Raz, R.: On Interpolation and Automatization for Frege Systems. SIAM J. Comput. 29(6), 1939–1967 (2000)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Chvátal, V.: Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Mathematics 4(4), 305–337 (1973)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Clegg, M., Edmonds, J., Impagliazzo, R.: Using the Groebner basis algorithm to find proofs of unsatisfiability. In: Proc. of the 28th Annual ACM Symposium on Theory of Computing (STOC), pp. 174–183 (1996)Google Scholar
  20. 20.
    Condon, A.: The Complexity of Stochastic Games. Information and Computation 96, 203–224 (1992)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Cook, S.A., Reckhow, R.A.: The Relative Efficiency of Propositional Proof Systems. J. Symbolic Logic 44(1), 36–50 (1979)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer (1999)Google Scholar
  23. 23.
    Ehrenfeucht, A., Mycielsky, J.: Positional strategies for mean payoff games. International Journal of Game Theory 8(2), 109–113 (1979)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Emerson, E.A., Jutla, C.S.: Tree Automata, Mu-Calculus and Determinacy. In: Proc. of 32nd IEEE Foundations of Computer Science (FOCS), pp. 368–377 (1991)Google Scholar
  25. 25.
    Esteban, J.L., Galesi, N., Messner, J.: On the Complexity of Resolution with Bounded Conjunctions. Theoretical Computer Science 321(2-3), 347–370 (2004)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Grigoriev, D.: Linear Lower Bound on Degrees of Positivstellensatz Calculus Proofs for the Parity. Theor. Comput. Sci. 259, 613–622 (2001)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Grigoriev, D., Hirsch, E.A., Pasechnik, D.V.: Complexity of semi-algebraic proofs. Moscow Mathematical Journal 2(4), 647–679 (2002)MathSciNetMATHGoogle Scholar
  28. 28.
    Huang, L., Pitassi, T.: Automatizability and Simple Stochastic Games. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 605–617. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  29. 29.
    Krajíček, J.: On the weak pigeonhole principle. Fundamenta Mathematicae 170(1-3), 123–140 (2001)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Krajíček, J.: Lower Bounds to the Size of Constant-Depth Propositional Proofs. J. of Symbolic Logic 59(1), 73–86 (1994)MATHCrossRefGoogle Scholar
  31. 31.
    Krajíček, J.: Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic. J. of Symbolic Logic 62(2), 457–486 (1997)MATHCrossRefGoogle Scholar
  32. 32.
    Laurent, M.: A comparison of the Sherali-Adams, Lovász-Schrijver and Lasserre relaxations for 0-1 programming. Mathematics of Operations Research 28, 470–496 (2001)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optimization 1, 166–190 (1991)MATHCrossRefGoogle Scholar
  34. 34.
    Marques-Silva, J., Lynce, I., Malik, S.: Conflict-Driven Clause Learning SAT Solvers. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability. IOS Press (2009)Google Scholar
  35. 35.
    Parrilo, P.A.: Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. Ph.D. Thesis, California Institute of Technology, Pasadena, CA (2000)Google Scholar
  36. 36.
    Pitassi, T., Segerlind, N.: Exponential Lower Bounds and Integrality Gaps for Tree-Like Lovász-Schrijver Procedures. SIAM J. Comput. 41(1), 128–159 (2012)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Pudlák, P.: On reducibility and symmetry of disjoint NP-pairs. Theor. Comput. Science 295, 323–339 (2003)MATHCrossRefGoogle Scholar
  38. 38.
    Pudlák, P.: On the complexity of propositional calculus. In: Sets and Proofs, Invited papers from Logic Colloquium 1997, pp. 197–218. Cambridge Univ. Press (1999)Google Scholar
  39. 39.
    Sherali, H.D., Adams, W.P.: A hierarchy of relaxations and convex hull characterizations for mixed-integer zero-one programming problems. Discrete Applied Mathematics 52(1), 83–106 (1994)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Stålmarck, G.: Short resolution proofs for a sequence of tricky formulas. Acta Informatica 33(3), 277–280 (1996)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Stengle, G.: A Nullstellensatz and a Positivstellensatz in Semialgebraic Geometry. Mathematische Annalen 207(2), 87–97 (1974)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Urquhart, A.: Hard examples for resolution. Journal of the ACM 34(1), 209–219 (1987)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Zwick, U., Paterson, M.S.: The complexity of mean payoff games on graphs. Theoretical Computer Science 158, 343–359 (1996)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Albert Atserias
    • 1
  1. 1.Universitat Politècnica de CatalunyaBarcelonaSpain

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