Theory of Computing Systems

, Volume 47, Issue 2, pp 491–506 | Cite as

On the Automatizability of Polynomial Calculus

  • Nicola Galesi
  • Massimo LauriaEmail author


We prove that Polynomial Calculus and Polynomial Calculus with Resolution are not automatizable, unless W[P]-hard problems are fixed parameter tractable by one-side error randomized algorithms. This extends to Polynomial Calculus the analogous result obtained for Resolution by Alekhnovich and Razborov (SIAM J. Comput. 38(4):1347–1363, 2008).


Automatizability Polynomial calculus Proof complexity Degree lower bound 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alekhnovich, M., Razborov, A.A.: Lower bounds for polynomial calculus: Non-binomial case. In: 42nd Annual Symposium on Foundations of Computer Science, pp. 190–199 (2001) Google Scholar
  2. 2.
    Alekhnovich, M., Razborov, A.A.: Resolution is not automatizable unless W[P] is tractable. SIAM J. Comput. 38(4), 1347–1363 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alon, N.: Tools from higher algebra. In: Handbook of Combinatorics, vol. 2, pp. 1749–1783. MIT Press, Cambridge (1995) Google Scholar
  4. 4.
    Atserias, A., Bonet, M.L.: On the automatizability of resolution and related propositional proof systems. Inf. Comput. 189(2), 182–201 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Beame, P., Pitassi, T.: Simplified and improved resolution lower bounds. In: 37th Annual Symposium on Foundations of Computer Science, pp. 274–282. IEEE Press, New York (1996) Google Scholar
  6. 6.
    Ben-Sasson, E., Wigderson, A.: Short proofs are narrow—resolution made simple. In: Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, pp. 517–526 (1999) Google Scholar
  7. 7.
    Bonet, M.L., Domingo, C., Gavaldà, R., Maciel, A., Pitassi, T.: Non-automatizability of bounded-depth frege proofs. Comput. Complex. 13(1–2), 47–68 (2004) zbMATHCrossRefGoogle Scholar
  8. 8.
    Bonet, M.L., Pitassi, T., Raz, R.: On interpolation and automatization for frege systems. SIAM J. Comput. 29(6), 1939–1967 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Clegg, M., Edmonds, J., Impagliazzo, R.: Using the Groebner basis algorithm to find proofs of unsatisfiability. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, pp. 174–183 (1996) Google Scholar
  10. 10.
    Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd edn. Springer, New York (2007) zbMATHGoogle Scholar
  11. 11.
    Downey, R., Fellows, M.: Parameterized Complexity. Springer, New York (1999) Google Scholar
  12. 12.
    Galesi, N., Lauria, M.: Degree lower bounds for a graph ordering principle. Submitted. See
  13. 13.
    Haken, A.: The intractability of resolution. Theor. Comput. Sci. 39, 297–308 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Impagliazzo, R., Pudlák, P., Sgall, J.: Lower bounds for the polynomial calculus and the Gröbner basis algorithm. Comput. Complex. 8(2), 127–144 (1999) zbMATHCrossRefGoogle Scholar
  15. 15.
    Jukna, S.: Extremal Combinatorics: with Applications in Computer Science. Springer, New York (2001) zbMATHGoogle Scholar
  16. 16.
    Krajícek, J.: Interpolation and approximate semantic derivations. Math. Log. Q. 48(4), 602–606 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Krajícek, J., Pudlák, P.: Some consequences of cryptographical conjectures for S 21 and ef. In: Leivant, D. (ed.) LCC. Lecture Notes in Computer Science, vol. 960, pp. 210–220. Springer, Berlin (1994) Google Scholar
  18. 18.
    Pitassi, T.: Algebraic propositional proof systems. In: Immerman, N., Kolaitis, P.G. (eds.) Descriptive Complexity and Finite Models. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 31, pp. 215–244. Am. Math. Soc., Providence (1996) Google Scholar
  19. 19.
    Pudlák, P.: On reducibility and symmetry of disjoint np-pairs. Theor. Comput. Sci. 295, 626–638 (2003) CrossRefGoogle Scholar
  20. 20.
    Pudlák, P., Sgall, J.: Algebraic models of computation and interpolation for algebraic proof systems. DIMACS Ser. Theor. Comput. Sci. 39, 279–296 (1998) Google Scholar
  21. 21.
    Razborov, A.A.: Lower bounds for the polynomial calculus. Comput. Complex. 7(4), 291–324 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    van Lint, J.H.: Introduction to Coding Theory, 3rd edn. Graduate Texts in Mathematics. Springer, New York (1998) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Computer ScienceSapienza—Università di RomaRomaItaly

Personalised recommendations