On Optimal Heuristic Randomized Semidecision Procedures, with Applications to Proof Complexity and Cryptography

Abstract

The existence of an optimal propositional proof system is a major open question in proof complexity; many people conjecture that such systems do not exist. Krajíček and Pudlák (J. Symbol. Logic 54(3):1063, 1989) show that this question is equivalent to the existence of an algorithm that is optimal on all propositional tautologies. Monroe (Theor. Comput. Sci. 412(4–5):478, 2011) recently presented a conjecture implying that such an algorithm does not exist.

We show that if one allows errors, then such optimal algorithms do exist. The concept is motivated by the notion of heuristic algorithms. Namely, we allow an algorithm, called a heuristic acceptor, to claim a small number of false “theorems” and err with bounded probability on other inputs. The amount of false “theorems” is measured according to a polynomial-time samplable distribution on non-tautologies. Our result remains valid for all recursively enumerable languages and can also be viewed as the existence of an optimal weakly automatizable heuristic proof system. The notion of a heuristic acceptor extends the notion of a classical acceptor; in particular, an optimal heuristic acceptor for any distribution simulates every classical acceptor for the same language.

We also note that the existence of a co-NP-language L with a polynomial-time samplable distribution on \(\overline{L}\) that has no polynomial-time heuristic acceptors is equivalent to the existence of an infinitely-often one-way function.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Alekhnovich, M., Ben-Sasson, E., Razborov, A.A., Wigderson, A.: Pseudorandom generators in propositional proof complexity. Technical Report 00-023, Electronic Colloquium on Computational Complexity (2000). Extended abstract appears in Proceedings of FOCS-2000

  2. 2.

    Bogdanov, A., Trevisan, L.: Average-case complexity. Found. Trends Theor. Comput. Sci. 2(1), 1–106 (2006)

    Article  MathSciNet  Google Scholar 

  3. 3.

    Cook, S.A., Krajíček, J.: Consequences of the provability of NPP/poly. J. Symb. Log. 72(4), 1353–1371 (2007)

    MATH  Article  Google Scholar 

  4. 4.

    Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. J. Symb. Log. 44(1), 36–50 (1979)

    MATH  Article  MathSciNet  Google Scholar 

  5. 5.

    Fortnow, L., Santhanam, R.: Hierarchy theorems for probabilistic polynomial time. In: Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, pp. 316–324 (2004)

    Google Scholar 

  6. 6.

    Fortnow, L., Santhanam, R.: Recent work on hierarchies for semantic classes. SIGACT News 37(3), 36–54 (2006)

    Article  Google Scholar 

  7. 7.

    Grigoriev, D., Hirsch, E.A., Pervyshev, K.: A complete public-key cryptosystem. Groups, Complexity, Cryptology 1(1), 1–12 (2009)

    MATH  Article  MathSciNet  Google Scholar 

  8. 8.

    Hirsch, E.A., Itsykson, D.M.: An infinitely-often one-way function based on an average-case assumption. St. Petersburg Math. J. 21(3), 459–468 (2010)

    MATH  Article  MathSciNet  Google Scholar 

  9. 9.

    Håstad, J., Impagliazzo, R., Levin, L.A., Luby, M.: A pseudorandom generator from any one-way function. SIAM J. Comput. 28(4), 1364–1396 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  10. 10.

    Harnik, D., Kilian, J., Naor, M., Reingold, O., Rosen, A.: On robust combiners for oblivious transfer and other primitives. In: Proc. of EUROCRYPT-2005 (2005)

    Google Scholar 

  11. 11.

    Itsykson, D.M.: Structural complexity of AvgBPP. Ann. Pure Appl. Log. 162(3), 213–223 (2010)

    MATH  Article  MathSciNet  Google Scholar 

  12. 12.

    Krajíček, J., Pudlák, P.: Propositional proof systems, the consistency of first order theories and the complexity of computations. J. Symb. Log. 54(3), 1063–1079 (1989)

    MATH  Article  Google Scholar 

  13. 13.

    Krajíček, J.: On the weak pigeonhole principle. Fundam. Math. 170(1–3), 123–140 (2001)

    MATH  Article  Google Scholar 

  14. 14.

    Krajíček, J.: Tautologies from pseudorandom generators. Bull. Symb. Log. 7(2), 197–212 (2001)

    MATH  Article  Google Scholar 

  15. 15.

    Levin, L.A.: Universal sequential search problems. Probl. Inf. Transm. 9, 265–266 (1973)

    Google Scholar 

  16. 16.

    Levin, L.A.: One-way functions and pseudorandom generators. Combinatorica 7, 357–363 (1987)

    MATH  Article  MathSciNet  Google Scholar 

  17. 17.

    McDiarmid, C.: In: Concentration. Algorithms and Combinatorics, vol. 16, pp. 195–248. Springer, Berlin (1998)

    Google Scholar 

  18. 18.

    Messner, J.: On optimal algorithms and optimal proof systems. In: Proceedings of the 16th Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 1563, pp. 361–372 (1999)

    Google Scholar 

  19. 19.

    Monroe, H.: Speedup for natural problems and noncomputability. Theor. Comput. Sci. 412(4–5), 478–481 (2011)

    MATH  Article  MathSciNet  Google Scholar 

  20. 20.

    Pervyshev, K.: On heuristic time hierarchies. In: Proceedings of the 22nd IEEE Conference on Computational Complexity, pp. 347–358 (2007)

    Google Scholar 

  21. 21.

    Pudlák, P.: On reducibility and symmetry of disjoint NP pairs. Theor. Comput. Sci. 295(1–3), 323–339 (2003)

    MATH  Article  Google Scholar 

  22. 22.

    Razborov, A.A.: On provably disjoint NP-pairs. Technical Report 94-006, Electronic Colloquium on Computational Complexity (1994)

  23. 23.

    Sadowski, Z.: On an optimal deterministic algorithm for SAT. In: Proceedings of CSL’98. Lecture Notes in Computer Science, vol. 1584, pp. 179–187. Springer, Berlin (1999)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Dmitry Itsykson.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hirsch, E.A., Itsykson, D., Monakhov, I. et al. On Optimal Heuristic Randomized Semidecision Procedures, with Applications to Proof Complexity and Cryptography. Theory Comput Syst 51, 179–195 (2012). https://doi.org/10.1007/s00224-011-9354-3

Download citation

Keywords

  • Propositional proof complexity
  • Optimal algorithm
  • i.o. one-way