Tuples of Disjoint NP-Sets

(Extended Abstract)
  • Olaf Beyersdorff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3967)


Disjoint NP-pairs are a well studied complexity theoretic concept with important applications in cryptography and propositional proof complexity. In this paper we introduce a natural generalization of the notion of disjoint NP-pairs to disjoint k-tuples of NP-sets for k ≥ 2. We define subclasses of the class of all disjoint k-tuples of NP-sets. These subclasses are associated with a propositional proof system and possess complete tuples which are defined from the proof system.

In our main result we show that complete disjoint NP-pairs exist if and only if complete disjoint k-tuples of NP-sets exist for all k ≥ 2. Further, this is equivalent to the existence of a propositional proof system in which the disjointness of all k-tuples is shortly provable. We also show that a strengthening of this conditions characterizes the existence of optimal proof systems.


Polynomial Time Proof System Propositional Formula Polynomial Size Propositional Representation 
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.
    Beyersdorff, O.: Representable disjoint NP-pairs. In: Proc. 24th Conference on Foundations of Software Technology and Theoretical Computer Science, pp. 122–134 (2004)Google Scholar
  2. 2.
    Beyersdorff, O.: Tuples of disjoint NP-sets. Technical Report TR05-123, Electronic Colloquium on Computational Complexity (2005)Google Scholar
  3. 3.
    Beyersdorff, O.: Disjoint NP-pairs from propositional proof systems. In: Proc. 3rd Conference on Theory and Applications of Models of Computation (2006)Google Scholar
  4. 4.
    Buss, S.R.: Bounded Arithmetic. Bibliopolis, Napoli (1986)MATHGoogle Scholar
  5. 5.
    Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. The Journal of Symbolic Logic 44, 36–50 (1979)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Glaßer, C., Selman, A.L., Sengupta, S.: Reductions between disjoint NP-pairs. In: Proc. 19th Annual IEEE Conference on Computational Complexity, pp. 42–53 (2004)Google Scholar
  7. 7.
    Glaßer, C., Selman, A.L., Sengupta, S., Zhang, L.: Disjoint NP-pairs. SIAM Journal on Computing 33(6), 1369–1416 (2004)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Glaßer, C., Selman, A.L., Zhang, L.: Survey of disjoint NP-pairs and relations to propositional proof systems. Technical Report TR05-072, Electronic Colloquium on Computational Complexity (2005)Google Scholar
  9. 9.
    Grollmann, J., Selman, A.L.: Complexity measures for public-key cryptosystems. SIAM Journal on Computing 17(2), 309–335 (1988)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Köbler, J., Messner, J., Torán, J.: Optimal proof systems imply complete sets for promise classes. Information and Computation 184, 71–92 (2003)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Krajíček, J.: Bounded Arithmetic, Propositional Logic, and Complexity Theory. Encyclopedia of Mathematics and Its Applications, vol. 60. Cambridge University Press, Cambridge (1995)CrossRefMATHGoogle Scholar
  12. 12.
    Krajíček, J.: Interpolation theorems, lower bounds for proof systems and independence results for bounded arithmetic. The Journal of Symbolic Logic 62(2), 457–486 (1997)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Krajíček, J.: Dual weak pigeonhole principle, pseudo-surjective functions, and provability of circuit lower bounds. The Journal of Symbolic Logic 69(1), 265–286 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Krajíček, J., Pudlák, P.: Propositional proof systems, the consistency of first order theories and the complexity of computations. The Journal of Symbolic Logic 54, 1963–1079 (1989)Google Scholar
  15. 15.
    Krajíček, J., Pudlák, P.: Quantified propositional calculi and fragments of bounded arithmetic. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 36, 29–46 (1990)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Krajíček, J., Pudlák, P.: Some consequences of cryptographical conjectures for \(S^{1\over2}\) and EF. Information and Computation 140(1), 82–94 (1998)Google Scholar
  17. 17.
    Ladner, R.E.: On the structure of polynomial-time reducibility. Journal of the ACM 22, 155–171 (1975)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Pudlák, P.: On reducibility and symmetry of disjoint NP-pairs. Theoretical Computer Science 295, 323–339 (2003)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Razborov, A.A.: On provably disjoint NP-pairs. Technical Report TR94-006, Electronic Colloquium on Computational Complexity (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Olaf Beyersdorff
    • 1
  1. 1.Institut für InformatikHumboldt-Universität zu BerlinBerlinGermany

Personalised recommendations