The complexity of generating and checking proofs of membership

  • Harry Buhrman
  • Thomas Thierauf
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1046)


We consider the following questions:
  1. 1.

    Can one compute satisfying assignments for satisfiable Boolean formulas in polynomial time with parallel queries to NP?

  2. 2.

    Is the unique optimal clique problem (UOCLIQUE) complete for PNP[O(log n)]?

  3. 3.

    Is the unique satisfiability problem (USAT) NP hard? We define a framework that enables us to study the complexity of generating and checking proofs of membership. We connect the above three questions to the complexity of generating and checking proofs of membership for sets in NP and PNP[O(log n)]. We show that an affirmative answer to any of the three questions implies the existence of coNP checkable proofs for PNP[O(log n)] that can be generated in FP NP . Furthermore, we construct an oracle relative to which there do not exist coNP checkable proofs for NP that are generated in FP NP . It follows that relative to this oracle all of the above questions are answered negatively.



Turing Machine Proof System Conjunctive Normal Form Boolean Formula Satisfying Assignment 
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.


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  1. [BDG-I&II]
    J. Balcázar, J. Díaz, and J. Gabarró. Structural Complexity I & II. EATCS Monographs on Theoretical Computer Science, Springer-Verlag (1988, 1991)Google Scholar
  2. [Be88]
    Beigel, R.: NP-hard sets are P-superterse unless R=NP. Technical Report 88-04, Dept. of Computer Science, The John Hopkins University (1988).Google Scholar
  3. [BG82]
    Blass, A., Gurevich, Y.: On the unique satisfiability problem. Information and Control 55 (1982) 80–88CrossRefGoogle Scholar
  4. [BKT94]
    Buhrman, H., Kadin, J., Thierauf, T.: On functions computable with nonadaptive queries to NP. Proc. 9th Structure in Complexity Theory Conference (1994) 43–52Google Scholar
  5. [CKR95]
    Chang, R., Kadin, J., Rohatgi, P.: On Unique Satisfiability and the threshhold behavior of randomized reductions. Journal of Computer and System Science 50 (1995) 359–373.CrossRefGoogle Scholar
  6. [Co71]
    Cook, S.: The Complexity of Theorem-Proving Procedures. Proc. 3rd ACM Symposium on Theory of Computing (1971) 151–158Google Scholar
  7. [CT91]
    Chen, Z., Toda, S.: On the Complexity of Computing Optimal Solutions. International Journal of Foundations of Computer Science 2 (1991) 207–220CrossRefGoogle Scholar
  8. [CT93]
    Chen, Z., Toda, S.: An Exact Characterization of FP NP. Manuscript (1993)Google Scholar
  9. [FHOS93]
    Fenner, S., Homer, S., Ogiwara, M., Selman, A.: On Using Oracles That Compute values. 10-th Annual Symposium on Theoretical Aspects of Computer Science, Springer Verlag LNCS 665 (1993) 398–407Google Scholar
  10. [Fo94]
    Fortnow, L.: Personal Communication. In the plane to Madras (India) (December 7, 1994)Google Scholar
  11. [H89]
    Hemachandra, L.: The strong exponential hierarchy collapses. Journal of Computer and System Sciences 39(3) (1989) 299–322CrossRefGoogle Scholar
  12. [HNOS94]
    Hemaspaandra, L., Naik, A., Ogihara, M., Selman, A.: Finding Satisfying Assignments Uniquely Isn't so Easy: Unique Solutions Collapes the Polynomial Hierarchy. Algorithms and Compuatation, International Symposium ISAAC '94, Springer Verlag LNCS 834 (1994) 56–64Google Scholar
  13. [HU79]
    Hopcroft, J., Ullman, J.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley (1979)Google Scholar
  14. [IT89]
    Impagliazzo, R., Tardos, G.: Decision Versus Search Problems in Super-Polynomial Time. Proc. 30th IEEE Annual Symposium on Foundations of Computer Science (1989) 222–227Google Scholar
  15. [Ka88]
    Kadin, J.: Restricted Turing Reducibilities and the Structure of the Polynomial Time Hierarchy. PhD thesis, Cornell University (1988)Google Scholar
  16. [Kr86]
    Krentel, M.: The Complexity of Optimization Problems. Proc. 18th ACM Symposium on Theory of Computing (1986) 69–76Google Scholar
  17. [Le73]
    Levin, L.: Universal Sorting Problems. Problems of Information Transmission 9 (1973) 265–266Google Scholar
  18. [Og95]
    Ogihara, M.: Functions Computable with Limited Access to NP. Technical Report 538, University of Rochester (1995)Google Scholar
  19. [P84]
    Papadimitriou, C.: On the complexity of unique solutions. Journal of the ACM 31(2) (1984) 392–400CrossRefGoogle Scholar
  20. [PY84]
    Papadimitriou, C., Yannakakis, M.: On the complexity of facets. Journal of Computer and System Sciences 28 (1984) 244–259CrossRefGoogle Scholar
  21. [PZ83]
    Papadimitriou, C., Zachos, D.: Two remarks on the power of counting. 6th GI Conference on TCS, Springer Verlag LNCS 145 (1983) 269–276Google Scholar
  22. [Se94]
    Selman, A.: A taxonomy of complexity classes of functions. Journal of Computer and System Science 48 (1994) 357–381.Google Scholar
  23. [To91]
    Toda, S.: On polynomial-time truth-table reducibilities of intractable sets to P-selective sets. Mathematical Systems Theory 24 (1991) 69–82.Google Scholar
  24. [VV86]
    Valiant, L., Vazirani, V.: NP is as easy as detecting unique solutions. Theoretical Computer Science 47(1) (1986) 85–93CrossRefGoogle Scholar
  25. [W86]
    Wagner, K.: More complicated questions about maxima and minima and some closure properties of NP. Proc. 13th International Colloquium on Automata, Languages, and Programming (ICALP), Springer Verlag LNCS 226 (1986) 53–80Google Scholar
  26. [W90]
    Wagner, K.: Bounded query classes. SIAM Journal on Computing 19(5) (1990) 833–846CrossRefGoogle Scholar
  27. [WT93]
    Watanabe, O., Toda, S.: Structural Analysis on the Complexity of Inverse Functions. Mathematical Systems Theory 26 (1993) 203–214CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Harry Buhrman
    • 1
  • Thomas Thierauf
    • 2
  1. 1.CWIGB AmsterdamThe Netherlands
  2. 2.Abt. Theoretische InformatikUniversität UlmUlmGermany

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