On checking versus evaluation of multiple queries

  • William I. Gasarch
  • Lane A. Hemachandra
  • Albrecht Hoene
Part of the Lecture Notes in Computer Science book series (LNCS, volume 452)


The distinction between computing answers and checking answers is fundamental to computational complexity theory, and is reflected in the relationship of NP to P. The plausibility of computing the answers to many membership queries to a hard set with few queries is the subject of the theory of terseness. In this paper, we develop companion theories—both complexity-theoretic and recursion-theoretic—of characteristic vector terseness, which ask whether the answers to many membership queries to a hard set can be checked with fewer queries.


Characteristic Vector SIAM Journal Random Oracle Multiple Query Membership Query 
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. [AG88]
    A. Amir and W. Gasarch. Polynomial terse sets. Information and Computation, 77:37–56, 1988.Google Scholar
  2. [AH89]
    E. Allender and L. Hemachandra. Lower bounds for the low hierarchy. In Automata, Languages, and Programming (ICALP 1989), pages 31–45. Springer-Verlag Lecture Notes in Computer Science #372, July 1989.Google Scholar
  3. [All86]
    E. Allender. The complexity of sparse sets in P. In Proceedings 1st Structure in Complexity Theory Conference, pages 1–11, Springer-Verlag Lecture Notes in Computer Science #223, June 1986.Google Scholar
  4. [AR88]
    E. Allender and R. Rubinstein. P-printable sets. SIAM Journal on Computing, 17(6):1193–1202, 1988.Google Scholar
  5. [Bar68]
    Y. Barzdin'. Complexity of programs to determine whether natural numbers not greater than n belong to a recursively enumerable set. Soviet Math. Dokl., 9:1251–1254, 1968.Google Scholar
  6. [BBJ+]
    [BBJ+] A. Bertoni, D. Bruschi, D. Joseph, M. Sitharam, and P. Young. Generalized boolean hierarchies and boolean hierarchies over RP. Manuscript, 1989. Preliminary version appears in Proceedings Fundamentals of Computation Theory, Springer-Verlag Lecture Notes in Computer Science.Google Scholar
  7. [BBS86]
    J. Balcázar, R. Book, and U. Schöning. Sparse sets, lowness, and highness. SIAM Journal on Computing, 15:739–747, 1986.Google Scholar
  8. [Bei87]
    R. Beigel. A structural theorem that depends quantitatively on the complexity of SAT. In Proceedings of the 2nd Annual Conference on Structure in Complexity Theory, pages 28–32. IEEE Computer Society Press, June 1987.Google Scholar
  9. [Bei88a]
    R. Beigel. Bounded queries to SAT and the Boolean hierarchy. Unpublished manuscript, August 1988.Google Scholar
  10. [Bei88b]
    R. Beigel. NP-hard sets are P-superterse unless R=NP. Technical Report 88-04, Johns Hopkins Department of Computer Science, August 1988.Google Scholar
  11. [BG82]
    A. Blass and Y. Gurevich. On the unique satisfiability problem. Information and Control, 55:80–88, 1982.Google Scholar
  12. [BGGO87]
    R. Beigel, W. Gasarch, J. Gill, and J. Owings. Terse, superterse, and verbose sets. Technical Report TR-1806, University of Maryland, Department of Computer Science, College Park, Maryland, 1987.Google Scholar
  13. [BGS75]
    T. Baker, J. Gill, and R. Solovay. Relativizations of the P=?NP question. SIAM Journal on Computing, 4(4):431–442, 1975.Google Scholar
  14. [Gai89]
    J. Cai. With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy. Journal of Computer and System Sciences, 38(1):68–85, 1989.Google Scholar
  15. [CGH+88]
    [CGH+88] J. Cai, T. Gundermann, J. Hartmanis, L. Hemachandra, V. Sewelson, K. Wagner, and G. Wechsung. The boolean hierarchy I: Structural properties. SIAM Journal on Computing, 17(6):1232–1252, December 1988.Google Scholar
  16. [CGH+89]
    [CGH+89] J. Cai, T. Gundermann, J. Hartmanis, L. Hemachandra, V. Sewelson, K. Wagner, and G. Wechsung. The boolean hierarchy II: Applications. SIAM Journal on Computing, 18(1):95–111, February 1989.Google Scholar
  17. [CH]
    J. Cai and L. Hemachandra. On the power of parity polynomial time. Mathematical Systems Theory. To appear.Google Scholar
  18. [Cha89]
    R. Chang. On the structure of bounded queries to arbitrary NP sets. In Proceedings of the 4th Conference on Structure in Complexity Theory, pages 250–258. IEEE Computer Science Press, June 1989.Google Scholar
  19. [CK89]
    R. Chang and J. Kadin. The boolean hierarchy and the polynomial hierarchy: A closer connection. Technical Report TR 89-1008, Department of Computer Science, Cornell University, Ithaca, NY, May 1989.Google Scholar
  20. [EHK81]
    R. Epstein, R. Haas, and R. Kramer. Hierarchies of sets and degrees below 0'. In Logic Year 1979–80, The University of Connecticut, Lecture Notes in Mathematics #859, pages 32–47. Springer-Verlag, Berlin, 1981.Google Scholar
  21. [Gil77]
    J. Gill. Computational complexity of probabilistic Turing machines. SIAM Journal on Computing, 6(4):675–695, December 1977.Google Scholar
  22. [GP86]
    L. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of Boolean functions. Theoretical computer Science, 43:43–58, 1986.Google Scholar
  23. [GW87]
    T. Gundermann and G. Wechsung. Counting classes with finite acceptance types. Computers and Artificial Types, 6(5):395–409, 1987.Google Scholar
  24. [Hau14]
    F. Hausdorff. Grundzüge der Mengenlehre. Leipzig, 1914.Google Scholar
  25. [Hem89]
    L. Hemachandra. The strong exponential hierarchy collapses. Journal of Computer and System Sciences, 39(3):299–322, 1989.Google Scholar
  26. [HH88]
    L. Hemachandra and A. Hoene. On checking versus evaluation of multiple queries: Characteristic vector terseness. Technical Report No. 88-21, Technische Universität, Berlin, October 1988.Google Scholar
  27. [HIS85]
    J. Hartmanis, N. Immerman, and V. Sewelson. Sparse sets in NP-P: EXPTIME versus NEXPTIME. Information and Control, 65(2/3):159–181, May/June 1985.Google Scholar
  28. [HU79]
    J. Hopcroft and J. Ullman. Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, 1979.Google Scholar
  29. [Joc68]
    C. Jockusch. Semirecursive sets and positive reducibility. Transactions of the AMS, 131(2):420–436, 1968.Google Scholar
  30. [Joc79]
    C. Jockusch. Recursion theory: Its generalizations and applications. In Proceedings of the Logic Colloquium, Leeds, pages 140–157. Cambridge University Press, 1979.Google Scholar
  31. [K88]
    [K88] J. Kämper. Non-uniform proof systems: a new framework to describe non-uniform and probabilistic complexity classes. In 8th Conference on Foundations of Software Technology and Theoretical Computer Science (FST-TCS 1988), pages 193–210. Springer-Verlag Lecture Notes in Computer Science #338, December 1988.Google Scholar
  32. [Kad88]
    J. Kadin. The polynomial time hierarchy collapses if the boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263–1282, 1988.Google Scholar
  33. [KS85]
    K. Ko and U. Schöning. On circuit-size complexity and the low hierarchy in NP. SIAM Journal on Computing, 14(1):41–51, 1985.Google Scholar
  34. [KSW87]
    J. Köbler, U. Schöning, and K. Wagner. The difference and truth-table hierarchies for NP. R.A.I.R.O. Informatique théorique et Applications, 21:419–435, 1987.Google Scholar
  35. [PY84]
    C. Papadimitriou and M. Yannakakis. The complexity of facets (and some facets of complexity). Journal of Computer and System Sciences, 28:244–259, 1984.Google Scholar
  36. [PZ83]
    C. Papadimitriou and S. Zachos. Two remarks on the power of counting. In Proceedings 6th GI Conference on Theoretical Computer Science, pages 269–276. Springer-Verlag Lecture Notes in Computer Science #145, 1983.Google Scholar
  37. [Rog67]
    H. Rogers, Jr. The Theory of Recursive Functions and Effective Computability. McGraw-Hill, 1967.Google Scholar
  38. [Rub88]
    R. Rubinstein. Structural Complexity Classes of Sparse Sets: Intractability, Data Compression and Printability. PhD thesis, Northeastern University, Boston, MA, August 1988.Google Scholar
  39. [Sch83]
    U. Schöning. A low and a high hierarchy in NP. Journal of Computer and System Sciences., 27:14–28, 1983.Google Scholar
  40. [Sto77]
    L. Stockmeyer. The polynomial-time hierarchy. Theoretical Computer Science, 3:1–22, 1977.Google Scholar
  41. [Val76]
    L. Valiant. The relative complexity of checking and evaluting. Information Processing Letters, 5:20–23, 1976.Google Scholar
  42. [Wag87]
    K. Wagner. Bounded query classes. Institut für Mathematik 157, Augsburg, Augsburg, W. Germany, October 1987.Google Scholar
  43. [Wag88]
    K. Wagner. Bounded query computation. In Proceedings 3rd Structure in Complexity Theory Conference, pages 260–277. IEEE Computer Society Press, June 1988.Google Scholar
  44. [Wec85]
    G. Wechsung. On the boolean closure of NP. In Proceedings of the International Conference on Fundamentals of Computation Theory, pages 485–493. Springer-Verlag, Lecture Notes in Computer Science, 1985.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • William I. Gasarch
    • 1
  • Lane A. Hemachandra
    • 2
  • Albrecht Hoene
    • 3
  1. 1.Institute for Advanced Computer Studies, Department of Computer ScienceUniversity of MarylandCollege Park
  2. 2.Department of Computer ScienceUniversity of RochesterRochester
  3. 3.Technische Universität Berlin, Fachbereich InformatikBerlin 10

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