Skip to main content
SpringerLink
Log in

On computing Boolean connectives of characteristic functions

  • Published:
Mathematical systems theory Aims and scope Submit manuscript

Abstract

This paper is a study of the existence of polynomial time Boolean connective functions for languages. A languageL has an AND function if there is a polynomial timef such thatf(x,y) εL ⇔x εL andy ε L. L has an OR function if there is a polynomial timeg such thatg(x,y) ε⇔xεL oryεL. While all NP complete sets have these functions, Graph Isomorphism, which is probably not complete, is also shown to have both AND and OR functions. The results in this paper characterize the complete sets for the classes Dp and pSAT[O(logn)] in terms of AND and OR and relate these functions to the structure of the Boolean hierarchy and the query hierarchies. Also, this paper shows that the complete sets for the levels of the Boolean hierarchy above the second level cannot have AND or OR unless the polynomial hierarchy collapses. Finally, most of the structural properties of the Boolean hierarchy and query hierarchies are shown to depend only on the existence of AND and OR functions for the NP complete sets.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Amir, R. Beigel, and W. I. Gasarch. Some connections between bounded query classes and non-uniform complexity.Proceedings of the 5th Structure in Complexity Theory Conference, pages 232–243, 1990.

  2. A. Amir and W. I. Gasarch. Polynomial terse sets.Information and Computation,77:37–56, April 1988.

    Google Scholar 

  3. A. Bertoni, D. Bruschi, D. Joseph, M. Sitharam, and P. Young. Generalized Boolean hierarchies and the hierarchy over RP. Technical Report 809, Department of Computer Sciences, University of Wisconsin-Madison, 1989.

  4. R. Beigel. NP-hard sets are p-supertese unless R = NP. Technical Report 4, Department of Computer Science, The Johns Hopkins University, 1988.

  5. R. Beigel. Bounded queries to SAT and the Boolean hierarchy.Theoretical Computer Science,84(2):199–223, July 1991.

    Google Scholar 

  6. J. Cai. Probability one separation of the Boolean hierarchy.Proceedings of the 4th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, pages 148–158, volume 247. Springer-Verlag, Berlin, 1987.

    Google Scholar 

  7. J. Cai, T. Gundermann, J. Hartmanis, L. Hemachandra, V Sewelson, K. Wagner, and G. Wechsung. The Boolean hierarchy, I: Structural properties.SLAM Journal on Computing,17(6):1232–1252, December 1988.

    Google Scholar 

  8. J. Cai and L. A. Hemachandra. The Boolean hierarchy: Hardware over NP. Technical Report TR 85-724, Department of Computer Science, Cornell University, December 1985.

  9. R. Chang. On the structure of bounded queries to arbitrary NP sets.Proceedings of the 4th Structure in Complexity Theory Conference, pages 250–258, June 1989. Revised paper inSIAM Journal on Computing,21(4):743–754, August 1992.

    Google Scholar 

  10. R. Chang and J. Kadin. The Boolean hierarchy and the polynomial hierarchy: a closer connection.Proceedings of the 5th Structure in Complexity Theory Conference, pages 169–178, July 1990. To appear inSIAM Journal on Computing.

  11. R. Chang, J. Kadin, and P. Rohatgi. Connections between the complexity of unique satisfiability and the threshold behavior of randomized reductions.Proceedings of the 6th Structure in Complexity Theory Conference, pages 255–269, July 1991. Revised paper to appear inJournal of Computer and System Sciences.

  12. J. Goldsmith, D. Joseph, and P. Young. Self-reducible, p-selective, near-testable, and p-cheatable sets: The effect of internal structure on the complexity of a set.Proceedings of the 2nd Structure in Complexity Theory Conference, pages 50–59, 1987.

  13. L. Hemachandra. The strong exponential hierarchy collapses.Journal of Computer and System Sciences,39(3):299–322, 1989.

    Google Scholar 

  14. C. M. Hoffman,Group-Theoretic Algorithms and Graph Isomorphism. Lecture Notes in Computer Science, volume 136. Springer-Verlag, Berlin, 1979.

    Google Scholar 

  15. J. Kadin. The polynomial time hierarchy collapses if the Boolean hierarchy collapses.SIAM Journal on Computing,17(6):1263–1282, December 1988.

    Google Scholar 

  16. J. Kadin, pNP[log n] and sparse Turing complete sets for NP.Journal of Computer and System Sciences,39:282–298, December 1989.

    Google Scholar 

  17. M. W. Krentel. The complexity of optimization problems.Journal of Computer and System Sciences,36(3):490–509, 1988.

    Google Scholar 

  18. J. Köbler, Uwe Schöning, and K. Wagner. The difference and truth-table hierarchies for NP.RAIRO Theoretical Informatics and Applications,21:419–435, 1987.

    Google Scholar 

  19. R. E. Ladner, N. A. Lynch, and A. L. Selman. A comparison of polynomial time reducibilities.Theoretical Computer Science,1(2):103–123, 1975.

    Google Scholar 

  20. C. Papadimitriou and M. Yannakakis. The complexity of facets (and some facets of complexity).Journal of Computer and System Sciences,28(2):244–259, April 1984.

    Google Scholar 

  21. U. Schöning. Graph isomorphism is in the low hierarchy.Journal of Computer and System Sciences,37:312–323, December 1988.

    Google Scholar 

  22. A. L. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP.Mathematical Systems Theory,13:55–65, 1979.

    Google Scholar 

  23. K. Wagner and G. Wechsung. On the Boolean closure of NP.Proceedings of the 1985 International Conference on Fundamentals of Computation Theory, Lecture Notes in Computer Science, pages 485–493, volume 199. Springer-Verlag, Berlin, 1985.

    Google Scholar 

Download references

Author information

Author notes

  1. R. Chang

    Present address: Computer Science Department, University of Maryland, Baltimore County, 21228, Baltimore, MD, USA

Authors and Affiliations

  1. Computer Science Department, Cornell University, 14853, Ithaca, NY, USA

    R. Chang

  2. Computer Science Department, University of Maine, 04469, Orono, ME, USA

    J. Kadin

Authors
  1. R. Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Kadin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The first author was supported in part by NSF Research Grants DCR-8520597 and CCR-88-23053, and by an IBM Graduate Fellowship.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chang, R., Kadin, J. On computing Boolean connectives of characteristic functions. Math. Systems Theory 28, 173–198 (1995). https://doi.org/10.1007/BF01303054

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01303054

Keywords

Search

Navigation