Skip to main content

On the amount of nondeterminism and the power of verifying

Extended abstract

  • Conference paper
  • First Online:
Mathematical Foundations of Computer Science 1993 (MFCS 1993)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 711))

Abstract

The relationship between nondeterminism and other computational resources is studied based on a special interactive-proof system model GC. Let s(n) be a function and C be a complexity class. Define GC(s(n), C) to be the class of languages that are accepted by verifiers in C that can make an extra O(s(n)) amount of nondeterminism. Our main results are (1) A systematic technique is developed to show that for many functions s(n) and for many complexity classes C, the class GC(s(n), C) has natural complete languages; (2) The class 0h of languages accepted by log-time alternating Turing machines making h alternations is precisely the class of languages accepted by uniform families of circuits of depth h; (3) The classes GC(s(n), II h 0), h ≥ 1, characterize precisely the fixed-parameter intractability of NP-hard optimization problems. In particular, the (2h)th level W[2h] of W-hierarchy introduced by Downey and Fellows collapses if and only if \(GC(s(n),\prod _{2h}^0 ) \subseteq P\) for some s(n)=ω(log n).

Supported by Engineering Excellence Award from Texas A&M University.

Supported by the National Science Foundation under Grant CCR-9110824.

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

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. K. Abrahamson, R. G. Downey and M. R. Fellows, Fixed-parameter intractability II, To appear in Proc. 10th Symposium on Theoretical Aspects of Computer Science, Wurzburg, Germany, (1993).

    Google Scholar 

  2. K. R. Abrahamson, J. A. Ellis, M. R. Fellows, and M. E. Mata, On the complexity of fixed parameter problems, Proc. 30th Annual Symposium on Foundations of Computer Science, (1989), pp. 210–215.

    Google Scholar 

  3. D. A. Barrington, N. Immerman, and H. Straubing, On uniformity within NC1, Journal of Computer and System Sciences 41, (1990), pp. 274–306.

    Google Scholar 

  4. S. R. Buss, The Boolean formula value problem is in ALOGTIME, Proc. 19th Annual ACM Symposium on Theory of Computing, (1987), pp. 123–131.

    Google Scholar 

  5. J. F. Buss and J. Goldsmith, Nondeterminism within P, Lecture Notes in Computer Science 480, (1991), pp. 348–359.

    Google Scholar 

  6. S. Buss, S. Cook, A. Gupta, and V. Ramachandran, An optimal parallel algorithm for formula evaluation, SIAM J. Comput. 21, (1992), pp. 755–780.

    Google Scholar 

  7. L. Cai and J. Chen, Fixed parameter tractability and approximability of NP-hard optimization problems, Proc. 2rd Israel Symposium on Theory of Computing and Systems, (1993), to appear.

    Google Scholar 

  8. J. Chen, Characterizing parallel hierarchies by reducibilities, Information Processing Letters 39, (1991), pp. 303–307.

    Google Scholar 

  9. J. Díaz, and J. Torán, Classes of bounded nondeterminism, Math. System Theory 23, (1990), pp. 21–32.

    Google Scholar 

  10. R. G. Downey and M. R. Fellows, Fixed-parameter intractability, Proc. 7th Structure in Complexity Theory Conference, (1992), pp. 36–49.

    Google Scholar 

  11. D. S. Johnson, The NP-completeness column: an ongoing guide, Journal of Algorithms 13, (1992), pp. 502–524.

    Google Scholar 

  12. C. Kintala and P. Fisher, Refining nondeterminism in relativized complexity classes, SIAM J. Comput. 13, (1984), pp. 329–337.

    Google Scholar 

  13. W. L. Ruzzo, On uniform circuit complexity, J. Comput. System Sci. 21, (1981), pp. 365–383.

    Google Scholar 

  14. Robert Szelepcsényi, βκ-complete problems and greediness, Proc. 9th British Colloquium for Theoretical Computer Science, (1993).

    Google Scholar 

  15. M. J. Wolf, Nondeterministic circuits, space complexity and quasigroups, Manuscript, (1992).

    Google Scholar 

  16. A. Yao, Separating the polynomial-time hierarchy by oracles, Proc. 26th Annual Symposium on Foundations of Computer Science, (1985), pp. 1–10.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Andrzej M. Borzyszkowski Stefan Sokołowski

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Cai, L., Chen, J. (1993). On the amount of nondeterminism and the power of verifying. In: Borzyszkowski, A.M., Sokołowski, S. (eds) Mathematical Foundations of Computer Science 1993. MFCS 1993. Lecture Notes in Computer Science, vol 711. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57182-5_23

Download citation

  • DOI: https://doi.org/10.1007/3-540-57182-5_23

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-57182-7

  • Online ISBN: 978-3-540-47927-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics