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.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
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).
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.
D. A. Barrington, N. Immerman, and H. Straubing, On uniformity within NC1, Journal of Computer and System Sciences 41, (1990), pp. 274–306.
S. R. Buss, The Boolean formula value problem is in ALOGTIME, Proc. 19th Annual ACM Symposium on Theory of Computing, (1987), pp. 123–131.
J. F. Buss and J. Goldsmith, Nondeterminism within P, Lecture Notes in Computer Science 480, (1991), pp. 348–359.
S. Buss, S. Cook, A. Gupta, and V. Ramachandran, An optimal parallel algorithm for formula evaluation, SIAM J. Comput. 21, (1992), pp. 755–780.
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.
J. Chen, Characterizing parallel hierarchies by reducibilities, Information Processing Letters 39, (1991), pp. 303–307.
J. Díaz, and J. Torán, Classes of bounded nondeterminism, Math. System Theory 23, (1990), pp. 21–32.
R. G. Downey and M. R. Fellows, Fixed-parameter intractability, Proc. 7th Structure in Complexity Theory Conference, (1992), pp. 36–49.
D. S. Johnson, The NP-completeness column: an ongoing guide, Journal of Algorithms 13, (1992), pp. 502–524.
C. Kintala and P. Fisher, Refining nondeterminism in relativized complexity classes, SIAM J. Comput. 13, (1984), pp. 329–337.
W. L. Ruzzo, On uniform circuit complexity, J. Comput. System Sci. 21, (1981), pp. 365–383.
Robert Szelepcsényi, βκ-complete problems and greediness, Proc. 9th British Colloquium for Theoretical Computer Science, (1993).
M. J. Wolf, Nondeterministic circuits, space complexity and quasigroups, Manuscript, (1992).
A. Yao, Separating the polynomial-time hierarchy by oracles, Proc. 26th Annual Symposium on Foundations of Computer Science, (1985), pp. 1–10.
Author information
Authors and Affiliations
Editor information
Rights 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