Abstract
We take a fresh look at CD complexity, where CD t(x) is the smallest program that distinguishes x from all other strings in time t(¦x¦). We also look at a CND complexity, a new nondeterministic variant of CD complexity.
We show several results relating time-bounded C, CD and CND complexity and their applications to a variety of questions in computational complexity theory including:
-
Showing how to approximate the size of a set using CD complexity avoiding the random string needed by Sipser. Also we give a new simpler proof of Sipser's lemma.
-
A proof of the Valiant-Vazirani lemma directly from Sipser's earlier CD lemma.
-
A relativized lower bound for CND complexity.
-
Exact characterizations of equivalences between C, CD and CND complexity.
-
Showing that a satisfying assignment can be found in output polynomial time if and only if a unique assignment can be found quickly. This answers an open question of Papadimitriou.
-
New Kolmogorov-based constructions of the following relativized worlds:
-
There exists an infinite set in P with no sparse infinite subsets in NP.
-
EXP=NEXP but there exists a nondeterministic exponential time Turing machine whose accepting paths cannot be found in exponential time.
-
Satisfying assignment cannot be found with nonadaptive queries to SAT.
Part of this research was done while visiting The University of Chicago. Partially supported by the Dutch foundation for scientific research (NWO) through NFI Project ALADDIN, under contract number NF 62-376 and SION project 612-34-002, and by the European Union through NeuroCOLT ESPRIT Working Group Nr. 8556, and HC&M grant nr. ERB4050PL93-0516.
Supported in part by NSF grant CCR 92-53582 and the Fulbright scholar program and NWO.
At http://www.cs.uchicago.edu/∼fortnow/papers/ a full version of this paper including complete proofs can be found.
Preview
Unable to display preview. Download preview PDF.
References
J. Balcázar, J. Díaz, and J. Gabarró. Structural Complexity I. Springer-Verlag, 1988.
H. Buhrman, J. Kadin, and T. Thierauf. On functions computable with nonadaptive queries to NP. In Proc. Structure in Complexity Theory 9th Annual Conference, pages 43–52. IEEE computer society press, 1994.
H. Buhrman and T. Thierauf. The complexity of generating and checking proofs of membership. In C. Pueach and R. Reischuk, editors, 13th Annual Symposium on Theoretical Aspects of Computer Science, number 1046 in Lecture Notes in Computer Science, pages 75–86. Springer, 1996.
S. Cook. The complexity of theorem-proving procedures. In Proc. 3rd ACM Symposium Theory of Computing, pages 151–158, Shaker Heights, Ohio, 1971.
J. Díaz and J. Torán. Classes of bounded nondeterminism. Math. Systems Theory, 23:21–32, 1990.
S. Even, A. L. Selman, and Y. Yacobi. The complexity of promise problems with applications to public-key cryptography. Information and Control, 61(2):159–173, May 1984.
L. Fortnow and M. Kummer. Resource-bounded instance complexity. Theoretical Computer Science A, 161:123–140, 1996.
J. Goldsmith, L. Hemachandra, and K. Kunen. Polynomial-time compression. Computational Complexity, 2(1):18–39, 1992.
L. Hemaspaandra, A. Naik, M. Ogihara, and A. Selman. Computing solutions uniquely collapses the polynomial hierarchy. SIAM J. Comput., 25(4):697–708, 1996.
A.E. Ingham. The Distribution of Prime Numbers. Cambridge Tracts in Mathematics and Mathematical Physics. Cambridge University Press, 1932.
R. Impagliazzo and G. Tardos. Decision versus search problems in superpolynomial time. In Proc. 30th IEEE Symposium on Foundations of Computer Science, pages 222–227, 1989.
Jenner and Toran. Computing functions with parallel queries to NP. Theoretical Computer Science, 141, 1995.
M. Krentel. The complexity of optimization problem. J. Computer and System Sciences, 36:490–509, 1988.
Ming Li and P.M.B. Vitányi. An Introduction to Kolmogorov Complexity and Its Applications. Springer-Verlag, 1993.
M. Ogihara. Functions computable with limited access to NP. Information Processing Letters, 58:35–38, 1996.
C. Papadimitriou. The complexity of knowledge representation. Invited Presentation at the Eleventh Annual IEEE Conference on Computational Complexity, May 1996.
M. Sipser. A complexity theoretic approach to randomness. In Proc. 15th ACM Symposium on Theory of Computing, pages 330–335, 1983.
L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85–93, 1986.
O. Watanabe and S. Toda. Structural analysis on the complexity of inverse functions. Mathematical Systems Theory, 26:203–214, 1993.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Buhrman, H., Fortnow, L. (1997). Resource-bounded kolmogorov complexity revisited. In: Reischuk, R., Morvan, M. (eds) STACS 97. STACS 1997. Lecture Notes in Computer Science, vol 1200. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023452
Download citation
DOI: https://doi.org/10.1007/BFb0023452
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-62616-9
Online ISBN: 978-3-540-68342-1
eBook Packages: Springer Book Archive