# Resource-bounded kolmogorov complexity revisited

## 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.

**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**.

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### References

- [BDG88]J. Balcázar, J. Díaz, and J. Gabarró.
*Structural Complexity I*. Springer-Verlag, 1988.Google Scholar - [BKT94]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.Google Scholar - [BT96]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.Google Scholar - [Coo71]S. Cook. The complexity of theorem-proving procedures. In
*Proc. 3rd ACM Symposium Theory of Computing*, pages 151–158, Shaker Heights, Ohio, 1971.Google Scholar - [DT90]J. Díaz and J. Torán. Classes of bounded nondeterminism.
*Math. Systems Theory*, 23:21–32, 1990.CrossRefGoogle Scholar - [ESY84]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.CrossRefGoogle Scholar - [FK96]L. Fortnow and M. Kummer. Resource-bounded instance complexity.
*Theoretical Computer Science A*, 161:123–140, 1996.CrossRefGoogle Scholar - [GHK92]J. Goldsmith, L. Hemachandra, and K. Kunen. Polynomial-time compression.
*Computational Complexity*, 2(1):18–39, 1992.CrossRefGoogle Scholar - [HNOS96]L. Hemaspaandra, A. Naik, M. Ogihara, and A. Selman. Computing solutions uniquely collapses the polynomial hierarchy.
*SIAM J. Comput.*, 25(4):697–708, 1996.CrossRefGoogle Scholar - [Ing32]A.E. Ingham.
*The Distribution of Prime Numbers*. Cambridge Tracts in Mathematics and Mathematical Physics. Cambridge University Press, 1932.Google Scholar - [IT89]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.Google Scholar - [JT95]Jenner and Toran. Computing functions with parallel queries to NP.
*Theoretical Computer Science*, 141, 1995.Google Scholar - [Kre88]M. Krentel. The complexity of optimization problem.
*J. Computer and System Sciences*, 36:490–509, 1988.CrossRefGoogle Scholar - [LV93]Ming Li and P.M.B. Vitányi.
*An Introduction to Kolmogorov Complexity and Its Applications*. Springer-Verlag, 1993.Google Scholar - [Ogi96]M. Ogihara. Functions computable with limited access to NP.
*Information Processing Letters*, 58:35–38, 1996.CrossRefGoogle Scholar - [Pap96]C. Papadimitriou. The complexity of knowledge representation. Invited Presentation at the Eleventh Annual IEEE Conference on Computational Complexity, May 1996.Google Scholar
- [Sip83]M. Sipser. A complexity theoretic approach to randomness. In
*Proc. 15th ACM Symposium on Theory of Computing*, pages 330–335, 1983.Google Scholar - [VV86]L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions.
*Theoretical Computer Science*, 47:85–93, 1986.CrossRefGoogle Scholar - [WT93]O. Watanabe and S. Toda. Structural analysis on the complexity of inverse functions.
*Mathematical Systems Theory*, 26:203–214, 1993.CrossRefGoogle Scholar