Randomness vs. completeness: On the diagonalization strength of resource-bounded random sets
We show that the question of whether the p-tt-complete or p-T-complete sets for the deterministic time classes E and EXP have measure 0 in these classes in the sense of Lutz's resource-bounded measure cannot be decided by relativizable techniques. On the other hand, we obtain the following absolute results if we bound the norm, i.e., the number of oracle queries of the reductions: For r = tt, T, etse465-01etse and etse465-02etse
In the second part of the paper we investigate the diagonalization strength of random sets in an abstract way by relating randomness to a new genericity concept. This provides an alternative, quite elegant and powerful approach for obtaining results on resource-bounded measures like the ones in the first part of the paper.
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- 1.E. Allender, M. Strauss. Measure on small complexity classes with applications for BPP. In Proceedings of the 35th Symposium on Foundations of Computer Science, 867–818, IEEE Computer Society Press, 1994.Google Scholar
- 2.K. Ambos-Spies. Resource-bounded genericity. In Computability, Enumerability, Unsolvability (S. B. Cooper et al., Eds.), London Mathematical Society Lecture Notes Series 224, 1–59, Cambridge University Press, 1996.Google Scholar
- 3.K. Ambos-Spies, H. Fleischhack, H. Huwig. Diagonalizations over deterministic polynomial time. In Proceedings of the First Workshop on Computer Science Logic, CSL'87, Lecture Notes in Computer Science 329, 1–16, Springer Verlag, 1988.Google Scholar
- 8.H. Buhrman, E. Mayordomo. An excursion to the Kolmogorov random strings. In Proceedings of the 10th IEEE Structure in Complexity Theory Conference, 197–205, IEEE Computer Society Press, 1995.Google Scholar
- 9.H. Buhrman, D. v. Melkebeek. Hard Sets are Hard to Find. In Proceedings of the 13th IEEE Conference on Comput. Complexity, IEEE Computer Society Press, 1998.Google Scholar
- 10.H. Buhrman, D. v. Melkebeek, K. W. Regan, D. Sivakumar, M. Strauss. A generalization of resource-bounded measure with an application. In Proceedings of the Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, Springer Verlag, 1998.Google Scholar
- 11.S. A. Fenner. Notions of resource-bounded category and genericity. In Proceedings of the 6th IEEE Structure in Complexity Theory Conference, 196-212, IEEE Computer Society Press, 1991.Google Scholar
- 13.J. H. Lutz. The quantitative structure of exponential time. In Complexity Theory Retrospective II (L.A. Hemaspaandra, A.L. Selman, eds.), Springer-Verlag, 1997.Google Scholar
- 15.C. P. Schnorr. Zufälligkeit und Wahrscheinlichkeit. Lecture Notes in Mathematics 218, Springer-Verlag, 1971.Google Scholar