Proceedings 35th Annual Symposium on Foundations of Computer Science (1994)

Santa Fe, NM, USA

Nov. 20, 1994 to Nov. 22, 1994

ISBN: 0-8186-6580-7

pp: 807-818

E. Allender , Dept. of Comput. Sci., Rutgers Univ., New Brunswick, NJ, USA

ABSTRACT

We present a notion of resource-bounded measure for P and other subexponential-time classes. This generalization is based on Lutz's notion of measure, but overcomes the limitations that cause Lutz's definitions to apply only to classes at least as large as E. We present many of the basic properties of this measure, and use it to explore the class of sets that are hard for BPP. Bennett and Gill showed that almost all sets are hard for BPP; Lutz improved this from Lebesgue measure to measure on ESPACE. We use our measure to improve this still further, showing that for all /spl epsiv/>0, almost every set in E/sub /spl epsiv// is hard for BPP, where E/sub /spl epsiv//=/spl cup//sub /spl delta/</spl epsiv//DTIME(2(n/sup /spl delta//)), which is the best that can be achieved without showing that BPP is properly contained in E. A number of related results are also obtained in this way.

INDEX TERMS

resource-bounded measure theory, small complexity classes, BPP, resource-bounded measure, subexponential-time classes, class of sets

CITATION

M. Strauss and E. Allender, "Measure on small complexity classes, with applications for BPP,"

*Proceedings 35th Annual Symposium on Foundations of Computer Science(FOCS)*, Santa Fe, NM, USA, 1994, pp. 807-818.

doi:10.1109/SFCS.1994.365713

CITATIONS