Measure on small complexity classes, with applications for BPP

E. Allender, M. Strauss
Proceedings 35th Annual Symposium on Foundations of Computer Science  
We present a notion of resource-bounded m e asure 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 de nitions to apply only to classes at least as large as E. W e p r esent 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 t o m e
more » ... measure t o m e asure o n ESPACE. We use our measure to improve this still further, showing that for all 0, almost every set in E is hard for BPP, where E = S DTIME2 n ; which is the best that can be achieved without showing that BPP is properly contained i n E. A number of related r esults are also obtained in this way.
doi:10.1109/sfcs.1994.365713 dblp:conf/focs/AllenderS94 fatcat:s52duvyslbf4rnbcxqsxsank4m