Measure in feasible complexity classes

Resource-bounded measure assigns sizes to subsets of (countable) complexity classes. This thesis provides resource-bounded measure on classes smaller than has been provided in previous work, classes as small as P. The analogs of the existing tools for resource-bounded measure, when applied to P or other subexponential time classes, involve sublinear-time machines, and these machines, which are not even allowed enough time to read all their input, are qualitatively weaker than their counterparts in previous work, so developing a useful definition of measure on P involves overcoming several obstacles. Since only subexponential time classes are considered feasible, it is important to extend the tool to them.

After overcoming these obstacles and defining the measure, the thesis examines the notion for robustness. We show that all variants we studied, with one important exception, yield the same notion of measure.

Using the new tool, measure-theoretic properties of P and other subexponential classes are explored. While it is known that almost-every language (in the Lebesgue sense) is hard for the bounded-error probabilistic class BPP, the thesis shows that almost every language (in a severely resource-bounded sense) is hard for BPP.

A recently-proposed notion of source for BPP is also examined in the thesis. A source is a pseudorandom string that can substitute for the stream of random bits used by a BPP machine in repeated runs. The thesis shows that such sources are precisely the normal numbers of Borel, abundant in P, and computable by very weak machines. Using the above result about sets that are hard for BPP, the thesis provides a new more satisfactory notion of source.