Determine the P-measure of BPP

Determine whether the polynomial-time resource-bounded measure of BPP is zero or not; equivalently, ascertain whether mu_P(BPP)=0 or BPP=EXP under Lutz’s resource-bounded measure framework.

Background

Resource-bounded measure (Lutz) assigns measure zero or one to complexity classes via polynomial-time computable martingales. For BPP, van Melkebeek’s zero–one law shows that either BPP has P-measure zero or BPP equals EXP, tying the measure question to a major derandomization question.

The paper develops counting-measure techniques and shows BPP has SharpP-strong dimension 0, but this does not resolve the P-measure question, which remains a central open problem.