Randomness and Nondeterminism are Incomparable for Read-Once Branching Programs

We extend the tools for proving lower bounds for randomized branching programs by presenting a new technique for the read-once case which is applicable to a large class of functions. This technique fills the gap between simple methods only applicable for OBDDs and the well-known “rectangle technique” of Borodin, Razborov and Smolensky which works for the quite general models of nondeterministic and randomized read-times branching programs, but which has the drawback that it could only be applied to very special functions so far. By an application of the new method, we resolve the remaining open problems concerning the relations of the most important probabilistic complexity classes for read-once branchings programs. We obtain that the analogues of the classes BPP and NP for read-once branching programs are incomparable and that RP is a proper subclass of NP.