Randomization and Nondeterminsm Are Incomparable for Ordered Read-once Branching Programs

In 3] we exhibited a simple boolean functions fn in n variables such that: 1) fn can be computed by polynomial size randomized ordered read-once branching program with one sided small error; 2) any nondeterministic ordered read-once branching program that computes fn has exponential size. In this paper we present a simple boolean function gn in n variables such that: 1) gn can be computed by polynomial size nondeterministic ordered read-once branching program; 2) any two-sided error randomized ordered read-once branching program that computes fn has exponential size. These mean that BP P and N P are incomparable in the context of ordered read-once branching program. 1 Preliminaries Branching programs is well known model of computation for discrete functions 14]. Many types of restricted branching programs have been investigated as important theoretical model of computations 9]. Ordered read-once branching program or ordered binary decision diagrams (OBDD) 4, 15] also important for practical computer science. They are used in circuits veriications. But many important functions cannot be computed by determinsitc read-once branching programs of polynomial size 4, 13, 8]. In 2] we introduced the model of randomized branching programs and showed that randomized ordered read-once branching programs can be more eeective than determinstic ones. In 3] we deened exclusive boolean function f n in n variables which can be computed by polynomial size randomized ordered read-once branching program, but any nondeterminstic ordered read-once branching program needs exponetial size to compute f n. Martin Sauerhoo 10] considered function from theorem 3 6]. He proved that this function needs (also as in the deterministic case) exponetial size randomized read-once branching programs for ? Work done in part while visiting Steklov Mathematical Institute in Moscow.