Lower Bounds for Testing Computability by Small Width OBDDs

We consider the problem of testing whether a function f : {0, 1} → {0, 1} is computable by a read-once, width-2 ordered binary decision diagram (OBDD), also known as a branching program. This problem has two variants: one where the variables must occur in a fixed, known order, and one where the variables are allowed to occur in an arbitrary order. We show that for both variants, any nonadaptive testing algorithm must make Ω(n) queries, and thus any adaptive testing algorithm must make Ω(logn) queries. We also consider the more general problem of testing computability by width-w OBDDs where the variables occur in a fixed order. We show that for any constant w ≥ 4, Ω(n) queries are required, resolving a conjecture of Goldreich [15]. We prove all of our lower bounds using a new technique of Blais, Brody, and Matulef [6], giving simple reductions from known hard problems in communication complexity to the testing problems at hand. Our result for width-2 OBDDs provides the first example of the power of this technique for proving strong nonadaptive bounds.