On Testing Computability by Small Width OBDDs

Abstract

We take another step in the study of the testability of small-width OBDDs, initiated by Ron and Tsur (Random’09). That is, we consider algorithms that, given oracle access to a function f:{0,1}ⁿ →{0,1}, need to determine whether f can be implemented by some restricted class of OBDDs or is far from any such function.

Ron and Tsur showed that testing whether a function f:{0,1}ⁿ →{0,1} is implementable by a width-2 OBDD has query complexity Θ(logn). Thus, testing width-2 OBDD functions is significantly easier than learning such functions (which requires Ω(n) queries). We show that such exponential gaps do not hold for several related classes. Specifically:

• 1 Testing whether f:{0,1}ⁿ →{0,1} is implementable by a width-4 OBDD requires $\Omega(\sqrt{n})$ queries.

• 1 Testing whether f:GF(3)ⁿ→GF(3) is a linear function with 0-1 coefficients requires $\Omega(\sqrt{n})$ queries. Note that this class of functions is a subset of the class of all linear functions over GF(3), and that each such linear function can be implemented by a width-3 OBDD.

• 1 There exists a subclass ${\mathcal C}$ of the linear functions from GF(2)ⁿ to GF(2) such that testing membership in ${\mathcal C}$ has query complexity Θ(n). Note that each linear function over GF(2) can be implemented by a width-2 OBDD.

Recall that each of these classes has a proper learning algorithm of query complexity O(n).