On testing computability by small width OBDDs

• Authors: Oded Goldreich
• Venue: APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
• Year: 2010

We take another step in the study of the testability of smallwidth 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 awidth-2OBDDhas query complexity Θ(log n). 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 Ω(√n) queries. 2. Testing whether f: GF(3)n → GF(3) is a linear function with 0-1 coefficients requires Ω(√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. 3. There exists a subclass C of the linear functions from GF(2)n to GF(2) such that testing membership in 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).