On Testing Computability by Small Width OBDDs

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).

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