On the nonapproximability of boolean functions by OBDDs and read-k-times branching programs

Abstract

Branching programs are considered as a nonuniform model of computation in complexity theory as well as a data structure for boolean functions in several applications. In many applications (e.g., verification), exact representations are required. For learning boolean functions f on the basis of classified examples, it is sufficient to produce the representation of a function g approximating f. This motivates the investigation of the size of the smallest branching program approximating f. Although several nonapproximability results are contained in the papers on randomized branching programs, these results often do not work for the uniform distribution (which is the most important one in applications). Here, the following nonapproximability results are presented. (1) It is proven that two simple and well-known functions from the branching program literature require exponential size to be approximated with respect to the uniform distribution by OBDDs, which are the most important type of branching programs in applications. (2) The first truly exponential lower bound on the size of approximating syntactic read-k-times branching programs with respect to the uniform distribution and error probability 1/2-2-Ω(n),, n the input size, is shown. In order to improve upon the best previous results for error probabilities smaller than 1/3, a strong combinatorial lemma from a paper of Ajtai on linear-length branching programs is exploited.