On the size of binary decision diagrams representing Boolean functions

Abstract

We consider the size of the representation of Boolean functions by several classes of binary decision diagrams (BDDs) (also called branching programs), namely the classes of arbitrary BDDs of real time BDD (RBDD) (i.e. BDDs where each computation path is limited to the number of variables), of free BDDs (FBDDs) (also called read-once-only branching programs), of ordered BDDs (OBDDS) i.e. FBDDs where variables are tested in the same order along all paths), and binary decision trees (BDTs). Using well-known techniques, we first establish asymptotically sharp bounds as a function of n on the minimum size of arbitrary BDDs representing almost all Boolean functions of n variables and provide asymptotic lower and upper bounds, differing only by a factor of two, on the minimum size OBDDs representing almost all Boolean functions of n variables. We then, using a method to obtain exponential lower bounds on complexity of computation of Boolean functions by RBDD, FBDD and OBDD that originated in (Breitbart, 1968), present the highest such bounds to date and also present improved bounds on the relative economy of description of particular Boolean functions by the above classes of BDDs. For each nontrivial pair of BDD classes considered, we exhibit infinite families of Boolean functions representable much more concisely by BDDs in one class than by BDDs in the other.