On the Structure of Counterexamples to Symmetric Orderings for BDD's

Abstract

Binary Decision Diagrams (BDDs) are used to represent boolean functions in a variety of applications. The size of a reduced ordered BDD depends on the ordering of variables. Several researchers have suggested grouping symmetric variables as a promising heuristic for finding good orderings. In this paper we study the conjecture which states that symmetric variables gather in at least one of the optimum variable orders. First, we prove some useful properties of partially symmetric functions. Next, we develop a faster procedure for finding counterexamples to this conjecture that exploits the partitioning of boolean functions into nn-equivalence classes. Third, we study the structure of counterexamples and devise a new and simple method to generate new counterexamples from given counterexamples. Finally, we present different kinds of counterexamples, which show that boolean functions are very diverse with respect to where symmetric orders can fall in the range from optimal orders to worst-case orders.