On the Relation Between BDDs and FDDs

Abstract

Data structures for Boolean functions form an essential component of design automation tools, especially in the area of logic synthesis. The slate of the art data structure is the ordered binary decision diagram (OBDD), which results from general binary decision diagrams (BDDs), also called branching programs, by the application of ordering restrictions. In the context of EXOR-based logic synthesis another type of decision diagram (DD), called (ordered) functional decision diagram ((O)FDD), becomes important. We study the relation between (ordered, free) BDDs and FDDs. Both BDDs and FDDs result from DDs by defining the represented function in different ways. If the underlying DD is complete, the relation between these two types of interpretation can be described by a Boolean transformation τ. This allows us to relate the FDD-size of ƒ and the BDD-size of τ(ƒ) also in the case where the corresponding DDs are free or ordered, but not (necessarily) complete. We use this property to derive several results on the computational power of OFDDs and OBDDs. Symmetric functions are shown to have efficient representations as OBDDs and OFDDs as well. Classes of functions are given that have exponentially more concise OFDDs than OBDDs, and vice versa. Finally, we determine the complexity of some standard operations if OFDDs are used for the representation of Boolean functions.