On the Relation Between BDDs and FDDs

Data structures for Boolean functions build an essential component of design automation tools, especially in the area of logic synthesis. The state 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 ordering restrictions. In the context of EXOR-based logic synthesis another type of decision diagram (DD), called (ordered) functional decision diagram ((O)FDD) becomes increasingly important. We study the relation between (ordered, free) BDDs and FDDs. Both, BDDs and FDDs, result from DDs by deening the represented function in diierent ways. If the underlying DD is complete, the relation between both types of interpretation can be described by a Boolean transformation. This allows us to relate the FDD-size of f and the BDD-size of (f) also in the case that 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 eecient representations as OBDDs and OFDDs as well. Classes of functions are given that have exponentially more concise OFDDs than OBDDs, and vice versa. We demonstrate how the lower bound techniques for OBDDs can be adapted to OFDDs: We prove that the hidden weighted bit function and multiplication as well require OFDDs of exponential size independent of the ordering of the variables. Finally, we determine the complexity of some standard operations if OFDDs are used for the representation of Boolean functions.