On the representational power of bit-level and word-level decision diagrams

Abstract| Several types of Decision Diagrams (DDs) have have been proposed in the area of Computer Aided Design (CAD), among them being bit-level DDs like OBDDs, OFDDs and OKFDDs. While the aforementioned types of DDs are suitable for representing Boolean functions at the bit-level and have proved useful for a lot of applications in CAD, recently DDs to represent integer-valued functions, like MTBDDs (=ADDs), EVBDDs, FEVBDDs, (*)BMDs, HDDs (=KBMDs), and K*BMDs, attract more and more interest, e.g., using *BMDs it was for the rst time possible to verify multipliers of bit length up to n = 256.| Several types of Decision Diagrams (DDs) have have been proposed in the area of Computer Aided Design (CAD), among them being bit-level DDs like OBDDs, OFDDs and OKFDDs. While the aforementioned types of DDs are suitable for representing Boolean functions at the bit-level and have proved useful for a lot of applications in CAD, recently DDs to represent integer-valued functions, like MTBDDs (=ADDs), EVBDDs, FEVBDDs, (*)BMDs, HDDs (=KBMDs), and K*BMDs, attract more and more interest, e.g., using *BMDs it was for the rst time possible to verify multipliers of bit length up to n = 256. In this paper we clarify the representational power of these DD classes. Several (inclusion) relations and (exponential) gaps between speci c classes di ering in the availability of additive and/or multiplicative edge weights and in the choice of decomposition types are shown. It turns out for example, that K(*)BMDs, a generalization of OKFDDs to the word-level, also \include" OBDDs, MTBDDs and (*)BMDs. On the other hand, it is demonstrated that a restriction of the K(*)BMD concept to subclasses, such as OBDDs, MTBDDs, (*)BMDs as well, results in families of functions which lose their e cient representation.