BDDs Naturally Represent Boolean Functions, and ZDDs Naturally Represent Sets of Sets

This paper studies a difference between Binary Decision Diagrams (BDDs) and Zero-suppressed BDDs (ZDDs) from a conceptual point of view. It is commonly understood that a BDD is a representation of a Boolean function, whereas a ZDD is a representation of a set of sets. However, there is a one-to-one correspondence between Boolean functions and sets of sets, and therefore we could also regard a BDD as a representation of a set of sets, and similarly for a ZDD and a Boolean function. The aim of this paper is to give an explanation why the distinction between BDDs and ZDDs mentioned above is made despite the existence of the one-to-one correspondence. To achieve this, we first observe that Boolean functions and sets of sets are equipped with non-isomorphic functor structures, and show that these functor structures are reflected in the definitions of BDDs and ZDDs. This result can be stated formally as naturality of certain maps. To the author's knowledge, this is the first formally stated theorem that justifies the commonly accepted distinction between BDDs and ZDDs. In addition, we show that this result extends to sentential decision diagrams and their zero-suppressed variant.