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This paper studies a difference between Binary Decision Diagrams (BDDs) andZero-suppressed BDDs (ZDDs) from a conceptual point of view. It is commonlyunderstood that a BDD is a representation of a Boolean function, whereas a ZDDis a representation of a set of sets. However, there is a one-to-onecorrespondence between Boolean functions and sets of sets, and therefore wecould also regard a BDD as a representation of a set of sets, and similarly fora ZDD and a Boolean function. The aim of this paper is to give an explanationwhy the distinction between BDDs and ZDDs mentioned above is made despite theexistence of the one-to-one correspondence. To achieve this, we first observethat Boolean functions and sets of sets are equipped with non-isomorphicfunctor structures, and show that these functor structures are reflected in thedefinitions of BDDs and ZDDs. This result can be stated formally as naturalityof certain maps. To the author's knowledge, this is the first formally statedtheorem that justifies the commonly accepted distinction between BDDs and ZDDs.In addition, we show that this result extends to sentential decision diagramsand their zero-suppressed variant.

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