A method to join the On-set and Off-set of an incompletely boolean function into a single BDD

Incompletely specified Boolean functions (a.k.a. Boolean Relations) are defined by their On-set, Off-set, and dc-set (don't care set). As the dc-set can be assigned either to the On-set or to the Off-set, a Boolean relation can contain several completely specified functions that satisfy the original Boolean Relation. This article proposes an algorithm for the assignment of don't cares for Boolean relations represented as a couple of Binary Decision Diagrams (BDDs). The algorithm consists of exploring the On-set and Off-set of the input Boolean relation represented as BDDs and joining them into a third BDD, which will have a lower number of nodes. Our algorithm showed an average reduction of 73.88% in the number of nodes compared to the On-sets of each function. In addition, the algorithm proved to be effective in reducing the number of input variables, where it presented an average reduction of 15.67%.