An Improved method to join BDDs for incompletely specified Boolean functions

An incompletely specified Boolean function, also known as a Boolean relation, can be described by any combination of its on-set, off-set, and don't care (DC) set pairs. To transform a Boolean relation into a Boolean function (fully specified), we have to assign the DCs from the original specification. Depending on the DCs assignment, a better function minimization may result from the original specification. This article presents a method for joining two BDDs by describing the on-set and off-set of an incompletely specified $f$ function into a single BDD for a fully specified coverage of the function. The proposed method is compared with the Restrict, Constrain and LICompaction methods implemented by the CUDD package, which are references for this problem using BDDs. In addition to these methods, we also compare our results with the Join method, which is a more recent proposal for the problem of assigning and minimizing incompletely specified Boolean functions. The LI-Compaction method presented the largest number of BDD nodes among the analyzed methods. Compared to the LICompaction method, the Restrict method produced 2.12% fewer nodes, the Constrain and Join methods produced 1.95% fewer nodes, while our method produced 24.80% fewer nodes.