An implicit formulation for exact BDD minimization

This paper addresses the problem of binary decision diagram (BDD) minimization in the presence of don’t care sets. Specifically, given an incompletely specified function and a fixed ordering of the variables, we propose an exact algorithm for selecting such that is a cover for and the binary decision diagram for is of minimum size. The approach described is the only known exact algorithm for this problem not based on the enumeration of all possible assignments to the points in the don’t care set. We also present a proof that this problem is NP-complete, a result that was also recently obtained in an independent way by other authors. We show that the BDD minimization problem can be formulated as a binate covering problem and solved using implicit enumeration techniques. In particular, we show that the minimum-sized binary decision diagram compatible with the specification can be found by solving a problem that is very similar to the problem of reducing incompletely specified finite state machines. We report experiments of an implicit implementation of our algorithm, by means of which a class of interesting examples was solved exactly. We compare it with existing heuristic algorithms, to measure the quality of the latter.