Anti-Links for Boolean Function Manipulation

Manipulation of Boolean functions has important applications in such fields as hardware verification, non-monotonic reasoning, and decision support. Most often, in such applications, the set of prime implicates/implicants of a large formula must be computed. Many algorithms have been proposed to compute the prime implicates of propositionalBoolean formulas, e.g. [4, 5]. Most of them assume that the input is in conjunctive/disjunctive normal form (CNF/DNF). The bottleneck of such algorithms is usually the large number of subsumed (i.e. non-minimal) disjunctive/conjunctive paths through the given formula. Therefore, it is crucial to remove as much redundancy of this kind as possible, before any explicit enumeration of prime implicates/implicants starts. Even if such explicit computation of prime implicates/implicants is not an immediate goal, removing any redundancies due to subsumed disjunctive/conjunctive paths is usually highly desirable. Among the most successful methods used to deal with this task are binary decision diagrams (BDDs) [2, 3]. It is, however, well known that BDD methods do not work well on some classes of formulas, in particular, when they are used to compute prime implicants. Another restriction of BDDs is that they (at least the efficient versions) essentially produce an XOR-AND normal form, but this is not always what is needed. Here, we present a technique based on anti-links for identifying and removing subsumed paths, that operates on formulas in negation normal form (NNF).1 It is not an alternative to BDDs and other methods for computing prime implicants, but is intended to be used as a supplementary pre-processing step that enhances the overall performance of the system. An anti-link fAX ; AY g consists of occurrences AX andAY of a literalA in an NNF formulaF . If the (largest) subformulasX andY ofF that containAX andAY , respectively, are disjunctivelyconnected, we call fAX ; AY g a disjunctive anti-link. If X and Y are conjunctively connected, then we call fAX ; AY g a conjunctive anti-link. If the formula F contains a disjunctive path p that is subsumed by a distinct path p0, then F contains a disjunctive or a conjunctive anti-link. Unfortunately, the converse is not true: the presence of anti-links does not imply the presence of subsumed paths. There is, however, a large class of anti-links, that we call redundant, that is always an indicator for subsumed paths: A disjunctive anti-link fAX ; AY g in F is redundant if either AX or AY is the argument of a conjunction. In that case certain disjunctive paths through X and Y that contain AX or AY are subsumed by a disjunctive path in F that contains the anti-link.