Exclusive and essential sets of implicates of Boolean functions

In this paper we study relationships between CNF representations of a given Boolean function f and certain sets of implicates of f . We introduce two definitions of sets of implicates which are both based on the properties of resolution. The first type of sets, called exclusive sets of implicates, is shown to have a functional property useful for decompositions. The second type of sets, called essential sets of implicates, is proved to posses an orthogonality property, which implies that every CNF representation and every essential set must intersect. The latter property then leads to an interesting conjecture, which we prove to be true for some special subclasses of Horn Boolean functions. Acknowledgements: The second author gratefully acknowledges the support by the the Czech Science Foundation (grant 201/07/0205). The fourth author gratefully acknowledges the support by the Czech Science Foundation (grant 201/07/P168).