Dual Subimplicants of Positive Boolean Functions∗

Given a positive Boolean function f and a subset ∆ of its variables, we give a combinatorial condition characterizing the existence of a prime implicant D̂ of the Boolean dual fd of f , having the property that every variable in ∆ appears in D̂. We show that the recognition of this property is an NP-complete problem, suggesting an inherent computational difficulty of Boolean dualization, independently of the size of the dual function. Finally it is shown that if the cardinality of ∆ is bounded by a constant, then the above recognition problem is polynomial. In particular, it follows that the co-occurance graph of the dual of a positive Boolean function can be always generated in time polynomial in the size of the function.