Boolean functions with long prime implicants

• We introduce a class of Boolean functions defined by a minimum length of its prime implicants.
• We show that given a DNF one can test in polynomial time whether it represents a function from this class.
• We describe a polynomial time algorithm for finding a shortest DNF representation of the functions belonging to this class.
• We also present a generalization of the above class for which we can approximate the Boolean minimization problem within a constant factor.

In this short note we introduce a hierarchy of classes of Boolean functions, where each class is defined by the minimum allowed length of prime implicants of the functions in the class. We show that for a given DNF and a given class in the hierarchy, it is possible to test in polynomial time whether the DNF represents a function from the given class. For the first class in the hierarchy we moreover present a polynomial time algorithm which for a given input DNF outputs a shortest logically equivalent DNF, i.e. a shortest DNF representation of the underlying function. This class is therefore a new member of a relatively small family of classes for which the Boolean minimization problem can be solved in polynomial time. For the second class and higher classes in the hierarchy we show that the Boolean minimization problem can be approximated within a constant factor.