On generating the irredundant conjunctive and disjunctive normal forms of monotone Boolean functions

Abstract

Let f:{0,1} n →{0,1} be a monotone Boolean function whose value at any point x∈{0,1} n can be determined in time t. Denote by c= ⋀ I∈C ⋁ i∈I x i the irredundant CNF of f, where C is the set of the prime implicates of f. Similarly, let d= ⋁ J∈D ⋀ j∈J x j be the irredundant DNF of the same function, where D is the set of the prime implicants of f. We show that given subsets C′⊆ C and D′⊆ D such that ( C′, D′)≠( C, D), a new term in ( C⧹ C′)∪( D⧹ D′) can be found in time O(n(t+n))+m o( log m) , where m=| C′|+| D′|. In particular, if f( x) can be evaluated for every x∈{0,1} n in polynomial time, then the forms c and d can be jointly generated in incremental quasi-polynomial time. On the other hand, even for the class of ∧,∨-formulae f of depth 2, i.e., for CNFs or DNFs, it is unlikely that uniform sampling from within the set of the prime implicates and implicants of f can be carried out in time bounded by a quasi-polynomial 2 polylog(·) in the input size of f. We also show that for some classes of polynomial-time computable monotone Boolean functions it is NP-hard to test either of the conditions D′= D or C′= C. This provides evidence that for each of these classes neither conjunctive nor disjunctive irredundant normal forms can be generated in total (or incremental) quasi-polynomial time. Such classes of monotone Boolean functions naturally arise in game theory, networks and relay contact circuits, convex programming, and include a subset of ∧,∨-formulae of depth 3.