On the Complexity of the Multiplication Method for Monotone CNF/DNF Dualization [chapter]

Khaled M. Elbassioni
2006 Lecture Notes in Computer Science  
Given the irredundant CNF representation φ of a monotone Boolean function f : {0, 1} n → {0, 1}, the dualization problem calls for finding the corresponding unique irredundant DNF representation ψ of f . The (generalized) multiplication method works by repeatedly dividing the clauses of φ into (not necessarily disjoint) groups, multiplying-out the clauses in each group, and then reducing the result by applying the absorption law. We present the first non-trivial upper-bounds on the complexity
more » ... this multiplication method. Precisely, we show that if the grouping of the clauses is done in an output-independent way, then multiplication can be performed in sub-exponential time (n|ψ|) O( √ |φ|) |φ| O(log n) . On the other hand, multiplication can be carried-out in quasi-polynomial time poly(n, |ψ|) · |φ| o(log |ψ|) , provided that the grouping is done depending on the intermediate outputs produced during the multiplication process.
doi:10.1007/11841036_32 fatcat:n52gy6g6rvhd5iq2psugmejdja