Complexity of DNF minimization and isomorphism testing for monotone formulas

We investigate the complexity of finding prime implicants and minimum equivalent DNFs for Boolean formulas, and of testing equivalence and isomorphism of monotone formulas. For DNF related problems, the complexity of the monotone case differs strongly from the arbitrary case. We show that it is DP-complete to check whether a monomial is a prime implicant for an arbitrary formula, but the equivalent problem for monotone formulas is in L. We show PP-completeness of checking if the minimum size of a DNF for a monotone formula is at most k, and for k in unary, we show that the complexity of the problem drops to coNP. In Christopher Umans [Christopher Umans, The minimum equivalent DNF problem and shortest implicants, Journal of Computer and System Sciences 63 (4) (2001) 597-611] a similar problem for arbitrary formulas was shown to be ∑₂^p-complete. We show that calculating the minimum equivalent DNF for a monotone formula is possible in output-polynomial time if and only if P = NP. Finally, we disprove a conjecture from Steffen Reith [Steffen Reith, On the complexity of some equivalence problems for propositional calculi, in: Proceedings of the 28th International Symposium on Mathematical Foundations of Computer Science (MFCS), vol. 2747, Lecture Notes in Computer Science, Springer, 2003, pp. 632-641] by showing that checking whether two formulas are isomorphic has the same complexity for arbitrary formulas as for monotone formulas.