On Constructing Minimal Formulae

Given a Boolean propositional formula, φ(Xn) over the basis Ω = {∧,∨,¬} we consider the following decision problem: is there a subset of literals, S, for which φ(Xn) ≡ ∧ y∈S y or φ(Xn) ≡ ∨ y∈S y? We prove that the “obvious” Σ p 2 upper bound is sub-optimal and that the problem is decidable in PNP || the class of languages decidable by polynomial time methods allowed to make non-adaptive queries to an NP oracle. We further show that the associated function problem of computing a witnessing such subset when one exists can be solved in FPNP || .