For the problem of finding all the maximal elements of $P$ over the lattice of subsets $2^{[n]}$, this amounts to exact inference of a positive boolean function of $n$ boolean variables. If you only care about the number of evaluations of $P$ (not the computational complexity), you can find a survey in Data Mining and Knowledge Discovery via Logic-Based Methods, Chapter 10, section 10.2.4, or in the last paragraph of section 6.1 of this article, which I was pointed to by this answer (beware, the rest of this article deals with computational complexity, not just the complexity of evaluating $P$).

For the problem of finding all the maximal elements of $P$ over the lattice of subsets $2^{[n]}$, this amounts to exact inference of a positive boolean function of $n$ boolean variables. If you only care about the number of evaluations of $P$ (not the computational complexity), you can find a survey in Data Mining and Knowledge Discovery via Logic-Based Methods, Chapter 10, section 10.2.4, or in the last paragraph of section 6.1 of this article, which I was pointed to by this answer (beware, the rest of this article deals with computational complexity, not just the complexity of evaluating $P$).