Minimal elements of a monotonic predicate over the powerset $2^{|n|}$ Consider a monotonic predicate $P$ over the powerset $2^{|n|}$ (ordered by inclusion). By "monotonic" I mean: $\forall x, y \in 2^{|n|}$ such that $x \subset y$, if $P(x)$ then $P(y)$. I am looking for an algorithm to find all the minimal elements of $P$, i.e., the $x \in 2^{|n|}$ such that $P(x)$ but $\forall y \subset x$, $\neg P(y)$. Since the width of $2^{|n|}$ is $n \choose n/2$, there could be exponentially many minimal elements, and therefore the running time of such an algorithm could be exponential in general. However, could there exist an algorithm for this task which is polynomial in the size of the output?

[Context: A more general question was asked but there was no attempt in the answers to evaluate the complexity of the algorithm in the size of the output. If I assume that there is only one minimal element, for instance, then I can perform a binary search following this answer and find it. However, if I want to continue finding more minimal elements, I need to maintain the current information I have about $P$ in a way which would make it tractable to continue the search without wasting time on what is already known. Is it possible to do this and find all the minimal elements in polynomial time in the size of the output?]

Ideally, I would like to understand if this can be done with general DAGs, but I already don't know how to answer the question for $2^{|n|}$.
Your problem is known in the learning literature as "learning monotone functions using membership queries". A class of monotone functions for which one can identify all minterms is known as "polynomially learnable using membership queries".

It seems that the existence of a polynomial time algorithm is still open. Schmulevich et al. prove that "Almost all monotone Boolean functions are polynomially learnable using membership queries". If we also require that the $t$th minterm be generated in time polynomial in $n$ and $t$, then the problem is equivalent to monotone dualization, as shown by Bioch and Ibaraki. Here is a survey addressing monotone dualization.