Complexity of constructing minimum depth decision trees
I think I can see a fairly easy reduction from 3DM. Let $B=\{0^J\}$, i.e., it is a singleton set with the only zero element. The points of $A$ correspond to the points of the 3DM that are to be matched. If a triple is matchable, then there is a coordinate where these 3 points are 1, while all other points are 0. The equivalence is straightforward.

I think an interesting question left open is if A is given (as part of the input), and our goal is to separate it from $\{0,1\}^J\setminus A$.