Polynomial-Time Recognition of 2-Monotonic Positive Boolean Functions Given by an Oracle

We consider the problem of identifying an unknown Boolean function f by asking an oracle the functional values f(a) for a selected set of test vectors a ∈ {0, 1}n. Furthermore, we assume that f is a positive (or monotone) function of n variables. It is not known yet whether the whole task of generating test vectors and checking if the identification is completed can be carried out in polynomial time in n and m or not, where m = |minT (f)| + |maxF (f)| and minT (f) (respectively, maxF (f)) denotes the set of minimal true (respectively, maximal false) vectors of f . To partially answer this question, we propose here two polynomial time algorithms that, given an unknown positive function f of n variables, decide whether f is 2-monotonic or not, and if f is 2-monotonic, output both sets minT (f) and maxF (f). The first algorithm uses O(nm2 + n2m) time and O(nm) queries, while the second one uses O(n3m) time and O(n3m) queries.