Identifying 2-monotonic positive boolean functions in polynomial time

Abstract

We consider to identify an unknown Boolean function f by asking an oracle the functional values f(a) for a selected set of test vectors a ε {0,1}ⁿ. If f is known to be a positive function of n variables, the algorithm by Gainanov can achieve the goal by issuing O(mn) queries, where m = ¦minT(f)¦ + ¦maxF(f)¦ and minT(f) (resp. maxF(f)) denotes the set of minimal true vectors (resp. maximal false vectors) of f. However, it is not known whether this whole task including the generation of test vectors can be carried out in polynomial time in n and m or not. To partially answer this question, we propose here two 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 sets min T(f) and max F(f). The first algorithm uses O(nm ²+n ² m) time and O(nm) queries while the second one uses O(n³m) time and O(n ³ m) queries.