The maximum latency and identification of positive Boolean functions

Abstract

Consider the problem of identifying min T(f) and max F(f) of a positive (i.e., monotone) Boolean function f, by using membership queries only, where min T(f) (maxF(f)) denotes the set of minimal true vectors (maximal false vectors) of f. It is known that an incrementally polynomial algorithm exists if and only if there is a polynomial time algorithm to check the existence of an unknown vector for given sets MT \(\subseteq \) min T(f) and MF \(\subseteq \) max F(f). Unfortunately, however, the complexity of this problem is still unknown. To answer this question partially, we introduce in this paper a measure for the difficulty of finding an unknown vector, which is called the maximum latency. If the maximum latency is constant, then an unknown vector can be found in polynomial time and there is an incrementally polynomial algorithm for identification. Several subclasses of positive functions are shown to have constant maximum latency, e.g., 2-monotonic positive functions, Δ-partial positive threshold functions and matroid functions, while the class of general positive functions has maximum latency not smaller than [n/4]+1 and the class of positive k-DNF functions has Ω(√n) maximum latency.