The Maximum Latency and Identification of Positive Boolean Functions

• Authors: Kazuhisa Makino;Toshihide Ibaraki
• Venue: SIAM Journal on Computing
• Year: 1997

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 $\min T(f)\,(\max F(f))$ 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 $u$ for given sets $MT \subseteq \min T(f)$ and $MF \subseteq \max F(f)$; that is, $u \in \{0,1\}^n \setminus (\{v | v \geq w {\rm for some } w \in MT \} \cup \{v | v \leq w {\rm for some } w \in MF \})$. This paper introduces a measure for the difficulty to find 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, $\Delta$-partial positive threshold functions, and matroid functions, while the class of general positive functions has $\lfloor n/4 \rfloor +1$ maximum latency and the class of positive $k$-DNF functions has $\Omega (\sqrt{n})$ maximum latency.