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

• Authors: Endre Boros;Peter L. Hammer;Toshihide Ibaraki;Kazuhiko Kawakami
• Venue: SIAM Journal on Computing
• Year: 1997

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 $a \in \{0,1\}^{n}$. Furthermore, we assume that $f$ is a positive (or monotone) function of $n$ variables. It is not yet known whether or not the whole task of generating test vectors and checking if the identification is completed can be carried out in polynomial time in $n$ and $m$, where $m=|\min T(f)| + |\max F(f)|$ and $\min T(f)$ (respectively, $\max F(f))$ 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 or not $f$ is 2-monotonic and, if $f$ is 2-monotonic, output both 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 $O(n^{3}m)$ time and $O(n^{3}m)$ queries.