# A fast and simple algorithm for identifying 2-monotonic 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)* (max *F(f)*) denotes the set of minimal true vectors (maximal false vectors) of *f*. As the existence of a polynomial total time algorithm (i.e., polynomial time in the length of input and output) for this problem is still open, we consider here a restricted problem: 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 min *T(f*) and max. *F(f)*. For this problem, we propose a simple algorithm, which is based on the concept of maximum latency, and show that it uses *O(n*^{2}m) time and *O(n*^{2}m) queries, where *m*=¦ min *T(f*)¦+¦ max *F(f)*¦. This answers affirmatively the conjecture raised in [3, 4], and is an improvement over the two algorithms discussed therein: one uses *O(n*^{3}m) time and *O(n*^{3}m) queries, and the other uses *O(nm*^{2}+n^{2}m) time and *O*(nm) queries.

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