Abstract
Consider the problem of identifying minT(f) and maxF(f) of a positive (i.e., monotone) Boolean functionf, by using membership queries only, where minT(f) (maxF(f)) denotes the set of minimal true vectors (maximum false vectors) off. Moreover, 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 functionfofnvariables, decide whetherfis 2-monotonic or not, and iffis 2-monotonic, output both minT(f) and maxF(f). For this problem, we propose a simple algorithm, which is based on the concept of maximum latency, and we show that it usesO(n2m) time andO(n2m) queries, wherem=|minT(f)|+|maxF(f)|. This answers affirmatively the conjecture raised in Boroset al.[Lecture Notes in Comput. Sci.557(1991), 104–115], Boroset al.[SIAM J. Comput.26(1997), 93–109], and is an improvement over the two algorithms discussed therein: one usesO(n3m) time andO(n3m) queries, and the other usesO(nm2+n2m) time andO(nm) queries.
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