# Identifying 2-monotonic positive boolean functions in polynomial time

## Abstract

We consider to identify an unknown Boolean function *f* by asking an oracle the functional values *f*(a) for a selected set of test vectors *a* ε {0,1}^{n}. If *f* is known to be a positive function of *n* variables, the algorithm by Gainanov can achieve the goal by issuing *O(mn)* queries, where *m* = ¦min*T(f)*¦ + ¦max*F(f)*¦ and min*T(f)* (resp. max*F(f)*) denotes the set of minimal true vectors (resp. maximal false vectors) of *f*. However, it is not known whether this whole task including the generation of test vectors can be carried out in polynomial time in *n* and *m* or not. To partially answer this question, we propose here two algorithms that, given an unknown positive function *f* of *n* variables, decide whether *f* is 2-monotonic or not, and if *f* is 2-monotonic, output 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.

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## References

- [1]D. Angluin, Queries and concept learning,
*Machine Learning*, 2 (1988), 319–342.Google Scholar - [2]P. Bertolazzi and A. Sassano, An
*O(mn)*time algorithm for regular set-covering problems,*Theoretical Computer Science*, 54 (1987), 237–247.Google Scholar - [3]Y. Crama, Dualization of regular Boolean functions,
*Discrete Applied Mathematics*, 16 (1987), 79–85.Google Scholar - [4]Y. Crama, P. L. Hammer and T. Ibaraki, Cause-effect relationships and partially defined boolean functions,
*Annals of Operations Research*, 16 (1988), 299–326.Google Scholar - [5]D. N. Gainanov, On one criterion of the optimality of an algorithm for evaluating monotonic Boolean functions,
*U.S.S.R. Computational Mathematics and Mathematical Physics*, 24 (1984), 176–181.Google Scholar - [6]A. V. Genkin and P. N. Dubner, Aggregation algorithm for finding the informative features,
*Automation and Remote Control*, (1988), 81–86.Google Scholar - [7]J. Hansel, On the number of monotonic Boolean functions of
*n*variables,*Cybernetics Collection*, 5 (1968), 53–58.Google Scholar - [8]S. Muroga,
*Threshold Logic and Its Applications*, John Wiley and Sons, 1971.Google Scholar - [9]U. N. Peled and B. Simeone, Polynomial-time algorithms for regular set-covering and threshold synthesis,
*Discrete Applied Mathematics*, 12 (1985), 57–69.Google Scholar - [10]U. N. Peled and B. Simeone, An
*O(nm)*-time algorithm for computing the dual of a regular Boolean function, Technical Report, University Illinois at Chicago (1990).Google Scholar - [11]N. A. Sokolov, On the optimal evaluation of monotonic Boolean functions,
*U.S.S.R. Computational Mathematics and Mathematical Physics*, 22 (1979), 207–220.Google Scholar