Boolean functions with a simple certificate for CNF complexity |
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Authors: | Ondřej Čepek Petr Kučera Petr Savický |
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Institution: | 1. Department of Theoretical Computer Science and Mathematical Logic, Faculty of Mathematics and Physics, Charles University, Malostranské náměstí 25, 118 00 Praha 1, Czech Republic;2. Institute of Finance and Administration, Estonska, 500 101 00 Praha 10, Czech Republic;3. Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodarenskou vezi, 271/2 182 07 Prague 8, Czech Republic |
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Abstract: | In this paper we study relationships between CNF representations of a given Boolean function and essential sets of implicates of . It is known that every CNF representation and every essential set must intersect. Therefore the maximum number of pairwise disjoint essential sets of provides a lower bound on the size of any CNF representation of . We are interested in functions, for which this lower bound is tight, and call such functions coverable. We prove that for every coverable function there exists a polynomially verifiable certificate (witness) for its minimum CNF size. On the other hand, we show that not all functions are coverable, and construct examples of non-coverable functions. Moreover, we prove that computing the lower bound, i.e. the maximum number of pairwise disjoint essential sets, is -hard under various restrictions on the function and on its input representation. |
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