Boolean functions with a simple certificate for CNF complexity

  • Authors:

  • Ondřej epek;Petr Kučera;Petr Savický

  • Affiliations:

  • Department of Theoretical Computer Science and Mathematical Logic, Faculty of Mathematics and Physics, Charles University, Malostranské námstí25, 118 00 Praha 1, Czech Republic and ...;Department of Theoretical Computer Science and Mathematical Logic, Faculty of Mathematics and Physics, Charles University, Malostranské námstí25, 118 00 Praha 1, Czech Republic;Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodarenskou vezi, 271/2 182 07 Prague 8, Czech Republic

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2012

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Abstract

In this paper we study relationships between CNF representations of a given Boolean function f and essential sets of implicates of f. It is known that every CNF representation and every essential set must intersect. Therefore the maximum number of pairwise disjoint essential sets of f provides a lower bound on the size of any CNF representation of f. 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 NP-hard under various restrictions on the function and on its input representation.