Exclusive and essential sets of implicates of Boolean functions

  • Authors:

  • Endre Boros;Ondřej epek;Alexander Kogan;Petr Kučera

  • Affiliations:

  • RUTCOR, Rutgers University, P.O. Box 5062, New Brunswick, NJ 08903, USA;Department of Theoretical Computer Science, Charles University, Malostranskénám. 25, 118 00 Praha 1, Czech Republic and Institute of Finance and Administration (VŠFS), Estonská ...;Department of Accounting, Business Ethics, and Information Systems, Rutgers Business School, Rutgers University, Newark, NJ 07102, USA and RUTCOR, Rutgers University, P.O. Box 5062, New Brunswick, ...;Department of Theoretical Computer Science, Charles University, Malostranskénám. 25, 118 00 Praha 1, Czech Republic

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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Abstract

In this paper we study relationships between CNF representations of a given Boolean function f and certain sets of implicates of f. We introduce two definitions of sets of implicates which are both based on the properties of resolution. The first type of sets, called exclusive sets of implicates, is shown to have a functional property useful for decompositions. The second type of sets, called essential sets of implicates, is proved to possess an orthogonality property, which implies that every CNF representation and every essential set must intersect. The latter property then leads to an interesting question, to which we give an affirmative answer for some special subclasses of Horn Boolean functions.