On the gap between ess(f) and cnf_size(f)

Lisa Hellerstein, Devorah Kletenik
2013 Discrete Applied Mathematics  
Horn functions Formula size a b s t r a c t Given a Boolean function f , the quantity ess(f ) denotes the largest set of assignments that falsify f , no two of which falsify a common implicate of f . Although ess(f ) is clearly a lower bound on cnf _size(f ) (the minimum number of clauses in a CNF formula for f ),Cepek et al. showed it is not, in general, a tight lower bound [6]. They gave examples of functions f for which there is a small gap between ess(f ) and cnf _size(f ). We demonstrate
more » ... gnificantly larger gaps. We show that the gap can be exponential in n for arbitrary Boolean functions, and Θ( √ n) for Horn functions, where n is the number of variables of f . We also introduce a natural extension of the quantity ess(f ), which we call ess k (f ), which is the largest set of assignments, no k of which falsify a common implicate of f .
doi:10.1016/j.dam.2012.07.004 fatcat:sje3ajdbn5eovcmiumlxlr2wgi