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

TL;DR

This paper investigates the relationship between the essential set size and the minimal CNF size of Boolean functions, revealing exponential gaps and introducing a generalized measure.

Contribution

It demonstrates that the gap between ess(f) and cnf_size(f) can be exponentially large, and introduces ess_k(f) as a natural extension of ess(f).

Findings

The gap can be exponential in n for arbitrary functions.

For Horn functions, the gap is Theta(sqrt{n}).

Introduces the concept of ess_k(f) as a generalized measure.

Abstract

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 that it is not, in general, a tight lower bound. They gave examples of functions f for which there is a small gap between ess(f) and cnf_size(f). We demonstrate significantly larger gaps. We show that the gap can be exponential in n for arbitrary Boolean functions, and Theta(sqrt{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.