Generalized Graph Colorability and Compressibility of Boolean Formulae

In this paper, we study the possibility of Occam’s razors for a widely studied class of Boolean Formulae : Disjunctive Normal Forms (DNF). An Occam’s razor is an algorithm which compresses the knowledge of observations (examples) in small formulae. We prove that approximating the minimally consistent DNF formula, and a generalization of graph colorability, is very hard. Our proof technique is such that the stronger the complexity hypothesis used, the larger the inapproximability ratio obtained. Our ratio is among the first to integrate the three parameters of Occam’s razors : the number of examples, the number of description attributes and the size of the target formula labelling the examples. Theoretically speaking, our result rules out the existence of efficient deterministic Occam’s razor algorithms for DNF. Practically speaking, it puts a large worst-case lower bound on the formulae’s sizes found by learning systems proceeding by rule searching.