Finding Minimal Cost Herbrand Models with Branch-Cut-and-Price

Abstract

Given (1) a set of clauses T in some first-order lan-guage L and (2) a cost function c : BL → R+,mapping each ground atom in the Herbrand baseBL to a non-negative real, then the problem offinding a minimal cost Herbrand model is to eitherfind a Herbrand model I of T which is guaranteedto minimise the sum of the costs of true groundatoms, or establish that there is no Herbrand modelfor T . A branch-cut-and-price integer program-ming (IP) approach to solving this problem is pre-sented. Since the number of ground instantiationsof clauses and the size of the Herbrand base areboth infinite in general, we add the correspondingIP constraints and IP variables ‘on the fly’ via ‘cut-ting’ and ‘pricing’ respectively. In the special caseof a finite Herbrand base we show that adding allIP variables and constraints from the outset can beadvantageous, showing that a challenging Markovlogic network MAP problem can be solved in thisway if encoded appropriatel