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

Given (1) a set of clauses $T$ in some first-order language $\cal L$ and (2) a cost function $c : B_{{\cal L}} \rightarrow \mathbb{R}_{+}$, mapping each ground atom in the Herbrand base $B_{{\cal L}}$ to a non-negative real, then the problem of finding a minimal cost Herbrand model is to either find a Herbrand model $\cal I$ of $T$ which is guaranteed to minimise the sum of the costs of true ground atoms, or establish that there is no Herbrand model for $T$. A branch-cut-and-price integer programming (IP) approach to solving this problem is presented. Since the number of ground instantiations of clauses and the size of the Herbrand base are both infinite in general, we add the corresponding IP constraints and IP variables `on the fly' via `cutting' and `pricing' respectively. In the special case of a finite Herbrand base we show that adding all IP variables and constraints from the outset can be advantageous, showing that a challenging Markov logic network MAP problem can be solved in this way if encoded appropriately

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

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