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

TL;DR

This paper introduces a branch-cut-and-price integer programming method for finding minimal cost Herbrand models in first-order logic, efficiently handling infinite bases by dynamic constraint and variable addition.

Contribution

It presents a novel IP approach that dynamically manages infinite Herbrand bases, enabling efficient minimal cost model computation and solving complex Markov logic network MAP problems.

Findings

Effective IP-based method for infinite Herbrand bases.

Demonstrated solution for Markov logic network MAP problem.

Dynamic constraint and variable addition improves scalability.

Abstract

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.