Finding Minimal Cost Herbrand Models with Branch-Cut-and-Price
This work addresses a computational bottleneck in logic and AI for researchers and practitioners dealing with optimization in first-order logic, though it is incremental as it builds on existing integer programming techniques.
The paper tackles the problem of finding minimal cost Herbrand models by developing a branch-cut-and-price integer programming approach that handles infinite constraints and variables on the fly, and demonstrates its effectiveness by solving a challenging Markov logic network MAP problem in the finite case.
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.