First-Order Mixed Integer Linear Programming

TL;DR

This paper introduces first-order programming (FOP), a novel framework that combines first-order logic and mixed integer linear programming to improve reasoning about objects, classes, and uncertainty in decision-making problems.

Contribution

The paper develops a formal foundation for FOP, including a sound and complete inference procedure, enabling more efficient reasoning than traditional first-order logic.

Findings

FOP subsumes both FOL and MILP.

FOP offers exponential savings in representation and proof size.

Inference in FOP is more tractable than in FOL.

Abstract

Mixed integer linear programming (MILP) is a powerful representation often used to formulate decision-making problems under uncertainty. However, it lacks a natural mechanism to reason about objects, classes of objects, and relations. First-order logic (FOL), on the other hand, excels at reasoning about classes of objects, but lacks a rich representation of uncertainty. While representing propositional logic in MILP has been extensively explored, no theory exists yet for fully combining FOL with MILP. We propose a new representation, called first-order programming or FOP, which subsumes both FOL and MILP. We establish formal methods for reasoning about first order programs, including a sound and complete lifted inference procedure for integer first order programs. Since FOP can offer exponential savings in representation and proof size compared to FOL, and since representations and proofs are never significantly longer in FOP than in FOL, we anticipate that inference in FOP will be more tractable than inference in FOL for corresponding problems.