Practical Tools for Reasoning About Linear Constraints

Abstract

We address the problem of building intelligent systems to reason about linear arithmetic constraints. We develop, along the lines of Logic Programming, a unifying framework based on the concept of Parametric Queries and a quasi-dual generalization of the classical Linear Programming optimization problem. Variable (quantifier) elimination is the key underlying operation which provides an oracle to answer all queries and plays a role similar to Resolution in Logic Programming. We discuss three methods for variable elimination, compare their feasibility, and establish their applicability. We then address practical issues of solvability and canonical representation, as well as dynamical updates and feedback. In particular, we show how the quasi-dual formulation can be used to achieve the discriminating characteristics of the classical Fourier algorithm regarding solvability, detection of implicit equalities and, in case of unsolvability, the detection of minimal unsolvable subsets. We illustrate the relevance of our approach with examples from the domain of spatial reasoning and demonstrate its viability with empirical results from two practical applications: computation of canonical forms and convex hull construction.