Solving sparse linear constraints

TL;DR: In this article, an efficient decision procedure for sparse linear arithmetic (SLA) constraints is proposed, by combining a solver for difference constraints with a solvers for general linear constraints.

Abstract: Linear arithmetic decision procedures form an important part of theorem provers for program verification. In most verification benchmarks, the linear arithmetic constraints are dominated by simple difference constraints of the form x ≤y + c. Sparse linear arithmetic (SLA) denotes a set of linear arithmetic constraints with a very few non-difference constraints. In this paper, we propose an efficient decision procedure for SLA constraints, by combining a solver for difference constraints with a solver for general linear constraints. For SLA constraints, the space and time complexity of the resulting algorithm is dominated solely by the complexity for solving the difference constraints. The decision procedure generates models for satisfiable formulas. We show how this combination can be extended to generate implied equalities. We instantiate this framework with an equality generating Simplex as the linear arithmetic solver, and present preliminary experimental evaluation of our implementation on a set of linear arithmetic benchmarks.