Fast Cube Tests for LIA Constraint Solving

Abstract

We present two tests that solve linear integer arithmetic constraints. These tests are sound and efficiently find solutions for a large number of problems. While many complete methods search along the problem surface for a solution, these tests use cubes to explore the interior of the problems. The tests are especially efficient for constraints with a large number of integer solutions, e.g., those with infinite lattice width. Inside the SMT-LIB benchmarks, we have found almost one thousand problem instances with infinite lattice width, and we have shown the advantage of our cube tests on these instances by comparing our implementation of the cube test with several state-of-the-art SMT solvers. Our implementation is not only several orders of magnitudes faster, but it also solves all instances, which most SMT solvers do not. Finally, we discovered an additional application for our cube tests: the extraction of equalities implied by a system of linear arithmetic inequalities. This extraction is useful both as a preprocessing step for linear integer constraint solving as well as for the combination of theories by the Nelson-Oppen method.

New techniques for linear arithmetic: cubes and equalities

“…All these solvers employ a branch-and-bound approach with an underlying dual simplex solver [14]. The only exception is mathsat5, which, subsequent to our first publication on the unit cube test [8], now also performs the unit cube test in advance. That is why we also test mathsat5 once with the unit cube test turned on (mathsat5-3.13+uc) and once with the test turned off (mathsat5-3.13).…”

“…We were able to develop several techniques based on this transformation that allow us to investigate linear arithmetic constraints in various ways. Here, we present our previous results [7,8] on the linear cube transformation in more detail as well as some new applications (e.g., quantifier elimination).…”

“…Based on the linear cube transformation, we present two tests tailored for SMT solvers: the largest cube test and the unit cube test [8]. The largest cube test finds a hypercube with maximum edge length contained in the input polyhedron, determines its rational valued center, and rounds it to a potential integer solution.…”

A Reduction from Unbounded Linear Mixed Arithmetic Problems into Bounded Problems

“…Even so, not all unbounded systems are equally difficult. For instance, a system where all directions are unbounded has always a mixed solution: Lemma 5 (Absolutely Unbounded [9]). If all directions are unbounded in a constraint system Ax ≤ b, then the constraint system has an integer solution.…”

Theorem recycling for Theorem Proving

“…Furthermore, in contrast to the previous solver, it now supports an integration with non-linear polynomial arithmetic which is complete for signature disjoint theory combinations. One of the deciding factors in leapfrogging the previous solver in number of benchmarks solved included so far an integration with the method by Bromberger and Weidenbach [3], [4] to detect integer feasible solutions from strengthened inequalities. We observed that the default strengthening proposed by Bromberger and Weidenbach can often be avoided: integer solutions can be guaranteed from weaker systems.…”