New techniques for linear arithmetic: cubes and equalities

Abstract

We present several new techniques for linear arithmetic constraint solving. They are all based on the linear cube transformation, a method presented here, which allows us to efficiently determine whether a system of linear arithmetic constraints contains a hypercube of a given edge length. Our first findings based on this transformation are two sound tests that find integer solutions for linear arithmetic constraints. 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. Experimental results confirm that our tests are superior on these instances compared to several state-ofthe-art SMT solvers. We also discovered that the linear cube transformation can be used to investigate the equalities implied by a system of linear arithmetic constraints. For this purpose, we developed a method that computes a basis for all implied equalities, i.e., a finite representation of all equalities implied by the linear arithmetic constraints. The equality basis has several applications. For instance, it allows us to verify whether a system of linear arithmetic constraints implies a given equality. This is valuable in the context of NelsonOppen style combinations of theories.

Spass-Satt

“…The next extension we discuss is the unit cube test (turn off with −C 0). It determines in polynomial time whether a polyhedron, i.e., the geometric representation of a system of inequalities, contains a hypercube parallel to the coordinate axes with edge length one [10,11]. The existence of such a hypercube guarantees a mixed/integer solution for the system of inequalities.…”

“…The unit cube test is only a sufficient and not a necessary test for the existence of a solution. There is at least one class of inequality systems, viz., absolutely unbounded inequality systems [10,11], where the unit cube test is also a necessary test and which are much harder for many complete decision procedures.…”

The Challenge of Unifying Semantic and Syntactic Inference Restrictions

“…The current state-of-the-art procedures don't use a CDCL-style approach but apply a relaxation of LIA to LRA (linear rational arithmetic) and use LRA solutions for a branch-and-bound approach. This general idea is complemented via simplifications and fast, sufficient tests for the existence of a solution [9,18,12,10].…”

A Reduction from Unbounded Linear Mixed Arithmetic Problems into Bounded Problems

“…We also recommend the equality basis method because it is incrementally efficient [10]. This incremental efficiency directly translates to determining bounded inequalities.…”