Speeding up Polyhedral Analysis by Identifying Common Constraints

AbstractSets of linear inequalities are an expressive reasoning tool for approximating the reachable states of a program. However, the most precise way to join two states is to calculate the convex hull of the two polyhedra that are represented by the inequality sets, an operation that is exponential in the dimension of the polyhedra. We investigate how similarities in the two input polyhedra can be exploited to improve the performance of this costly operation. In particular, we discuss how common equalities and certain inequalities can be omitted from the calculation without affecting the result. We expose a maximum of common equalities and inequalities by converting the polyhedra into a normal form and give experimental evidence of the merit of our method

New Techniques for Linear Arithmetic: Cubes and Equalities

International audienceWe 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-of-the-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 Nelson-Oppen style combinations of theories

Decision procedures for linear arithmetic

In this thesis, we present new decision procedures for linear arithmetic in the context of SMT solvers and theorem provers: 1) CutSat++, a calculus for linear integer arithmetic that combines techniques from SAT solving and quantifier elimination in order to be sound, terminating, and complete. 2) The largest cube test and the unit cube test, two sound (although incomplete) tests that find integer and mixed solutions in polynomial time. The tests are especially efficient on absolutely unbounded constraint systems, which are difficult to handle for many other decision procedures. 3) Techniques for the investigation of equalities implied by a constraint system. Moreover, we present several applications for these techniques. 4) The Double-Bounded reduction and the Mixed-Echelon-Hermite transformation, two transformations that reduce any constraint system in polynomial time to an equisatisfiable constraint system that is bounded. The transformations are beneficial because they turn branch-and-bound into a complete and efficient decision procedure for unbounded constraint systems. We have implemented the above decision procedures (except for Cut- Sat++) as part of our linear arithmetic theory solver SPASS-IQ and as part of our CDCL(LA) solver SPASS-SATT. We also present various benchmark evaluations that confirm the practical efficiency of our new decision procedures.In dieser Arbeit präsentieren wir neue Entscheidungsprozeduren für lineare Arithmetik im Kontext von SMT-Solvern und Theorembeweisern: 1) CutSat++, ein korrekter und vollständiger Kalkül für ganzzahlige lineare Arithmetik, der Techniken zur Entscheidung von Aussagenlogik mit Techniken aus der Quantorenelimination vereint. 2) Der Größte-Würfeltest und der Einheitswürfeltest, zwei korrekte (wenn auch unvollständige) Tests, die in polynomieller Zeit (gemischt-)ganzzahlige Lösungen finden. Die Tests sind besonders effizient auf vollständig unbegrenzten Systemen, welche für viele andere Entscheidungsprozeduren schwer sind. 3) Techniken zur Ermittlung von Gleichungen, die von einem linearen Ungleichungssystem impliziert werden. Des Weiteren präsentieren wir mehrere Anwendungsmöglichkeiten für diese Techniken. 4) Die Beidseitig-Begrenzte-Reduktion und die Gemischte-Echelon-Hermitesche- Transformation, die ein Ungleichungssystem in polynomieller Zeit auf ein erfüllbarkeitsäquivalentes System reduzieren, das begrenzt ist. Vereint verwandeln die Transformationen Branch-and-Bound in eine vollständige und effiziente Entscheidungsprozedur für unbeschränkte Ungleichungssysteme. Wir haben diese Techniken (ausgenommen CutSat++) in SPASS-IQ (unserem theory solver für lineare Arithmetik) und in SPASS-SATT (unserem CDCL(LA) solver) implementiert. Basierend darauf präsentieren wir Benchmark-Evaluationen, die die Effizienz unserer Entscheidungsprozeduren bestätigen