Precise Widening Operators for Convex Polyhedra

Convex polyhedra constitute the most used abstract domain among those capturing numerical relational information. Since the domain of convex polyhedra admits infinite ascending chains, it has to be used in conjunction with appropriate mechanisms for enforcing and accelerating convergence of the fixpoint computation. Widening operators provide a simple and general characterization for such mechanisms. For the domain of convex polyhedra, the original widening operator proposed by Cousot and Halbwachs amply deserves the name of standard widening since most analysis and verification tools that employ convex polyhedra also employ that operator. Nonetheless, there is demand for more precise widening operators that still has not been fulfilled. In this paper, after a formal introduction to the standard widening where we clarify some aspects that are often overlooked, we embark on the challenging task of improving on it. We present a framework for the systematic definition of new and precise widening operators for convex polyhedra. The framework is then instantiated so as to obtain a new widening operator that combines several heuristics and uses the standard widening as a last resort so that it is never less precise. A preliminary experimental evaluation has yielded promising results. We also suggest an improvement to the well-known widening delay technique that allows to gain precision while preserving its overall simplicity.