Donut Domains: Efficient Non-convex Domains for Abstract Interpretation

Program analysis using abstract interpretation has been successfully applied in practice to find runtime bugs or prove software correct. Most abstract domains that are used widely rely on convexity for their scalability. However, the ability to express non-convex properties is sometimes required in order to achieve a precise analysis of some numerical properties. This work combines already known abstract domains in a novel way in order to design new abstract domains that tackle some non-convex invariants. The abstract objects of interest are encoded as a pair of two convex abstract objects: the first abstract object defines an over-approximation of the possible reached values, as is done customarily. The second abstract object under-approximates the set of impossible values within the state-space of the first abstract object. Therefore, the geometrical concretization of our objects is defined by a convex set minus another convex set (or hole). We thus call these domains donut domains.