Backward under-approximations in numeric abstract domains to automatically infer sufficient program conditions

International audienceIn this article, we discuss the automatic inference of sufficient preconditions by abstract interpretation and sketch the construction of an under-approximating backward analysis. We focus on numeric properties of variables and revisit three classic numeric abstract domains: intervals, octagons, and polyhedra, with new under-approximating backward transfer functions, including the support for non-deterministic expressions, as well as lower widenings to handle loops. We show that effective under-approximation is possible natively in these domains without necessarily resorting to disjunctive completion nor domain complementation. Applications include the derivation of sufficient conditions for a program to never step outside an envelope of safe states, or dually to force it to eventually fail. We built a proof-of-concept prototype implementation and tried it on simple examples. Our construction and our implementation are very preliminary and mostly untried; our hope is to convince the reader that this constitutes a worthy avenue of research