An Abstract Domain to Infer Octagonal Constraints with Absolute Value

The octagon abstract domain, devoted to discovering octagonal constraints (also called Unit Two Variable Per Inequality or UTVPI constraints) of a program, is one of the most commonly used numerical abstractions in practice, due to its quadratic memory complexity and cubic time complexity. However, the octagon domain itself is restricted to express convex sets and has limitations in handling non-convex properties which are sometimes required for proving some numerical properties in a program. In this paper, we intend to extend the octagon abstract domain with absolute value, to infer certain non-convex properties by exploiting the absolute value function. More precisely, the new domain can infer relations of the form {±X ± Y ≤ c,±X ± |Y | ≤ d,±|X| ± |Y | ≤ e}. We provide algorithms for domain operations such that the new domain still enjoys the same asymptotic complexity as the octagon domain. Moreover, we present an approach to support strict inequalities over rational or real-valued variables in this domain, which also fits for the octagon domain. Experimental results of our prototype are encouraging; The new domain is scalable and able to find non-convex invariants of interest in practice but without too much overhead (compared with that using octagons).