Ranking Abstraction as a Companion to Predicate Abstraction,

Predicate abstraction has become one of the most successful methodologies for proving safety properties of programs. Recently, several abstraction methodologies have been proposed for proving liveness properties. This paper studies “ranking abstraction” where a program is augmented by a nonconstraining progress monitor, and further abstracted by predicate-abstraction, to allow for automatic verification of progress properties. Unlike most liveness methodologies, the augmentation does not require a complete ranking function that is expected to decrease with each step. Rather, the inputs are component rankings from which a complete ranking function may be formed. The premise of the paper is an analogy between the methods of ranking abstraction and predicate abstraction, one ingredient of which is refinement: When predicate abstraction fails, one can refine it. When ranking abstraction fails, one must determine whether the predicate abstraction, or the ranking abstraction, need be refined. The paper presents strategies for determining which case is at hand. The other part of the analogy is that of automatically deriving deductive proof constructs: Predicate abstraction is often used to derive program invariants for proving safety properties as a boolean combination of the given predicates. Deductive proof of progress properties requires well-founded ranking functions instead of invariants. We show how to obtain concrete global ranking functions from abstract programs. We demonstrate the various methods on examples with nested loops, including a bubble sort algorithm on linked lists.