Transition predicate abstraction and fair termination

Abstract

Predicate abstraction is the basis of many program verification tools. Until now, the only known way to overcome the inherent limitation of predicate abstraction to safety properties was to manually annotate the finite-state abstraction of a program. We extend predicate abstraction to transition predicate abstraction. Transition predicate abstraction goes beyond the idea of finite abstract-state programs (and checking the absence of loops). Instead, our abstraction algorithm transforms a program into a finite abstract-transition program. Then, a second algorithm checks fair termination. The two algorithms together yield an automated method for the verification of liveness properties under full fairness assumptions (justice and compassion). In summary, we exhibit principles that extend the applicability of predicate abstraction-based program verification to the full set of temporal properties.

Size-Change Termination and Transition Invariants

“…Transition predicate abstraction [32] is a method to compute transition invariants, just as predicate abstraction is a method to compute invariants. The method takes as input a selection of finitely many binary relation over states.…”

“…Our notation P # for the set of abstract transitions computed by Algorithm TPA stems from [32]. There, P # is called an abstract transition program.…”

“…Related work and further developments include, e.g., [7,24,27,36]). A branch of this work is based on the concept of transition invariants from [31]; see, e.g., [11,14,15,18,26,32]). The motivation behind the work in [31] was to carry over the ideas of [27] to verification methods in the style of software model checking [3,4].…”

Proving Liveness by Backwards Reachability

“…The main technique of software model checking, using finite-state abstractions [16] has been difficult to apply when proving liveness properties, since abstractions may introduce spurious loops [33] that do not preserve liveness. Podelski and Rybalchenko therefore extended the framework of predicate abstraction to that of transition predicate abstraction [32], which involves constructing an abstraction of the transition relation and its transitive closure.…”

“…Podelski and Rybalchenko extend the framework of predicate abstraction to that of transition predicate abstraction [32,33,29,20], which can be applied on arbitrary programs. The transitive closure of the transition relation is harder to compute or approximate than the set of reachable states.…”