Extending the Limits of Sequentially Phased Reasoning

Verification via abstraction is the topic of concluding correctness of a complicated program from correctness of an abstraction of this program. Principles for sequentially phased reasoning are examples of verification via abstraction. They use permutation equivalences to abstract concurrent programs which logically work in phases to considerably simpler variants which perform these phases sequentially. It is shown in this paper that there are classes of concurrent programs, such as self-stabilizing ones, which logically work in phases but cannot be abstracted to a sequentialized version using permutation equivalence. A new class of so-called Ω-equivalences is defined which generalizes permutation equivalence and stutter equivalence. It is proved that both equivalences are indeed special cases of Ω-equivalences. A verification strategy based on Ω-equivalences is formulated and compared with existing approaches for sequentially phased reasoning. The strategy is applied to a self-stabilizing coloring program that can neither be abstracted by permutation nor by stutter equivalences to a sequentialized version.