Generalized Strong Preservation by Abstract Interpretation

TL;DR

This paper extends abstract interpretation to create more general models than traditional Kripke structures, enabling a unified approach to strong preservation and behavioral equivalences in model checking.

Contribution

It introduces a novel abstract interpretation framework for strong preservation, relating it to completeness and fixed points, and unifies various behavioral equivalences under this approach.

Findings

Refined abstract models always exist and are characterized as greatest fixed points.

Bisimulation, simulation, and stuttering equivalences are characterized as completeness properties.

Partition refinement algorithms are interpreted as domain refinements in abstract interpretation.

Abstract

Standard abstract model checking relies on abstract Kripke structures which approximate concrete models by gluing together indistinguishable states, namely by a partition of the concrete state space. Strong preservation for a specification language L encodes the equivalence of concrete and abstract model checking of formulas in L. We show how abstract interpretation can be used to design abstract models that are more general than abstract Kripke structures. Accordingly, strong preservation is generalized to abstract interpretation-based models and precisely related to the concept of completeness in abstract interpretation. The problem of minimally refining an abstract model in order to make it strongly preserving for some language L can be formulated as a minimal domain refinement in abstract interpretation in order to get completeness w.r.t. the logical/temporal operators of L. It turns out that this refined strongly preserving abstract model always exists and can be characterized as a greatest fixed point. As a consequence, some well-known behavioural equivalences, like bisimulation, simulation and stuttering, and their corresponding partition refinement algorithms can be elegantly characterized in abstract interpretation as completeness properties and refinements.