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 amounts to the equivalence of concrete and abstract model checking of formulas in . We show how abstract interpretation can be used to design generic abstract models that allow to view standard abstract Kripke structures as particular instances. 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 can be formulated as a minimal domain refinement in abstract interpretation in order to get completeness w.r.t. the logical/temporal operators of . 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.
Highlights
The design of an abstract model checking framework always includes a preservation result, roughly stating that for any formula φ specified in some temporal language L, if φ holds on an abstract model φ holds on the concrete model
This work shows how the abstract interpretation technique allows to generalize the notion of strong preservation from standard abstract models specified as abstract Kripke structures to generic domains in abstract interpretation
For any inductively defined language L, it turns out that strong preservation of L in a standard abstract model checking framework based on partitions of the space state Σ becomes a particular instance of the property of forward completeness of abstract domains w.r.t. the semantic operators of the language L
Summary
The design of an abstract model checking framework always includes a preservation result, roughly stating that for any formula φ specified in some temporal language L , if φ holds on an abstract model φ holds on the concrete model. A number of algorithms for solving this problem exist, like those by Paige and Tarjan [42] for CTL, by Henzinger et al [35], Bustan and Grumberg [5] and Tan and Cleaveland [48] for ACTL, and Groote and Vaandrager [32] for CTL-X These are coarsest partition refinement algorithms: given a language L and a partition P of States, which is determined by a state labeling, these algorithms can be viewed as computing the coarsest partition PL that refines P and strongly preserves L. It turns out that ADL coincides with the forward complete shell for the operators of L of a basic abstract domain determined by the state labeling This characterization provides an elegant generalization of partition refinement algorithms used in standard abstract model checking. It turns out that a partition P is a bisimulation on some Kripke structure K if and only if the corresponding partitioning abstract domain ad(P ) is forward complete for the standard predecessor transformer pre→ in K
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