Managing LTL properties in Event-B refinement

Refinement in Event-B supports the development of systems via proof based step-wise refinement of events. This refinement approach ensures safety properties are preserved, but additional reasoning is required in order to establish liveness and fairness properties. In this paper we present results which allow a closer integration of two formal methods, Event-B and linear temporal logic. In particular we show how a class of temporal logic properties can carry through a refinement chain of machines. Refinement steps can include introduction of new events, event renaming and event splitting. We also identify a general liveness property that holds for the events of the initial system of a refinement chain. The approach will aid developers in enabling them to verify linear temporal logic properties at early stages of a development, knowing they will be preserved at later stages. We illustrate the results via a simple case study

Generalized Strong Preservation by Abstract Interpretation

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

Generalizing the Paige-Tarjan Algorithm by Abstract Interpretation

The Paige and Tarjan algorithm (PT) for computing the coarsest refinement of a state partition which is a bisimulation on some Kripke structure is well known. It is also well known in model checking that bisimulation is equivalent to strong preservation of CTL, or, equivalently, of Hennessy-Milner logic. Drawing on these observations, we analyze the basic steps of the PT algorithm from an abstract interpretation perspective, which allows us to reason on strong preservation in the context of generic inductively defined (temporal) languages and of possibly non-partitioning abstract models specified by abstract interpretation. This leads us to design a generalized Paige-Tarjan algorithm, called GPT, for computing the minimal refinement of an abstract interpretation-based model that strongly preserves some given language. It turns out that PT is a straight instance of GPT on the domain of state partitions for the case of strong preservation of Hennessy-Milner logic. We provide a number of examples showing that GPT is of general use. We first show how a well-known efficient algorithm for computing stuttering equivalence can be viewed as a simple instance of GPT. We then instantiate GPT in order to design a new efficient algorithm for computing simulation equivalence that is competitive with the best available algorithms. Finally, we show how GPT allows to compute new strongly preserving abstract models by providing an efficient algorithm that computes the coarsest refinement of a given partition that strongly preserves the language generated by the reachability operator.Comment: Keywords: Abstract interpretation, abstract model checking, strong preservation, Paige-Tarjan algorithm, refinement algorith

PLTL Partitioned Model Checking for Reactive Systems under Fairness Assumptions

We are interested in verifying dynamic properties of finite state reactive systems under fairness assumptions by model checking. The systems we want to verify are specified through a top-down refinement process. In order to deal with the state explosion problem, we have proposed in previous works to partition the reachability graph, and to perform the verification on each part separately. Moreover, we have defined a class, called Bmod, of dynamic properties that are verifiable by parts, whatever the partition. We decide if a property P belongs to Bmod by looking at the form of the Buchi automaton that accepts the negation of P. However, when a property P belongs to Bmod, the property f => P, where f is a fairness assumption, does not necessarily belong to Bmod. In this paper, we propose to use the refinement process in order to build the parts on which the verification has to be performed. We then show that with such a partition, if a property P is verifiable by parts and if f is the expression of the fairness assumptions on a system, then the property f => P is still verifiable by parts. This approach is illustrated by its application to the chip card protocol T=1 using the B engineering design language