Abstract
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 arbitrary (temporal) languages and of generic abstract models, possibly different from standard state partitions, 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 deal with strong preservation of new languages by providing an efficient algorithm that computes the coarsest refinement of a given partition that strongly preserves a language generated by the reachability operator.
Highlights
Paige and Tarjan algorithm (PT) is used in model checking for reducing the state space of a Kripke structure K because the quotient of K w.r.t. bisimulation equivalence strongly preserves temporal languages like CTL∗, CTL and the whole μ-calculus [2,4]
This abstract interpretation-based view of the PT algorithm leads us to generalize PT to: (1) a generic domain A of abstract models that generalizes the role played in PT by the domain of state partitions Part(Σ); (2) a generic set Op of operators on ℘(Σ) that provides the semantics of some language LOp and generalizes the role played in PT by the set OpHML of operators of HML
GPT may be systematically instantiated to classes of abstract models and inductive languages that satisfy some conditions
Summary
The Paige and Tarjan [26] algorithm — in the paper denoted by PT — for efficiently computing the coarsest refinement of a given partition which is stable for a given state transition relation is well known. PT is used in model checking for reducing the state space of a Kripke structure K because the quotient of K w.r.t. bisimulation equivalence strongly preserves temporal languages like CTL∗, CTL and the whole μ-calculus [2,4]. This means that logical specifications can be checked on the abstract quotient model of K with no loss of precision.
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