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A type reduction theory for systems with replicated components
The Parameterised Model Checking Problem asks whether an implementation
Impl(t) satisfies a specification Spec(t) for all instantiations of parameter
t. In general, t can determine numerous entities: the number of processes used
in a network, the type of data, the capacities of buffers, etc. The main theme
of this paper is automation of uniform verification of a subclass of PMCP with
the parameter of the first kind, i.e. the number of processes in the network.
We use CSP as our formalism. We present a type reduction theory, which, for a
given verification problem, establishes a function \phi that maps all
(sufficiently large) instantiations T of the parameter to some fixed type T^
and allows us to deduce that if Spec(T^) is refined by \phi(Impl(T)), then
(subject to certain assumptions) Spec(T) is refined by Impl(T). The theory can
be used in practice by combining it with a suitable abstraction method that
produces a t-independent process Abstr that is refined by {\phi}(Impl(T)) for
all sufficiently large T. Then, by testing (with a model checker) if the
abstract model Abstr refines Spec(T^), we can deduce a positive answer to the
original uniform verification problem. The type reduction theory relies on
symbolic representation of process behaviour. We develop a symbolic operational
semantics for CSP processes that satisfy certain normality requirements, and we
provide a set of translation rules that allow us to concretise symbolic
transition graphs. Based on this, we prove results that allow us to infer
behaviours of a process instantiated with uncollapsed types from known
behaviours of the same process instantiated with a reduced type. One of the
main advantages of our symbolic operational semantics and the type reduction
theory is their generality, which makes them applicable in a wide range of
settings