Weak Bisimulation and Open Maps
Marcelo P. Fiore
May 1999 |

## Abstract:
A general treatment of weak bisimulation and observation
congruence on presheaf models is presented. It extends previous work on
bisimulation from open maps and answers a fundamental question there. The
consequences of the general theory are derived for two key models for
concurrency, the interleaving model of synchronisation trees and the
independence model of labelled event structures. Starting with a ``hiding''
functor from a category of paths to observable paths, it is shown how to
derive a monad on the category of presheaves over the category of paths. The
Kleisli category of the monad is shown to be equivalent to that of presheaves
with weak morphisms, which roughly are only required to preserve observable
paths. The monad is defined through an intermediary view of processes as
bundles in
Cat. This view, which seems important in its own right,
leads directly to a hiding operation on processes; when the hiding operation
is translated back to presheaves through an adjunction it yields the monad.
The monad is shown to automatically preserve open-map bisimulation,
generalising the expectation that strongly bisimilar processes should be
weakly bisimilar. However, two alternative general ways to define weak
bisimulation and observation congruence (in the Kleisli category itself or in
the presheaf category after application of the monad) do not coincide in
general. However a necessary and sufficient condition for their equivalence
is proved; from this it is sufficient to show a fill-in condition applies to
category of paths with weak morphisms. The fill-in condition is shown to hold
for the two traditional models, synchronisation trees and event
structures
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