Abstract. The paper introduces the notion of a weak bisimulation for coalgebras whose type is a monad satisfying some extra properties. In the first part of the paper we argue that systems with silent moves should be modelled coalgebraically as coalgebras whose type is a monad. We show that the visible and invisible part of the functor can be handled internally inside a monadic structure. In the second part we introduce the notion of an ordered saturation monad, study its properties, and show that it allows us to present two approaches towards defining weak bisimulation for coalgebras and compare them. We support the framework presented in this paper by two main examples of models: labelled transition systems and simple Segala systems.
We introduce a general categorical framework for the definition of weak behavioural equivalences, building on and extending recent results in the field. This framework is based on special order enriched categories, i.e. categories whose homsets are endowed with suitable complete orders. Using this structure we provide an abstract notion of saturation, which allows us to define various (weak) behavioural equivalences. We show that the Kleisli categories of many common monads are categories of this kind. On one hand, this allows us to instantiate the abstract definitions to a wide range of existing systems (weighted LTS, Segala systems, calculi with names, etc.), recovering the corresponding notions of weak behavioural equivalences; on the other, we can readily provide new weak behavioural equivalences for more complex behaviours, like those definable on presheaves, topological spaces, measurable spaces, etc.
Abstract. The aim of this paper is to introduce a coalgebraic setting in which it is possible to generalize and compare the two known approaches to defining weak bisimulation for labelled transition systems. We introduce two definitions of weak bisimulation for coalgebras over ordered functors, show their properties and give sufficient conditions for them to coincide. We formulate a weak coinduction principle.
Abstract. In the first part of the paper we recall the coalgebraic approach to handling the so-called invisible transitions that appear in different state-based systems semantics. We claim that these transitions are always part of the unit of a certain monad. Hence, coalgebras with internal moves are exactly coalgebras over a monadic type. The rest of the paper is devoted to supporting our claim by studying two important behavioural equivalences for state-based systems with internal moves, namely: weak bisimulation and trace semantics. We continue our research on weak bisimulations for coalgebras over order enriched monads. The key notions used in this paper and proposed by us in our previous work are the notions of an order saturation monad and a saturator. A saturator operator can be intuitively understood as a reflexive, transitive closure operator. There are two approaches towards defining saturators for coalgebras with internal moves. Here, we give necessary conditions for them to yield the same notion of weak bisimulation. Finally, we propose a definition of trace semantics for coalgebras with silent moves via a uniform fixed point operator. We compare strong and weak bisimilation together with trace semantics for coalgebras with internal steps.
Timed transition systems are behavioural models that include an explicit treatment of time flow and are used to formalise the semantics of several foundational process calculi and automata. Despite their relevance, a general mathematical characterisation of timed transition systems and their behavioural theory is still missing. We introduce the first uniform framework for timed behavioural models that encompasses known behavioural equivalences such as timed bisimulations, timed language equivalences as well as their weak and time-abstract counterparts. All these notions of equivalences are naturally organised by their discriminating power in a spectrum. We prove that this result does not depend on the type of the systems under scrutiny: it holds for any generalisation of timed transition system. We instantiate our framework to timed transition systems and their quantitative extensions such as timed probabilistic systems.
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