Parameterised Bisimulations: Some Application

Zenodo (CERN European Organization for Nuclear Research), 2006

Bisimulation can be defined in a simple way using coinductive methods, and has rather pleasant properties. Ready similarity was proposed by Meyer et al. as a way to weakening the bisimulation equivalence thus getting a semantics defined in a similar way, but supported for more reasonable (weaker) observational properties. Global bisimulations were introduced by Frutos et al. in order to study different variants of non-determinism getting, in particular, a semantics under which the internal choice operator becomes associative. Global bisimulations are defined as plain bisimulations but allowing the use of new moves, called global transitions, that can change the processes not only locally in its head, but anywhere. Now we are continuing the study of global bisimulation but focusing on the way different semantics can be characterised as global bisimulation semantics. In particular, we have studied ready similarity, on the one hand because it was proposed as the strongest reasonable semantics weaker than bisimulation; on the other hand, because ready similarity was not directly defined as an equivalence relation but as the nucleus of an order relation, and this open the question whether it is also possible to define it as a symmetric bisimulation-like semantics. We have got a simple and elegant characterisation of ready similarity as a global bisimulation semantics, that provides a direct symmetric characterisation of it as an equivalence relation, without using any order as intermediate concept. Besides, we have found that it is not necessary to start from a simulation based semantics to get an equivalent global bisimulation. What has proved to be very useful is the axiomatic characterisation of the semantics. Following these ideas we have got also global bisimulation for several semantics, including refusals and traces. That provides a general framework that allows to relate both intensional and extensional semantics.

Journal of Universal Computer …, 2006

Bisimulation can be defined in a simple way using coinductive methods, and has rather pleasant properties. Ready similarity was proposed by Meyer et al. as a way to weakening the bisimulation equivalence thus getting a semantics defined in a similar way, but supported for more reasonable (weaker) observational properties. Global bisimulations were introduced by Frutos et al. in order to study different variants of non-determinism getting, in particular, a semantics under which the internal choice operator becomes associative. Global bisimulations are defined as plain bisimulations but allowing the use of new moves, called global transitions, that can change the processes not only locally in its head, but anywhere. Now we are continuing the study of global bisimulation but focusing on the way different semantics can be characterised as global bisimulation semantics. In particular, we have studied ready similarity, on the one hand because it was proposed as the strongest reasonable semantics weaker than bisimulation; on the other hand, because ready similarity was not directly defined as an equivalence relation but as the nucleus of an order relation, and this open the question whether it is also possible to define it as a symmetric bisimulation-like semantics. We have got a simple and elegant characterisation of ready similarity as a global bisimulation semantics, that provides a direct symmetric characterisation of it as an equivalence relation, without using any order as intermediate concept. Besides, we have found that it is not necessary to start from a simulation based semantics to get an equivalent global bisimulation. What has proved to be very useful is the axiomatic characterisation of the semantics. Following these ideas we have got also global bisimulation for several semantics, including refusals and traces. That provides a general framework that allows to relate both intensional and extensional semantics.

2005

The question of when two systems are behaviourally equal has occupied a large part of the literature on verification and has yielded various equivalences (and congruences). These equivalence relations are most useful in comparing systems whose executions are not necessarily finite. An axiomatization of these equivalences gives us both, a nice algebraic handle on processes, and a proof system for checking the equality of two processes. Comparison of efficiency of non-terminating processes like an operating system has been largely untackled. We have presented here, an axiomatization for a certain subset of ordering induced bisimilarities. This axiomatization yields the axiomatization for equivalences like observational equivalence and inefficiency bisimulation as special cases. The axiomatization has been proven to be complete for finite state processes, and can be used as a proof system for checking the equality of systems.

Perspectives in Concurrency Theory

The question of when two nondeterministic concurrent systems are behaviourally related has occupied a large part of the literature on process algebra and has yielded a variety of equivalences (and congruences) and preorders (and precongruences) all based on the notion of bisimulations. Recently one of the authors has tried to unify a class of these bisimulation based relations by a parametrised notion of bisimu- lation and shown that the properties of the bisimilarity relations are often inherited from those of the underlying relationships between the observables. In addition to the usual strong and weak bisimilarity relations, it is possible to capture some other bisimilarity relations – those sensitive to costs, performance, dis- tribution or locations etc – by parametrised bisimulations. In this paper we present an equational axiomatization of all equivalence relations that fall in the class of parametrised bisimilarities without empty observables. Our axiomatization has been inspired by the axiomatization of observational congruence by Bergstra and Klop and attempts to extend it for parametrised bisimilarities. The axiomatization has been proven to be complete for finite process graphs relative to a complete axiomatization for the relations on observables. In the process, we also show that in the absence of empty observables, all preorders and equivalence relations are also precongruences and congruences, respectively.

Information and Computation, 1995

We study three notions of bisimulation equivalence for concurrent processes. Bisimulation equivalences are based on an operational interpretation of processes as labelled transition systems, and constitute the strongest notion of equivalence one may adopt for such systems: two systems are equivalent if and only if they have the same step-by-step behaviour. We focus first on Milner's notion of weak bisimulation (also known as observational equivalence) and propose an alternative formulation for it. More specifically, we show that Milner's notion may be redefined as one of reducibility to a same system-via a reduction function called abstraction homorriorphism. We use our characterisation to derive a complete set of reduction rules for observational equivalence on finite processes. We also show how abstraction homomorphisms may be extended to labelled event structures: however we do not consider the possibility of unobservable events here. We look then for notions of bisimulation which account for the concurrent aspects of processes. Traditional transition systems-evolving via successive elementary actions-only provide an interleaving semantics for concurrency. We suggest two generalisations of the notion of transition system: distributed transition systems, obtained by generalising the residual of a transition, and pornset transition systems, obtained by extending the notion of action labelling a transition (an action being now a partially ordered multiset). For the latter we find a corresponding notion of bisimulation on labelled event structures. Based on these new kinds of transitions, we obtain two bisimulation equivalences-one stronger than the other-which are both more discriminating than Milner's equivalence. For both of them we present an algebraic characterisation by means of a complete set of axioms.

2005

The focus of process calculi is interaction rather than computation, and for this very reason: (i) their operational semantics is conveniently expressed by labelled transition systems (LTSs) whose labels model the possible interactions with the environment; (ii) their abstract semantics is conveniently expressed by observational congruences. However, many current-day process calculi are more easily equipped with reduction semantics, where the notion of observable action is missing. Recent techniques attempted to bridge this gap by synthesising LTSs whose labels are process contexts that enable reactions and for which bisimulation is a congruence. Starting from Sewell's set-theoretic construction, category-theoretic techniques were defined and based on Leifer and Milner's relative pushouts, later refined by Sassone and the fourth author to deal with structural congruences given as groupoidal 2-categories. Building on recent works concerning observational equivalences for tile logic, the paper demonstrates that double categories provide an elegant setting in which the aforementioned contributions can be studied. Moreover, the formalism allows for a straightforward and natural definition of weak observational congruence.

1998

We define a class of process algebras with silent step and a generalised operation $\gsum{}$ that allows explicit treatment of \emph{alternative quantification} over data, and we investigate the specific subclass formed by the algebras of finite processes modulo rooted branching bisimulation. We give a ground complete axiomatisation for those branching bisimulation algebras of which the data part has built-in equality and Skolem functions.

… of Networks and …, 2002

The fundamental notion of bisimulation has inspired various notions of system equivalences in concurrency theory. Many notions of bisimulation for various discrete systems have been recently unified in the abstract category theoretical formulation of bisimulation due to Joyal, Nielsen and Winskel. In this paper, we adopt their framework and unify the notions of bisimulation equivalences for discrete, continuous dynamical and control systems. This shows that our equivalence notion is on the right track, but also confirms that abstract bisimulation is general enough to capture equivalence notions in the domain of continuous systems. We believe that the unification of the bisimulation relation for labelled transition systems and dynamical systems under the umbrella of abstract bisimulation, as achieved in this work, is a first step towards a unified approach to modeling of and reasoning about the dynamics of discrete and continuous structures in computer science and control theory.

ArXiv, 2020

We investigate how various forms of bisimulation can be characterised using the technology of logical relations. The approach taken is that each form of bisimulation corresponds to an algebraic structure derived from a transition system, and the general result is that a relation $R$ between two transition systems on state spaces $S$ and $T$ is a bisimulation if and only if the derived algebraic structures are in the logical relation automatically generated from $R$. We show that this approach works for the original Park-Milner bisimulation and that it extends to weak bisimulation, and branching and semi-branching bisimulation. The paper concludes with a discussion of probabilistic bisimulation, where the situation is slightly more complex, partly owing to the need to encompass bisimulations that are not just relations.