Bisimulations, Games and Logic

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.

2009

Based on a simple axiomatization of concurrent behaviour we define two ways of observing parallel computations and show that in each case they are dual to conflict and causality, respectively. We give a logical characterization to those dualities and show that natural fixpoint modal logics can be extracted from such a characterization. We also study the equivalences induced by such logics and prove that they are decidable and can be related with well-known bisimulations for interleaving and noninterleaving concurrency. Moreover, by giving a game-theoretical characterization to the equivalence induced by the main logic, which is called Separation Fixpoint Logic (SFL), we show that the equivalence SFL induces is strictly stronger than a history-preserving bisimulation (hpb) and strictly weaker than a hereditary history-preserving bisimulation (hhpb). Our study considers branching-time models of concurrency based on transition systems and petri net structures.

Electronic Notes in Theoretical Computer Science, 1998

Event structures have come to play an important role in the formal study of the behaviour of distributed systems. The advantage of event structures is that they explicitly exhibit the interplay between concurrency and nondeterminism. This paper is contributed to develop a numberof new bisimulations which are natural and nicely t with the concept of event structures. We establish closed relationships between the bisimulations, resulting in a lattice of implications. Some logics with a C T L avour, beinginterpreted over event structures, are further introduced to characterize the proposed bisimulations logically.

Theoretical Computer Science, 1993

Log. Methods Comput. Sci., 2017

Weak bisimulations are typically used in process algebras where silent steps are used to abstract from internal behaviours. They facilitate relating implementations to specifications. When an implementation fails to conform to its specification, pinpointing the root cause can be challenging. In this paper we provide a generic characterisation of branching-, delayed-, $\eta$- and weak-bisimulation as a game between Spoiler and Duplicator, offering an operational understanding of the relations. We show how such games can be used to assist in diagnosing non-conformance between implementation and specification. Moreover, we show how these games can be extended to distinguish divergences.

Logical Methods in Computer Science, 2017

Weak bisimulations are typically used in process algebras where silent steps are used to abstract from internal behaviours. They facilitate relating implementations to specifications. When an implementation fails to conform to its specification, pinpointing the root cause can be challenging. In this paper we provide a generic characterisation of branching-, delay-, η-and weak-bisimulation as a game between Spoiler and Duplicator, offering an operational understanding of the relations. We show how such games can be used to assist in diagnosing non-conformance between implementation and specification. Moreover, we show how these games can be extended to distinguish divergences.

Foundations of Software Science and Computation Structures, 2018

The hiding operation, crucial in the compositional aspect of game semantics, removes computation paths not leading to observable results. Accordingly, games models are usually biased towards angelic non-determinism: diverging branches are forgotten. We present here new categories of games, not suffering from this bias. In our first category, we achieve this by avoiding hiding altogether; instead morphisms are uncovered strategies (with neutral events) up to weak bisimulation. Then, we show that by hiding only certain events dubbed inessential we can consider strategies up to isomorphism, and still get a category-this partial hiding remains sound up to weak bisimulation, so we get a concrete representations of programs (as in standard concurrent games) while avoiding the angelic bias. These techniques are illustrated with an interpretation of affine nondeterministic PCF which is adequate for weak bisimulation; and may, must and fair convergences.

We show h o w to use a wide class of Structural Operational Semantics speci cations for Truly Concurrent Semantics, whereas the classical approach is rather based on interleaving models. The models we consider are Asynchronous Transition Systems and their behaviors are de ned by means of a classical truly concurrent bisimulation : ST-bisimulation. However, the de nition of ST-bisimulation over Asynchronous Transition Systems is new and makes this equivalence better understood. Moreover, we explain how our result holds of many other truly concurrent bisimulations of the literature.

2000

This thesis is concerned with formal semantics and models for concurrent computational systems, that is, systems consisting of a number of parallel computing sequential systems, interacting with each other and the environment. A formal semantics gives meaning to computational systems by describing their behaviour in a mathematical model. For concurrent systems the interesting aspect of their computation is often how they interact with the environment during a computation and not in which state they terminate, indeed they may not be intended to terminate at all. For this reason they are often referred to as reactive systems, to distinguish them from traditional calculational systems, as e.g. a program calculating your income tax, for which the interesting behaviour is the answer it gives when (or if) it terminates, in other words the (possibly partial) function it computes between input and output. Church's thesis tells us that regardless of whether we choose the lambda calculus, Turing machines, or almost any modern programming language such as C or Java to describe calculational systems, we are able to describe exactly the same class of functions. However, there is no agreement on observable behaviour for concurrent reactive systems, and consequently there is no correspondent to Church's thesis. A result of this fact is that an overwhelming number of different and often competing notions of observable behaviours, primitive operations, languages and mathematical models for describing their semantics, have been proposed in the litterature on concurrency.

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.