## Objective

One of the problems with the usual formal semantics of concurrency is that it does not reflect the way in which processes are concurrently composed. CEDISYS aims to develop a theory of concurrency in which the distributed nature of processes is properly taken into account (ie truly concurrent semantics).

In addition, the Action planned to develop appropriate models, languages and logics; these will be compositional or syntax-directed in order to support top-down development of programs. They will also support abstraction. The Action also experimented withtechniques and tools for supporting the implementation and animation of the proposed formalisms.

A fundamental understanding of the nature of concurrency and a formal framework for describing concurrent and distributed systems were developed. The framework should lead to methodologies for proving systems correct and, more generally, for deriving their properties.

A web of strong formal connections has been developed, expressed as reflections and coreflections in categorical terms, among a number of simple transition based models. A new model for describing the operational semantics of parallel processes, the Chemical Absract Machine, was introduced. It is well suited for a semantics based on the notion of reduction.

In the categorical approach, computations of a transition system appear to be equipped with a rich algebraic structure. Petri nets can be conveniently handled in this way, and their arrows turn out to be isomorphic to concatenable variants of Petri nonsequential processes. Similarly, when the concurrent transition system of the lambda calculus is considered, its arrows coincide with the classes of Levy's permutation equivalence. The constructions can be generalized using the concepts of internal graph and category. The theory can be applied to logic programming.

A refine version of observation for calculus of communicating systems (CCS) has been introduced which allows the observer to see the distributed nature of processes and a comparison with distributed bisimulation is given. An equational characterization of bisimulation equivalence has been introduced, which is parametric on the actual algebra of observations.

A language of communication, restriction and recursion is defined, whichcontains a notion of action refinement related to the ST-operational semantics. The idea is to view a nonatomic action as a pair of related actions, namely the beginning and the end. A compositional proof system has been defined based on a category of labelled transition systems. Assertions are formulae in a modal mu calculus.

APPROACH AND METHODS

Of the various models of concurrent distributed behaviour proposed, CEDISYS employed two as the preferred semantic domain: Petri-nets and event structures. Using algebraic and categorical techniques, Petri-nets have been extended with compositional and observational mechanisms.

Process description languages originally defined for the interleaving approach (in which a computation is viewed as a sequence of interleaved events) will be equipped with truly concurrent semantics by adapting structural operational semantics and algebraic techniques. Several notions of testing and observational equivalence of processes are being introduced and compared. The issue of the atomicity of actions (the property of being decomposable but non-interruptible) and of action refinement are being studied as well, with the objective of providing process description languages for hierarchical system design and specification.

Reasoning about distributed systems is an important goal in many computer science applications. Modal and temporal logical languages are being used in this Action to do this. The close relationships between particular behavioural equivalences and assertion languages are being recorded by stating which results in one theory can be adequately expressed in the other.

PROGRESS AND RESULTS

A web of strong formal connections has been developed, expressed as reflections and co-reflections in categorical terms, among a number of simple transition-based models. A chapter by Nielsen & Winskel in the forthcoming Handbook of Logic in Computer Science will show the complete picture. A new model for describing the operational semantics of parallel processes, the Chemical Abstract Machine, was introduced. It is well-suited for a semantics based on the notion of reduction.

In the categorical approach, computations of a transition system appear to be equipped with a rich algebraic structure. Petri-nets can be conveniently handled in this way, and their arrows turn out to be isomorphic to concatenable variants of Petri nonsequential processes. Similarly, when the concurrent transition system of the lambda-calculus is considered, its arrows coincide with the classes of Levy's permutation equivalence. The constructions can be generalised using the concepts of internal graph andcategory. The theory can be applied to logic programming.

A refined version of observation for CCS has been introduced which allows the observer to see the distributed nature of processes. It is shown that the resulting theory of location equivalence can be characterised by a simple modal logic. A comparison wit h distributed bisimulation is also given. An alternative to defining a particular noninterleaving semantics is to develop parametric constructions. An equational characterization of bisimulation equivalence has been introduced, which is parametric on the actual algebra of observations. Allowed observations contain interesting special cases, like spatial pomsets. A good level of generality is also achieved by proved trees (essentially causal trees with proved transitions).

A language of communication, restriction and recursion is defined, which contains a notion of action refinement related to the ST-operational semantics. The idea is to view a non-atomic action as a pair of related actions, namely the beginning and the end . A compositional proof system has been defined based on a category of labelled transition systems. Assertions are formulae in a modal mu-calculus. The method consists of applying a sequence of reductions, transforming satisfaction problems for composite process into equivalent satisfaction problems for their immediate subcomponents. The approach yields the most efficient local model checking algorithm published.

POTENTIAL

Most of the existing practical methodologies and tools based on interleaving models of concurrency could be transferred to the true concurrency context. However, a more substantial breakthrough is possible, as completely new methods and tools will become feasible, taking advantage of the superior descriptive power of true concurrency. They will provide the ground for direct improvements in at least three areas: design methods for distributed systems; expert systems designed for reasoning about time; and the architecture of distributed systems.

### Coordinator

56100 Pisa

Italy

### Participants (3)

8000 Aarhus

06565 Valbonne

BN1 9QT Brighton