Section: Scientific Foundations

Models of concurrency: nets, scenarios, event structures, and their variants

For Finite State Machines (FSM), a large body of theory has been developed to address problems such as: observation (the inference of hidden state trajectories from incomplete observations), control, diagnosis, and learning. These are difficult problems, even for simple models such as FSM's. One of the research tracks of DistribCom consists in extending such theories to distributed systems involving concurrency, i.e., systems in which both time and states are local, not global. For such systems, even very basic concepts such as “trajectories” or “executions” need to be deeply revisited. Computer scientists have for a long time recognized this topic of concurrent and distributed systems as a central one. In this section, we briefly introduce the reader to the models of scenarios, event structures, nets, languages of scenarios, graph grammars, and their variants.

Scenarios.

The simplest concept related to concurrency is that of a finite execution of a distributed machine. To this end, scenarios have been informally used by telecom engineers for a long time. In scenarios, so-called “instances” exchange asynchronous messages, thus creating events that are totally ordered on a given instance, and only partially ordered by causality on different instances (emission and reception of a message are causally related). The formalization of scenarios was introduced by the work done in the framework of ITU and OMG on High-level Message Sequence Charts and on UML Sequence Diagrams in the last ten years, see  [52] , [57] . This allowed in particular to formally define infinite scenarios, and to enhance them with variables, guards, etc  [61] , [59] , [60] . Today, scenarios are routinely offered by UML and related software modeling tools.

Event structures.

Event structures were invented by Glynn Winskel and co-authors in 1980 [56] , [62] . Executions are sets of events that are partially ordered by a causality relation. Event structures collect all the executions by superimposing shared prefixes. Events not belonging to a same execution are said in conflict. Events that are neither causally related nor in conflict are called concurrent. Concurrent processes model the “parallel progress” of components.

Categories of event structures have been defined, with associated morphisms, products, and co-products, see [63] . Products and co-products formalize the concepts of parallel composition and “union” of event structures, respectively. This provides the needed apparatus for composing and projecting (or abstracting) systems. Event structures have been mostly used to give the semantics of various formalisms or languages, such as Petri nets, CCS, CSP, etc  [56] , [62] . We in DistribCom make a nonstandard use of these, e.g., we use them as a structure to compute and express the solutions of observation or diagnosis problems, for concurrent systems.

Nets and languages of scenarios.

The next step is to have finite representations of systems having possibly infinite executions. In DistribCom, we use two such formalisms: Petri nets  [58] , [45] and languages of scenarios such as High-level Message Sequence Charts (HMSC)  [52] , [60] . Petri nets are well known, at least in their basic form, we do not introduce them here. We use so-called safe Petri Nets, in which markings are boolean (tokens can be either 0 or 1); and we use also variants, see below.

Extensions and variants.

Two extensions of the basic concepts of nets or scenario languages are useful for us. Nets or scenario languages enriched with variables, actions, and guards, are useful to model general concurrent and distributed dynamical systems in which a certain discrete abstraction of the control is represented by means of a net or a scenario language. Manipulating such symbolic nets requires using abstraction techniques. Time Petri nets and network of timed automata are particular cases of symbolic nets. Probabilistic Nets or event structures: Whereas a huge literature exists on stochastic Petri nets or stochastic process algebras (in computer science), randomizing concurrent models, i.e., with $\omega$'s being concurrent trajectories, not sequential ones, has been addressed only since the 21st century. We have contributed to this new area of research.

Handling dynamic changes in the systems.

The last and perhaps most important issue, for our applications, is the handling of dynamic changes in the systems model. This is motivated by the constant use of dynamic reconfigurations in management systems. Extensions of net models have been proposed to capture this, for example the dynamic nets of Vladimiro Sassone  [44] and net systems  [46] . For the moment, such models lack a suitable theory of unfoldings.