## Section: New Results

### Fundamentals results and algorithms: distributed unfoldings

Participants : Eric Fabre, Blaise Genest.

This is a joint work with our former postdoc Agnes Madalinski, now assistant professor at the University of Santiago in Chile.

Distributed systems can be modeled as networks of interacting components, for example networks of Petri nets, or networks of automata. The simplest way to combine components into larger systems is to take their product. It is well known that the unfolding of a product Petri net can be expressed as the product (in a specific sense) of the unfoldings of the components. The factorized form of unfoldings is generally more compact than the expanded product, because each factor only represents its local conflicts and does not have to display the choices that are made by other components. This property was the basis of our previous results about distributed diagnosis.

With Agnes Madalinski, we explored the construction of finite complete prefixes (FCP) for distributed systems [45] , [14] . More precisely, our goal was to obtain a FCP in factorized form, when the underlying system is expressed as a product of components. FCP represent in a compact manner all possible behaviors, and states, that a concurrent system can reach, which makes them a central tool for model checking applications for example. It is likely that factorized forms will be more compact, and will thus open the way to new distributed model checking techniques. A trivial but impractical solution consists in computing a FCP of the global system, and then deriving from it its factorized form. We explored a fully decentralized method to obtain directly an FCP of each component, such that the product of all these local FCP would yield an FCP of the global system. A solution was proposed in some limited situations, through the notion of extended stopping point. This is a first step to solving problems like distributed reachability analysis, distributed planning, etc., as they are now explored in the FAST cooperation, and in the DISC European project.

This year, we also developped another technique, similar in its goal to unfolding but very different in its methodology. Instead of considering a true concurrency model which does not exhibit redundancy by construction, we consider a priori several interleavings resulting in the same distributed execution. Then, we perform on the fly partial order reduction, such that the redudancy is kept as small as possible [10] . We showed that performing a breadth first search, no redundant path needs to be considered, whereas in depth first search (DFS), some limited redudancy needs to be used. It is quite similar to unfolding. The experiment we performed showed that the technique is simple yet very efficient, and that the amount of redudancy needed in DFS is very limited in practice.