Section: New Results

Petri Nets

Over the past two decades, numerous verification tools have been successfully used for verifying complex concurrent systems, modelled using various formalisms. However, it is still hard to coordinate these tools since they rely on such a large number of formalisms. Having a proper syntactical mechanism to interrelate them through variability would increase the capability of effective integrated formal methods. In [28] , we propose a modular approach for defining new formalisms by reusing existing ones and adding new features and/or constraints. Our approach relies on standard XML technologies; their use provides the capability of rapidly and automatically obtaining tools for representing and validating models. It thus enables fast iterations in developing and testing complex formalisms. As a case study, we applied our modular definition approach on families of Petri nets and timed automata.

In [38] , we study the following summarization problem: given a parallel composition A = A1 || ... || An of labelled transition systems communicating with the environment through a distinguished component Ai, efficiently compute a summary Si such that E || A and E || Si are trace-equivalent for every environment E. While Si can be computed using elementary automata theory, the resulting algorithm suffers from the state-explosion problem. We present a new, simple but subtle algorithm based on net unfoldings, a partial-order semantics, give experimental results. Our algorithm can also handle divergences and compute weighted summaries with minor modifications.

At the end of the eighties, continuous Petri nets were introduced for: (1) alleviating the combinatory explosion triggered by discrete Petri nets and, (2) modelling the behaviour of physical systems whose state is composed of continuous variables. Since then several works have established that the computational complexity of deciding some standard behavioural properties of Petri nets is reduced in this framework. In [39] , we first establish the decidability of additional properties like boundedness and reachability set inclusion. We also design new decision procedures for the reachability and lim-reachability problems with a better computational complexity. Finally we provide lower bounds characterising the exact complexity class of the boundedness, the reachability, the deadlock freeness and the liveness problems.

In [45] , we integrate two compact data structures for representing state spaces of Petri nets: merged processes and contextual prefixes. The resulting data structure, called contextual merged processes (CMP), combines the advantages of the original ones and copes with several important sources of state space explosion: concurrency, sequences of choices, and concurrent read accesses to shared resources. In particular, we demonstrate on a number of benchmarks that CMPs are more compact than either of the original data structures. Moreover, we sketch a polynomial (in the CMP size) encoding into SAT of the model-checking problem for reachability properties.

Under maximal semantics, the occurrence of an event $a$ in a concurrent run of an occurrence net may imply the occurrence of other events, not causally related to $a$, in the same run. In recent works, we have formalized this phenomenon as the reveals relation, and used it to obtain a contraction of sets of events called facets in the context of occurrence nets. In [36] , we extend this idea to propose a canonical contraction of general safe Petri nets into pieces of partial-order behaviour which can be seen as “macro-transitions” since all their events must occur together in maximal semantics. On occurrence nets, our construction coincides with the facets abstraction. Our contraction preserves the maximal semantics in the sense that the maximal processes of the contracted net are in bijection with those of the original net.