At the end of the eighties, continuous Petri nets were introduced for: (1) alleviating the combinatory explosion triggered by discrete Petri nets (i.e. usual 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. Here we first establish the decidability of additional properties like coverability, boundedness and reachability set inclusion. We also design new decision procedures for reachability and lim-reachability problems with a better computational complexity. Finally we provide lower bounds characterising the exact complexity class of the reachability, the coverability, the boundedness, the deadlock freeness and the liveness problems. A small case study is introduced and analysed with these new procedures. ‡ This work has been partially supported by ImpRo (ANR-2010-BLAN-0317). 2 E. Fraca and S. Haddad / Complexity Analysis of Continuous Petri Nets speed of firings. In the former case, such nets are called autonomous CPNs while in the latter they are called timed CPNs. In both cases, the evolution is due to a fractional transition firing (infinitesimal and simultaneous in the case of timed CPNs). Modelling with CPNs. CPNs have been used in several significant application fields. In [3], a method based on CPNs is proposed for the fault diagnosis of manufacturing systems while such a diagnosis is intractable with discrete Petri nets (for modelling of manufacturing systems see also [18]). In [16], the authors introduce a bottom-up modelling methodology based on CPNs to represent cell metabolism and solve in this framework the regulation control problem. Combining discrete and continuous Petri nets yields hybrid Petri nets, with applications to modelling and simulation of water distribution systems [9] and to the analysis of traffic in urban networks [17]. Analysis of CPNs. While several analysis methods have been developed for timed CPNs, there is no hope for fully automatic techniques in the general case since standard problems of dynamic systems are known to be undecidable even for bounded nets [14].Due to the semantics of autonomous CPNs, a marking can be the limit of the markings visited along an infinite firing sequence. Thus most of the usual properties are duplicated depending on whether these markings are considered or not. Taking into account these markings, reachability (resp. liveness, deadlock-freeness) becomes lim-reachability (resp. lim-liveness, lim-deadlock-freeness).Contrary to the timed case, the analysis of autonomous CPNs (that we simply call CPNs in the sequel) appears to be less complex than the one of discrete Petri nets. In [10], exponential time decision procedures are proposed for the reachability and lim-reachability problems for general CPNs. In [15], assuming additional hypotheses on the net, the authors design polynomial time decision procedures for (lim-)reac...

Abstract. 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. Here 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.

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