Abstract We propose a semi-decision procedure for checking generalized reachability properties, on generalized Petri nets, that is based on the Property Directed Reachability (PDR) method. We actually define three different versions, that vary depending on the method used for abstracting possible witnesses, and that are able to handle problems of increasing difficulty. We have implemented our methods in a model-checker called SMPT and give empirical evidences that our approach can handle problems that are difficult or impossible to check with current state of the art tools.

Numerous real-world systems can be modeled with Petri nets, which allow a combination of concurrency with synchronizations and conflicts. To alleviate the difficulty of checking their behaviour, a common approach consists in studying specific subclasses. In the converse problem of Petri net synthesis, a Petri net of some subclass has to be constructed efficiently from a given specification, typically from a labelled transition system (lts) describing the behaviour of the desired net. In this paper, we focus on a notorious subclass of persistent Petri nets, the weighted marked graphs (WMGs), also called generalised (or weighted) event (or marked) graphs or weighted T-nets. In such nets, edges have multiplicities (weights) and each place has at most one ingoing and one outgoing transition. Although extensively studied in previous works and benefiting from strong results, both their analysis and synthesis can be further investigated. We provide new behavioural properties of WMGs expressed on their reachability graph, notably backward persistence and strong similarities between any two sequences sharing the same starting state and the same destination state. Besides, we design a general synthesis procedure aiming at the WMG class. Finally, when no solution to the synthesis problem exists, i.e., when the given lts is not WMG-solvable, we show how to construct a WMG whose reachability graph is a minimal over-approximation of the given lts.

Liveness, (non-)deadlockability and reversibility are behavioral properties of Petri nets that are fundamental for many real-world systems. Such properties are often required to be monotonic, meaning preserved upon any increase of the marking. However, their checking is intractable in general and t heir monotonicity is not always satisfied. To simplify the analysis of these features, structural approaches have been fruitfully exploited in particular subclasses of Petri nets, deriving the behavior from the underlying graph and the initial marking only, often in polynomial time. In this paper, we further develop these efficient structural methods to analyze deadlockability, liveness, reversibility and their monotonicity in weighted Petri nets. We focus on the join-free subclass, which forbids synchronizations, and on the homogeneous asymmetric-choice subclass, which allows conflicts and synchronizations in a restricted fashion. For the join-free nets, we provide several structural conditions for checking liveness, (non-)deadlockability, reversibility and their monotonicity. Some of these methods operate in polynomial time. Furthermore, in this class, we show that liveness, non-deadlockability and reversibility, taken together or separately, are not always monotonic, even under the assumptions of structural boundedness and structural liveness. These facts delineate more sharply the frontier between monotonicity and non-monotonicity of the behavior in weighted Petri nets, present already in the join-free subclass. In addition, we use part of this new material to correct a flaw in the proof of a previous characterization of monotonic liveness and boundedness for homogeneous asymmetric-choice nets, published in 2004 and left unnoticed.

Weighted Petri nets provide convenient models of many man-made systems. Real applications are often required to possess the fundamental Petri net properties of liveness and reversibility, as liveness preserves all the functionalities (fireability of all transitions) of the system and reversibility lets the system return to its initial state (marking) using only internal operations. Characterizations of both behavioral properties, liveness and reversibility, are known for well-formed weighted Choice-Free and ordinary Free-Choice Petri nets, which are special cases of Equal-Conflict Petri nets. However, reversibility is not well understood for this larger class, where choices must share equivalent preconditions, although characterizations of liveness are known. In this paper, we provide the first characterization of reversibility for all live Equal-Conflict Petri nets by extending, in a weaker form, a known condition that applies to the Choice-Free and Free-Choice subclasses. We deduce the monotonicity of reversibility in the live Equal-Conflict class. We also give counter-examples for other classes where the characterization does not hold. Finally, we focus on well-formed Equal-Conflict Petri nets, for which we offer the first polynomial sufficient conditions for liveness and reversibility, contrasting with the previous exponential time conditions.

Join-Free Petri nets, whose transitions have at most one input place, model systems without synchronizations, while Choice-Free Petri nets, whose places have at most one output transition, model systems without conflicts. These classes respectively encompass the state machines (S-systems) and the marked graphs (T-systems). Whereas a structurally bounded and structurally live Petri net is said to be “well-formed”, a bounded and live Petri net is said to be “well-behaved”. Necessary and sufficient conditions for the well-formedness of Join-Free and Choice-Free nets have been known for some time, yet the behavioral properties of these classes are still not well understood. In particular polynomial sufficient conditions for liveness, that is, polynomial in time and with a polynomial initial number of tokens, have not been found until now. Besides, home markings , which can be reached from every reachable marking thus allowing for the construction of systems that can return to their initial data distribution, are not well apprehended either for these subclasses. We extend results on weighted T-systems to the class of weighted Petri nets and present transformations which preserve the language of the system and reduce the initial marking. We introduce a notion of balancing that makes possible the transformation of conservative systems into so-called “token-conservative” systems, whose number of tokens is invariant, while retaining the feasible transition sequences. This transformation is pertinent for all well-formed Petri nets and leads to polynomial sufficient conditions of liveness for well-formed Join-Free and Choice-Free nets. Finally, we also provide polynomial live and home markings for Fork-Attribution systems.

Cyclo-Static DataFlow Graphs (CSDFG in short) is a formalism commonly used to model parallel applications composed by actors communicating through buffers. The liveness of a CSDFG ensures that all actors can be executed infinitely often. This property is clearly fundamental for the design of embedded applications.