Polynomial Sufficient Conditions of Well-Behavedness for Weighted Join-Free and Choice-Free Systems

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 (or S-systems) and the marked graphs (or T-systems).Whereas a structurally bounded and structurally live Petri net graph 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 efficient sufficient conditions for liveness have not been found until now. In this paper, we extend results on weighted T-systems to the class of weighted Petri nets and present transformations which preserve the feasible sequences of transitions and reduce the initial marking. We introduce a notion of "balancing" that makes possible the transformation of conservative systems into so-called "1-conservative systems" while retaining the feasible transition sequences. This transformation leads to polynomial sufficient conditions of liveness for well-formed Join-Free and Choice-Free nets.