On the fluidization of Petri nets and marking homothecy

The analysis of Discrete Event Dynamic Systems suffers from the well known state explosion problem. A classical technique to overcome it is to relax the behavior by partially removing the integrality constraints and thus to deal with hybrid or continuous systems. In the Petri nets framework, continuous net systems (technically hybrid systems) are the result of removing the integrality constraint in the firing of transitions. This relaxation may highly reduce the complexity of analysis techniques but may not preserve important properties of the original system. This paper deals with the basic operation of fluidization. More precisely, it aims at establishing conditions that a discrete system must satisfy so that a given property is preserved by the continuous relaxation. These conditions will be mainly based on the marking homothetic behavior of the system. The focus will be on logical properties as boundedness, B-fairness, deadlock-freeness, liveness and reversibility. Furthermore, testing homothetic monotonicity of some properties in the discrete systems is also studied, as well as techniques to improve the quality of the fluid relaxation by removing spurious solutions.