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Further study on liveness of time Petri net

Liveness is the most important properties of the Petri nets, for the classic Petri nets, many scholars have done a lot of work and obtained satisfactory judging result, but most literatures only focus on concrete application for Time Petri Net (TPN for short). We have shown that TPN's liveness and boundedness behavior has no relation with its corresponding classic Petri Net through and give two sufficient and necessary conditions in time interval of TPN preserving liveness and boundedness in previous study. But this is for special subclasses of TPN and its adaption is very small. In this paper, we give the structural and dynamic behavior conditions of TPN preserving liveness. It offers the theory foundation for using the classic Petri Net's property judgment result to judge a TPN's liveness.