How Many Live Minimal Structural Traps Are Required to Make a Minimal Deadlock Locally Live in General Petri Nets ?

Petri nets are one of useful models for discrete event systems in which liveness problem as well as reachability problem is one of big issues. But, it has not been completely solved from the point of view of useful initial-marking-based liveness conditions in general Petri nets. In this paper, to guarantee localliveness (i. e. ,liveness under MoD) for each minimal deadlock (MSDL). ND=(SD. TD.FD.MoD). with real deadlock-trap structure, it is shown that the minimum number of required live minimal structural traps (MSTRs). NT=(ST. TT, FT, MoT) s. t. SD⊇ ST, is conditionally (which means that the conditions of Lemma 4-9 are fulfilled for a bounded MSDL ND containing at least one MSTR NT s. t. SD⊇ ST and see also Remarks 4-2 (3) in Subsection 4.3) "one". Note that this local liveness for ND s. t SD⊇ST is one of useful necessary conditions for liveness condition of general Petri nets N=(S, T, F, Mo) s. t S⊇SD . However, because this has not been discussed in literature and is not trivial, some new concepts such as T -cornucopias and return paths are introduced into the real deadlock-trap structure s. t SD⊇ST in Nand this is proven by dividing it into two cases: ND s. t S D⊇S T is live and unbounded under MaD and ND s. t. SD⊇ST is live and bounded under MaD. Moreover, another related problem is also discussed. Usefulness for the results obtained is also discussed