In: IEICE Trans. on Fundamentals of Electronics, Communications and Computer Science, Vol. E78-A, No. 12, pages 1875-1889. 1996.
Abstract: If a general Petri net N = (S,T,F,M(o)) is transition-live under M(o), it is evident that each maximal structural deadlock SDL ((N) over cap(D)) in N as well as each minimal structural deadlock MSDL (N-D) in each (N) over cap(D) is also transition-live under M. However, since the converse of the latter of the above is not always true, it is important to obtain the conditions for this converse to be true if we want to have a useful necessary and sufficient `initial-marking-based' or `structural' liveness condition for N. Up to now, useful and well-known structural or initial-marking-based necessary and sufficient liveness conditions of Petri nets have only been those of an asymmetric choice (AC) net and its subclasses such as an EFC net, an FC net, an FCF net, MG, and SR?. However, all the above subclasses are activated only by real or virtual deadlock-trap properties which are local liveness for each minimal deadlocks; in other words, the above topics of this paper are unconditionally satisfied in those subclasses because of their special structure of nets. In this paper, a necessary and sufficient structural liveness condition for a general Petri net N with globally structural live minimal structural deadlocks is presented as follows: The next (I) or (II) is satisfied. (I) N has no SDL (N) over cap(D). (II) If N has at least one SDL (N) over cap(D), (i) or (ii) is satisfied under the condition that each MSDL No in N is transition-live under nl,. (i) N has no singular MSDL (alpha) (i.e., (alpha-I) and (alpha-II)). (ii) If N has at least one singular MSDL (alpha-I) (alpha-II), resp.), every semi-MSDL (I) ((II), resp.) N-DS = (S-DS, T-DS, F-DS, M(oDS)) with respect to each singular MSDL (alpha-I) (alpha-II), resp.), is transition-live under the M(oDS) under the condition of `the condition (**)', where the locally structural liveness for this N-DS means (1) or (2) ((3), resp.) of Lemma 4-4 and `the condition (**)' is defined in Lemma 4-7 of this paper. The relationship between the above results and the liveness problem for N is also shown.
Keywords: Petri nets, concurrent systems, semi-minimal deadlocks, singular minimal deadlocks, structural liveness.