A novel liveness condition for S 3 PGR 2
Daniel Yuh Chao, Jiun‐Ting Chen, Fang Yu · Transactions of the Institute of Measurement and Control · 2012
Systems of simple sequential processes with resources (S 3 PR), modeled by ordinary Petri nets (OPNs) are live if and only if no siphons (a set of places) can ever become empty of tokens. Systems of simple sequential processes with general resources requirement (S 3 PGR 2 ), modeled by general Petri nets (GPNs), are a generalization of S 3 PR. It has been a hot research topic to find the sufficient and necessary condition of liveness for S 3 PGR 2 . When an OPN (respectively, GPN) is deadlocked, the set of all unmarked (respectively, non-max-marked) places forms a siphon, which is said to be deadly marked. However, when an S 3 PGR 2 is livelocked but not deadlocked, e.g., both live and dead transitions exist, the set of non-max-marked places, denoted by [Formula: see text], may not form a siphon. Thus, it is limited using deadly marked or max-controlled siphons to characterize liveness of S 3 PGR 2 . The first author proposed earlier max’ -controlled siphons to characterize liveness of S 3 PGR 2 . However, the presence of a set of non-max’ -marked places is not a sufficient condition for an S 3 PGR 2 to be non-live and cannot be relied on to devise mixed integer programming (MIP) testing. In this paper, we present the sufficient and necessary liveness condition of S 3 PGR 2 . We define pivot markings, under which we prove that [Formula: see text] forms a siphon. To detect livelocks, the number of disabled arcs must be maximal under the pivot marking. This result is useful to reorient the MIP test of S 3 PR for S 3 PGR 2 .