Segment theory to compute elementary siphons in Petri nets for deadlock control

Unlike other techniques, Li and Zhou add control nodes and arcs for only elementary siphons greatly reducing the number of control nodes and arcs (implemented by costly hardware of I/O devices and memory) required for deadlock control in Petri net supervisors. Li and Zhou propose that the number of elementary siphons is linear to the size of the net. An elementary siphon can be synthesized from a resource circuit consisting of a set of connected segments. We show that the total number of elementary siphons, |Pi(E)|, is upper bounded by the total number of resource places |PR|lower than that min(|P|, |T|) by Li and Zhou where |P|(|T|) is the number of places (transitions) in the net. Also, we claim that the number of elementary siphons |Pi(E)|equals that of independent segments (simple paths) in the resource subnet of an (SPR)-P-3 (systems of simple sequential processes with resources). Resource circuits for the elementary siphons can be traced out based on a graph-traversal algorithm. During the traversal process, we can also identify independent segments (i.e. their characteristic T-vectors are independent) along with those segments for elementary siphons. This offers us an alternative and yet deeper understanding of the computation of elementary siphons. Also, it allows us to adapt the algorithm to compute elementary siphons in [2] for a subclass of (SPR)-P-3 (called (SPR)-P-4) to more complicated (SPR)-P-3 that contains weakly dependent siphons.