Characterizing Liveness Monotonicity for Weighted Petri Nets in Terms of Siphon-Based Properties

Abstract

A Petri net (N, M0) is monotonically live (m-live) if it remains live when the values of its initial marking M0 are increased. N is structurally m-live if there exists an initial marking M0 such that (N, M0) is m-live. Three new siphon-based characterizations for these properties are obtained: (1) For a weighted net N, the ST-property (i.e., every siphon contains a trap) is a necessary but not sufficient condition for N to be structurally m-live. (2) For a weighted net N, a necessary but not sufficient condition for (N, M0) to be m-live is that every siphon of N contains an M0-controlled trap (i.e., for every reachable marking M, the trap contains a place whose token value is not smaller than the least weight of its outgoing arcs). (3) A homogeneous asymmetric choice net (N, M0) is m-live if and only if every minimal siphon of N contains an M0-controlled trap. Characterization (3) is a generalization of Commoner's Theorem from ordinary liveness for ordinary free choice nets to m-liveness for homogeneous asymmetric choice nets.