Transformation From Legal-marking Set to Admissible-marking Set of Petri Nets With Uncontrollable Transitions

1  Abstract—Linear constraint transformation is an essential step to solve the forbidden state problem in Petri nets that contain uncontrollable transitions. This work studies the equivalent transformation from a legal-marking set to its admissible-marking set given such a net. First, the concepts of an escaping-marking set and a transforming marking set are defined. Based on them, two algorithms are given to compute the admissible-marking set and the transforming marking set, which establish the theoretical foundation for the equivalent transformation of linear constraints. Second, the theory about the equivalent transformation of a disjunction of linear constraints imposed to Petri nets with uncontrollable transitions is established. Third, two rules are given to decide the priority of transitions for transformation. Finally, the transformation procedure from a given linear constraint to a logic expression of linear constraints that can describe its entire admissible-marking set is illustrated via two examples. the set of controllable transitions Tu the set of uncontrollable transitions N (P, T, F)=(P, T u ∪T c , F), an ordinary Petri net M the set of all possible markings of N MF a forbidden-marking set L a legal-marking set A the admissible marking set with respect to L MWF the weakly forbidden-marking set with respect to L u a supervisory policy u zero the least permissive supervisory policy that disables all controllable transitions R(N, m0) the set of all reachable markings of N from m0 R t (N, m) the set of all reachable markings of N from m by firing t only R(N, m0, u) the set of all reachable markings under the supervision of a policy u in N from m0 R(N, m, u zero) the set of all reachable markings of N from m by firing uncontrollable transitions only Q a marking set Г(Q, t) the escaping-marking set of Q via t Q t the transforming marking set of Q via t m|P' the restriction of marking m to P' Q|P' {m| P' | m Q}, the restriction of the marking set Q to P' Q* the all-place-restricted marking set of Q (ω, k) a linear constraint W