  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.
