Finding legal firing sequences of Petri nets by means of dynamic programming included linear programming

There is no general useful method, except the reachability tree, to find legal firing sequences even for a bounded Petri net. An LP-based method with semi-polynomial-time-complexity for this problem with the m×1 known firing count vector u has been recently reported by Fujii et al (1994), but its disadvantage is that the size of constraints becomes exhaustive. For improving the above point, this paper proposes a DP-based method which can reduce about d^-1 times the size of constraints for each subprocess and can also avoid the space explosion in DP by using some properties between a Petri net and its reversed net, where d is the length of firing sequence and is defined by d=^ΔΣ_i=1^mu_i and u=^Δ[u_i]. Moreover, it is shown that each suboptimization can also be solved by LP with the polynomial time complexity. As a result of the above, this proposed method based on DP including d LPs is the algorithm with about d^-3 times semi-polynomial time complexity compared with that given by Fujii et al. in which only one LP is applied to the given reachability problem