A method for finding firing sequences for the reachability problem of Petri nets with mathematical programming technique

Petri nets (PNs) are important modeling tools for many practical control problems. Analysis of reachability is important to the characterization of PN models. Very little work has been reported that combines optimization theory with PNs. The fundamental problem of deciding the reachability with a prespecified firing count vector u and the total firing instance d is considered in this dissertation.
Many feasible optimization techniques are investigated and some turned out to be extremely elegant. A set of problems related to the reachability analysis in PNs is identified. Starting with the basic reachability equations, the straightforward Single step LP method that uses one iteration of linear programming (LP) is derived. Another important method is based on the Optimality Principle (OP), which is combined with the LP technique to derive the OP-based LP method. Dynamic Programming(DP)-based techniques are powerful computational methods for optimization problems. Starting from the elementary net equations, the basic DP algorithm to compute optimal legal firing sequences (LFSs) is derived. To avoid the state-space-explosion-problem, the reversed net of an original net is used and the algorithm is modified to compute on a reversed net. The original problem is subdivided into d recursive parts (d is the length of the LFS) and each part is solved using one LP. This is the principle of the DP-based LP method. To improve computation time, a forward iterative DP-based LP method is derived. The computation deadlocks due to the firing of the sink transition of a critical siphon. To avoid this, an approximation algorithm involving Pontryagin's Minimum Principle(PMP)-based LP method for known u and d is introduced and discussed. This method computes on an original PN N with an initial state x (0). A heuristic algorithm based on the properties of siphons is developed, which avoids the firing of the sink transitions of the potential critical siphons if used with the PMP-based LP method.
Analysis of Complexity is equally important for computer implementations. Therefore, time and space complexities are considered in order to compute LFSs of PNs. The identified problems are addressed. Future research issues are also outlined in the dissertation.

Optimization algorithms applied to large petri nets

Communications to, or between, low-end microprocessors within a product always comes at a cost. This paper develops a new, economic solution that will be useful in a variety of cost-sensitive applications. This paper starts by identifying the properties of an inter-microprocessor communications system that adds minimal cost to a product and enables the use of lower price microprocessors. This leads us to introduce a new category of communications called time independent asynchronous (TIA) communications. An economic 2-wire TIA communications protocol is developed and described using timing diagrams. The protocol is modeled using signal transition graphs (STGs), which are found to have some limitations, and so a modification is developed called STG for threads (STG-FT). Two-wire TIA is simulated to confirm livelock and deadlock properties. An implementation is created that verifies the simulation results, and the performance is reported. Finally, a novel application of 2-wire TIA is discussed.

A New Time Independent Asynchronous Protocol and Its Applications

A new method based on a feedback maximization network for finding legal firing sequences of a P1 net, Petri net models covered by basic P-invariant set equal to one, is proposed. In this work the outline of this algorithm and a simple numerical application example of the proposed method is presented.

A new method for finding firing sequences of Petri nets

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/spl times/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/sup -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=/sup /spl Delta///spl Sigma//sub i=1//sup m/u/sub i/ and u=/sup /spl Delta//[u/sub 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/sup -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.

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

Finding firing sequences for the reachability problem can be regarded as one of the most important issues associated with Petri nets. The purpose of this study is to develop a new method for finding firing sequences by defining an extended state equation. The proposed method is formulated as an integer programming problem whose solution can provide sufficient information on the firing sequences. First, the optimization problem is approached with a linear programming technique. Evaluating the reduced-cost and activeness of every variable, one can systematically sort out the irrelevant variables to be neglected in the integer programming problem of the succeeding stage. In this paper, the usefulness of the proposed method is demonstrated with the evaluation of a couple of simple examples. >

Y. Fujii

Papers

A new method for finding firing sequences for the reachability problem of Petri nets by using a linear programming technique