**6. Conclusions**

436 Petri Nets – Manufacturing and Computer Science

Carrying out the same process, the conclusion is as follows: Md is not reachable under **V**2.

Carrying out the same process, the conclusion is as follows: Md is not reachable under **V**2.

Carrying out the same process shown in Figure 7, the conclusion is as follows: Md is reachable from M0 under **V**1+**U. V**1+**U** is an executable solution in Xe, and the firing

As a result of calculating each element of the sufficient test space Xe={**V**1, **V**2, **V**3, **V**1+**U**, **V**1+2**U**, **V**3+**U**} individually, a firing sequence is finally found at the fourth element (**V**1+**U**) of Xe**.** Therefore, the elements **V**1+2**U** and **V**3+**U** don't need to be calculated. Consequently, the structure of the Petri net (Figure 5) is shown to possess at least one reachable firing sequence.

**Step 2.** For **X**=**V**2=(2,2,1,2,0,0,0,0)

**Step 3.** For **X**=**V**3=(1,2,1,1,1,1,0,0)

**Step 4.** For **X**=**V**1+**U**=(0,2,1,0,2,2,1,1)

sequence is t5\*t7\*t6\*t2\*t3\*t5\*t6\*t2\*t8.

**Figure 6.** Firing path tree for **V**1.

In this chapter, a new general criterion has been created to solve the reachability problems for ordinary Petri nets. This criterion is based on two processes: (i) Calculating the sufficient test space. (ii) Testing whether or not the destination marking is reachable from the initial marking under the sufficient test space. The sufficient test space significantly reduces the quantity of computation needed to search for an executable solution in X. The firing path tree shows the firing sequence of an executable solution. Consequently, if the destination marking is reachable from the initial marking, this method gives at least one firing sequence that leads from the initial marking to the destination marking. Some examples are given to illustrate how to use this method to solve the reachability problem. This algorithm can be utilized in the following fields: Path searching, auto routing, and reachability between any places in a complicated network.
