# An Automatic Algorithm to Generate a Reachability Tree for Large-Scale Fuzzy Petri Net by And/Or Graph

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## Abstract

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## 1. Introduction

- It difficult to directly describe the dependence relationships (‘and’ or ‘or’ relationship) among the places in the neighbouring layers;
- It difficult to generate automatically a large reachability tree for a rather complex expert system.

## 2. Modified Reachability Tree for Different Fuzzy Petri Net (FPN) Models

#### 2.1. FPN and Fuzzy Production Rule (FPR)

**Type 1: Simple rule**

**Type 2: And rule**

**Type 3: Or rule**

#### 2.2. A Modified Reachability Tree Based on And/Or Graph

- The symbol ‘︶’ is used to mark the ‘and’ relation for the ‘and’ type FPR in the modified reachability tree.
- All parameters’ values are marked in the modified reachability tree.
- The modified reachability reverse records the flow relationship between the input place(s) and the output place(s) to aid in decomposition and reasoning operations.

## 3. Generation Algorithm of Reachability Tree for FPN

#### 3.1. Introduction to Breadth-First Search (BFS)

#### 3.2. Algorithm to Generate the Reachability Tree for FPN

- Phase 1 is used to redefine and optimize the FPRs, which includes two operations. First, redefine all FPRs based on the following formalism one by one:$$<type(simple,or,and?),\{condition(s)\},conclusion,\left\{parameters\right\}>\text{}$$Then, settle the conclusion-sharing case in KBS. In KBS, a common situation that exists is that many FPRs could obtain the same conclusion. The conclusion-sharing situation will lead to chaotic and complex flow-relationships in the reachability tree. Hence, in this algorithm, the conclusion-sharing situations will be checked first and the FPRs with the conclusion-sharing situations will be divided into two FPRs by using a virtual conclusion as follows.$$<type(simple,or,and?),\{condition(s)\},\begin{array}{cc}virtual& conclusion\end{array},\left\{parameters\right\}>\text{}$$$$<type(simple,or,and?),\{\begin{array}{cc}virtual& conclusion\end{array}(s)\},\begin{array}{cc}previous& conclusion\end{array},\left\{parameters\right\}>\text{}$$
- Phase 2 is used to generate all reachability trees for each FPR and to calculate the number of all root-nodes.
- Phase 3 is to expand the reachability tree of an any given root-node from the root-nodes’ set by using the BFS mechanism repeatedly until the final completed reachability tree is gained, and to execute the BFS operation for each root-node.

## 4. Experiments and Related Analysis

- FPRs with ‘and’ and ‘or’ relationships;
- FPRs with sharing condition situation;
- FPRs with sharing conclusion situation;
- FPR with multi-conclusions.

#### 4.1. Experimental Data and Simulation Results

**Experiment one**: in this experiment, four FPRs, includes one simple type, two ‘and’ rules and a ‘or’ rule, are used to generate the corresponding reachability tree. The FPRs are listed in Table 1 and the simulation result is demonstrated in Figure 5.

**Experiment two**: in this experiment, two FPRs with sharing conditions are used to generate the corresponding reachability tree. The FPRs are listed in Table 2 and the simulation result is demonstrated in Figure 6.

#### 4.2. Analysis of Simulation Results

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Corresponding fuzzy Petri net (FPN) and (

**b**) modified reachability tree of ‘simple’ rule.

**Figure 5.**The obtained reachability tree from Table 1.

**Figure 6.**The reachability tree obtained from Table 2.

**Figure 7.**The reachability tree obtained from Table 3.

**Figure 8.**The reachability tree obtained from Table 4.

No | FPR |
---|---|

Rule 1 | $\begin{array}{ccccccccc}if& {R}_{1}& and& {R}_{2}& and& {R}_{3}& then& {Q}_{1}& ({w}_{11},{w}_{12},{w}_{13},{\mu}_{1},C{F}_{1})\end{array}$ |

Rule 2 | $\begin{array}{ccccc}if& {R}_{3}& then& {Q}_{2}& ({w}_{2}=1,{\mu}_{2},C{F}_{2})\end{array}$ |

Rule 3 | $\begin{array}{ccccccc}if& {Q}_{1}& or& {Q}_{2}& then& {Q}_{3}& ({w}_{31}=1,{w}_{32}=1,{\mu}_{31},{\mu}_{32},C{F}_{31},C{F}_{32})\end{array}$ |

Rule 4 | $\begin{array}{ccccccccc}if& {R}_{7}& and& {R}_{6}& and& {R}_{5}& then& {R}_{1}& \begin{array}{ccccccc}if& {Q}_{1}& or& {Q}_{2}& then& {Q}_{3}& ({w}_{15},{w}_{16},{w}_{17},{\mu}_{15},C{F}_{16})\end{array}\end{array}$ |

No | FPR |
---|---|

Rule 1 | $\begin{array}{ccccccccc}if& {R}_{1}& and& {R}_{2}& and& {R}_{3}& then& {Q}_{1}& ({w}_{11},{w}_{12},{w}_{13},{\mu}_{11},C{F}_{11})\end{array}$ |

Rule 2 | $\begin{array}{ccccccc}if& {R}_{1}& or& {R}_{2}& then& {Q}_{2}& ({w}_{21}=1,{w}_{22}=1,{\mu}_{21},{\mu}_{22},C{F}_{21},C{F}_{22})\end{array}$ |

No | FPR |
---|---|

Rule 1 | $\begin{array}{ccccccccc}if& {R}_{1}& and& {R}_{2}& and& {R}_{3}& then& {Q}_{1}& ({w}_{11},{w}_{12},{w}_{13},{\mu}_{11},C{F}_{11})\end{array}$ |

Rule 2 | $\begin{array}{ccccccccc}if& {R}_{1}& or& {R}_{2}& or& {R}_{3}& then& {Q}_{1}& ({w}_{21}=1,{w}_{22}=1,{w}_{23}=1,{\mu}_{21},{\mu}_{22},{\mu}_{23},C{F}_{21},C{F}_{22},C{F}_{23})\end{array}$ |

No | FPR |
---|---|

Rule 1 | $\begin{array}{ccccccccc}if& {R}_{1}& and& {R}_{2}& and& {R}_{3}& then& \begin{array}{ccc}{Q}_{1}& and& {Q}_{2}\end{array}& ({w}_{11},{w}_{12},{w}_{13},{\mu}_{11},C{F}_{11},C{F}_{12})\end{array}$ |

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**MDPI and ACS Style**

Zhou, K.-Q.; Mo, L.-P.; Ding, L.; Gui, W.-H.
An Automatic Algorithm to Generate a Reachability Tree for Large-Scale Fuzzy Petri Net by And/Or Graph. *Symmetry* **2018**, *10*, 454.
https://doi.org/10.3390/sym10100454

**AMA Style**

Zhou K-Q, Mo L-P, Ding L, Gui W-H.
An Automatic Algorithm to Generate a Reachability Tree for Large-Scale Fuzzy Petri Net by And/Or Graph. *Symmetry*. 2018; 10(10):454.
https://doi.org/10.3390/sym10100454

**Chicago/Turabian Style**

Zhou, Kai-Qing, Li-Ping Mo, Lei Ding, and Wei-Hua Gui.
2018. "An Automatic Algorithm to Generate a Reachability Tree for Large-Scale Fuzzy Petri Net by And/Or Graph" *Symmetry* 10, no. 10: 454.
https://doi.org/10.3390/sym10100454