# A Bidirectional Diagnosis Algorithm of Fuzzy Petri Net Using Inner-Reasoning-Path

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## Abstract

**:**

## 1. Introduction

## 2. Fuzzy Petri Net

#### 2.1. Fuzzy Petri Net and Related Definitions

**Definition**

**1.**

- 1.
- $P=\{{p}_{1},{p}_{2},\cdots ,{p}_{n}\}$is a finite set of places. Moreover,$X={({x}_{1},{x}_{2},\cdots ,{x}_{n})}^{T}$indicates a place vector, where$\left|X\right|=\left|P\right|$. If${p}_{i}$is the goal place or a place which has a direct or indirect relationship with the goal place,${x}_{i}=1$. Else,${x}_{i}=0$.
- 2.
- $T=\{{t}_{1},{t}_{2},\cdots ,{t}_{m}\}$is a finite set of transitions. Moreover,$Y={({y}_{1},{y}_{2},\cdots ,{y}_{m})}^{T}$indicates a transit vector, where$\left|Y\right|=\left|T\right|$. If${t}_{j}$is the transition which has a direct or indirect relationship of the goal place,${y}_{j}=1$. Else,${y}_{j}=0$.
- 3.
- $I:P\times T\to {(I({p}_{i},{t}_{j}))}_{n\times m}$is an input matrix. Here,$I({p}_{i},{t}_{j})$records whether a directed arc from${p}_{i}$to${t}_{j}(i=1,2,\cdots ,n;j=1,2,\cdots ,m)$exists, where$$I({p}_{i},{t}_{j})=\{\begin{array}{cc}1& \mathrm{if}\text{}\mathrm{there}\text{}\mathrm{is}\text{}\mathrm{an}\text{}\mathrm{arc}\text{}\mathrm{from}\text{}{p}_{i}{\text{}\mathrm{to}\text{}\mathrm{t}}_{j}\\ 0& otherwise\end{array}$$
- 4.
- $O:T\times P\to {(O({t}_{j},{p}_{i}))}_{m\times n}$is an output matrix. Here,$O{({t}_{j},{p}_{i})}_{n\times m}$records whether a directed arc from${t}_{j}$to${p}_{i}(j=1,2,\cdots ,m;i=1,2,\cdots ,n)$exists, where$$O({t}_{j},{p}_{i})=\{\begin{array}{cc}1& {\mathrm{if}\text{}\mathrm{there}\text{}\mathrm{is}\text{}\mathrm{an}\text{}\mathrm{arc}\text{}\mathrm{from}\text{}\mathrm{t}}_{j}\text{}\mathrm{to}\text{}{p}_{i}\\ 0& otherwise\end{array}$$
- 5.
- $M={({m}_{1},{m}_{2},\cdots ,{m}_{n})}^{T}$is a vector of fuzzy marking, where${m}_{i}\in [0,1]$means the truth degree of corresponding place${p}_{i}(i=1,2,\cdots ,n)$. The initial truth degree vector is denoted by${M}_{0}$.
- 6.
- $\mu :\mu \to (0,1]$,${\mu}_{i}$is the threshold of${t}_{j}$. Moreover,$D={({\mu}_{1},{\mu}_{2},\cdots ,{\mu}_{m})}^{T}$is a threshold vector, where${\mu}_{j}\in (0,1]\text{\hspace{0.17em}}(j=1,2,\cdots ,m)$;
- 7.
- $W(i,j)$is the weight of the arc from${p}_{i}$to${t}_{j}$.$w(i,j)\in [0,1]$indicates how much the place${p}_{i}$impacts its following transition${t}_{j}$;
- 8.
- $CF$is the belief strength, where$C{F}_{ij}\in (0,1]$indicates how much of a transition${t}_{j}$impacts its output places${p}_{i}$.

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

#### 2.2. Proposed Operators of the Proposed Algorithm

**Definition**

**6.**

**Definition**

**7.**

## 3. Bidirectional Adaptive Reasoning Algorithm

#### 3.1. The Proposed Algorithm

#### 3.2. Implementation Steps

## 4. Analysis

#### 4.1. Correctness

#### 4.2. Algorithm Complexity

- In the worst situation, all places and transitions appear in the reasoning path. This means that the backward reasoning mechanism is out of work. The algorithm complexity of the proposed algorithm is $O(n\times m)$.
- In other situations, the number of unconcerned places and transitions are $r$ and $p$, and the algorithm complexity of the proposed algorithm is $O((n-r)\times (m-p))$.

## 5. Case Study

#### 5.1. Relevant Experimental Data of the Case Study

#### 5.2. Experiments

#### 5.2.1. Experiment One

#### 5.2.2. Experiment Two

#### 5.3. Analysis of Experiments 1 and 2

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**The meaning of each place of Figure 2.

Place | Meaning |
---|---|

P1 | molecular pump is not in proper position |

P2 | pressure exerted is too high |

P3 | temperature of cooling water is high |

P4 | cooling system failures |

P5 | pump is not drying enough |

P6 | air exhaust is not enough |

P7 | compressor operates in magnetic field |

P8 | roller bearing wears |

P9 | compressor is noisy |

P10 | temperature of bump is high |

P11 | blade of turbine wears |

P12 | blade of compressor is broken |

P13 | pressurization ratio of compressor is low |

P14 | blade of turbine is scales |

P15 | compressor is in turbulence |

P16 | blade of turbine breaks down |

Repeat | Output Strength Vector |
---|---|

1st | ${(0.85,0.9,0.865,0,0,0,0,0,0,0)}^{T}$ |

2nd | ${(0.85,0.9,0.865,0.3825,0.72,0.7785,0,0,0,0)}^{T}$ |

3rd | ${(0.85,0.9,0.865,0.3825,0.72,0.7785,0.306,0.70065,054495,0)}^{T}$ |

4th | ${(0.85,0.9,0.865,0.3825,0.72,0.7785,0.306,0.70065,054495,0.613069)}^{T}$ |

5th | ${(0.85,0.9,0.865,0.3825,0.72,0.7785,0.306,0.70065,054495,0.613069)}^{T}$ |

Repeat | M′ |
---|---|

1st | ${(0.85,0.9,0.85,0.85,0,0.765,0.72,0.7785,0,0,0,0,0,0)}^{T}$ |

2nd | ${(0.85,0.9,0.85,0.85,0,0.765,0.72,0.7785,0.306,070065,054495,0,0,0)}^{T}$ |

3rd | ${(0.85,0.9,0.85,0.85,0,0.765,0.72,0.7785,0.306,0.70065,0.54495,0.665618,0.490455,0)}^{T}$ |

4th | ${(0.85,0.9,0.85,0.85,0,0.765,0.72,0.7785,0.306,0.70065,0.54495,0.665618,0.490455,0.582415)}^{T}$ |

5th | ${(0.85,0.9,0.85,0.85,0,0.765,0.72,0.7785,0.306,0.70065,0.54495,0.665618,0.490455,0.582415)}^{T}$ |

Repeat | Output Strength Vector |
---|---|

1st | ${(0.745,0,0,0)}^{T}$ |

2nd | ${(0.745,0.6705,0,0)}^{T}$ |

3rd | ${(0.745,0.6705,0.46935,0)}^{T}$ |

4th | ${(0.745,0.6705,0.46935,0.37548)}^{T}$ |

5th | ${(0.745,0.6705,0.46935,0.37548)}^{T}$ |

Repeat | M′ |
---|---|

1st | ${(0.7,0.75,0.8,0.6075,0,0,0)}^{T}$ |

2nd | ${(0.7,0.75,0.8,0.6075,0.46935,0,0)}^{T}$ |

3rd | ${(0.7,0.75,0.8,0.6075,0.46935,0.37548,0)}^{T}$ |

4th | ${(0.7,0.75,0.8,0.6075,0.46935,0.37548,0.300384)}^{T}$ |

5th | ${(0.7,0.75,0.8,0.6075,0.46935,0.37548,0.300384)}^{T}$ |

Related Matrices | Experiment One | Experiment Two | |
---|---|---|---|

Original Matrices | $H$,$A$ and $B$ | 16 × 11 | 16 × 11 |

$-{H}^{T}$ | 11 × 16 | 11 × 16 | |

In backward reasoning phase | ${A}^{\prime}$ and ${B}^{\prime}$ | 16 × 11 | 16 × 11 |

${{A}^{\prime}}^{T}$ | 11 × 16 | 11 × 16 | |

In forward reasoning phase | ${A}^{\prime}$ and ${B}^{\prime}$ | 14 × 10 | 7 × 4 |

${{A}^{\prime}}^{T}$ | 10 × 14 | 4 × 7 |

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**MDPI and ACS Style**

Zhou, K.-Q.; Gui, W.-H.; Mo, L.-P.; Zain, A.M.
A Bidirectional Diagnosis Algorithm of Fuzzy Petri Net Using Inner-Reasoning-Path. *Symmetry* **2018**, *10*, 192.
https://doi.org/10.3390/sym10060192

**AMA Style**

Zhou K-Q, Gui W-H, Mo L-P, Zain AM.
A Bidirectional Diagnosis Algorithm of Fuzzy Petri Net Using Inner-Reasoning-Path. *Symmetry*. 2018; 10(6):192.
https://doi.org/10.3390/sym10060192

**Chicago/Turabian Style**

Zhou, Kai-Qing, Wei-Hua Gui, Li-Ping Mo, and Azlan Mohd Zain.
2018. "A Bidirectional Diagnosis Algorithm of Fuzzy Petri Net Using Inner-Reasoning-Path" *Symmetry* 10, no. 6: 192.
https://doi.org/10.3390/sym10060192