# SCIENCE CHINA Information Sciences, Volume 61, Issue 1: 010205(2018) https://doi.org/10.1007/s11432-017-9283-7

## Modeling and analysis of colored petri net based on the semi-tensor product of matrices • AcceptedSep 28, 2017
• PublishedDec 11, 2017
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### Abstract

This paper applies the model petri net method based on the semi-tensor product of matrices to colored petri net. Firstly, we establish the marking evolution equation for colored petri net by using the semi-tensor product of matrices. Then we define the concept of controllability and the control-marking adjacency matrix for colored petri net. Based on the marking evolution equation and control-marking adjacency matrix, we give the necessary and sufficient condition of reachability and controllability for colored petri net. The algorithm to verify the reachability of colored petri net is given, and we analyze the computational complexity of the algorithm. Finally, an example is given to illustrate the effectiveness of the proposed theory. The significance of the paper lies in the application of the model petri net method based on the semi-tensor product of matrices to colored petri net. This is a convenient way of verifying whether one marking is reachable from another one as well as finding all firing sequences between any two reachable markings. Additionally, the method lays the foundations for the analysis of other properties of colored petri net.

### Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61573199, 61573200), Tianjin Natural Science Foundation of China (Grant Nos. 14JCYBJC18700, 13JCYBJC17400).

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• Figure 1

The colored petri net of dining philosophers.

• Figure 2

The state space of the colored petri net of dining philosophers.

• Figure 3

The colored petri net used to verify the reachability and controllability.

• Figure 4

The state space of the colored petri net used to verify the reachability and controllability.

•

Algorithm 1 Verifying the reachability of colored petri net

Consider a colored petri net $\langle~{\rm~CPN},M_{1}~\rangle$ with $m$ transitions, $n$ variables, and $s$ markings, and where $M_{i}=\delta_{s}^{i}$ and $M_{j}=\delta_{s}^{j}$ are two markings. We can verify whether $M_{j}$ is reachable from $M_{i}$ by $t$ steps by the following method. Step 1: Construct the marking evolution equation of $\langle~{\rm~CPN},M_{1}~\rangle$. Step 2: Calculate the matrix $(LW_{[s,mh]})^{t}\delta_{s}^{i}$. If $\delta_{s}^{j}\notin~{\rm~Col}((LW_{[s,mh]})^{t}\delta_{s}^{i})$ then $\delta_{s}^{j}$ is not reachable from $\delta_{s}^{i}$ in $t$ steps. Step 3: Find all $k$ such that $\delta_{s}^{j}={\rm~Col}_{k}((LW_{[s,mh]})^{t}\delta_{s}^{i})$. Now $\delta_{(mh)^{t}}^{k}=\ltimes_{j=1}^{t}\delta_{mh}^{k_{j}}$ and we can obtain $k_j$, $(j=1,2,\ldots,t)$ by decomposing the semi-tensor product of matrices. Thus, we can find the firing sequence $\delta_{mh}^{k_{1}}$,$\delta_{mh}^{k_{2}}$,$\delta_{mh}^{k_{3}}$,…,$\delta_{mh}^{k_{t}}$. For each $\delta_{mh}^{k_{j}},(j=1,2,\ldots,t)$, we can also obtain the firing transition and the binding for all variables by decomposing the semi-tensor product of matrices.

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