### Behavior Policy
Summarizing our introductory example, we will demonstrate how to use the state space from Figure 2 with transition functions definitions listed in Table 4 to determine all sequences of actions that lead to the state denoted as s2. Each sequence is equivalent to behavior policy that, when applied, results in moving both agents to the area of type \(B\). To determine such sequences, we create two matrices, a matrix of transitions \(\mathbb{M}^{t}_{F}\) and matrix of initial state \(\mathbb{M}^{0}_{K}\). Having both of them, we can multiply subsequent \(\mathbb{M}^{t}_{K}\) matrices by corresponding \(\mathbb{M}^{t}_{F}\) matrices and check whether the third state (recall that numbering starts from \(0\)) is reachable. By reachable, we understand having a value other than \(0\) in the specified column of the \(\mathbb{M}^{t}_{K}\) matrix. Definitions of both matrices are listed below:  
\[\mathbb{M}^{t}_{F}= \begin{bmatrix}f_{null}&f_{1}+f_{2}&f_{3}\\ f_{null}&f_{null}&f_{4}\\ f_{null}&f_{null}&f_{null}\end{bmatrix} \]  
\[\mathbb{M}^{0}_{K}=\left[\left\{\left(1,0),\left(2,0\right)\right),\varnothing \right\}\quad 0\quad 0\right]\]  
The \(\left\langle\left(1,0\right),\left(2,0\right)\right\rangle\) tuple in the first column of \(\mathbb{M}^{0}_{K}\) matrix denotes that we have two objects. The zeros in both tuples indicate that each object starts the mission at the same moment. Subsequent \(\mathbb{M}^{t}_{K}\) matrices allow us to determine how a system changes when a specified number of actions occur. For example, \(\mathbb{M}^{1}_{K}\) gives us information about how the system evolves when one action occurs (analogously \(\mathbb{M}^{2}_{K}\) for two actions etc.). In our example \(\mathbb{M}_{K}^{1}\) and \(\mathbb{M}_{K}^{2}\) are of the form:  
\[\mathbb{M}_{K}^{1}=\mathbb{M}_{K}^{0}\cdot\mathbb{M}_{F}^{0}= \begin{bmatrix} \langle(1,0),(2,0)\rangle,\mathcal{O}\end{bmatrix} \quad 0\quad 0\end{bmatrix}\cdot \begin{bmatrix}f_{ mult}(c,0)&f_{1}(c,0)+f_{2}(c,0)&f_{3}(c,0)\\ f_{mult}(c,0)&f_{mult}(c,0)&f_{4}(c,0)\\ f_{mult}(c,0)&f_{null}(c,0)&f_{null}(c,0)\end{bmatrix} \]  
\[\mathbb{M}_{K}^{1}= \begin{bmatrix}0&\langle(2,0),(1,2)\rangle,\{r1_{1}\} \end{bmatrix} +\left[\langle(1,0),(2,2)\rangle,\{r1_{1}\}\right]\quad\left[ \langle(1,1),(2,1)\rangle,\{r2_{1}\}\right]\end{bmatrix}\]  
\[\mathbb{M}_{K}^{2}=\mathbb{M}_{K}^{1}\cdot\mathbb{M}_{F}^{1}=\mathbb{M}_{K}^{ 1}\cdot \begin{bmatrix}f_{mult}(c,1)&f_{1}(c,1)+f_{2}(c,1)&f_{3}(c,1)\\ f_{mult}(c,1)&f_{null}(c,1)&f_{4}(c,1)\\ f_{mult}(c,1)&f_{null}(c,1)&f_{null}(c,1)\end{bmatrix} \]  
\[\mathbb{M}_{K}^{2}= \begin{bmatrix}0&0&\langle(1,2),(2,2)\rangle,\{r1_{1},r1_{ 2}\}\end{bmatrix} +\left[\langle(2,2),(1,2)\rangle,\{r1_{1},r1_{2}\}\right] \end{bmatrix}\]  
The interpretation of each of the above \(\mathbb{M}_{K}^{1}\) matrices is as follows. The \(\mathbb{M}_{K}^{1}\) matrix indicates that with just one action there are two ways for the system to be in the state where one of the agents move to the area of type \(B\) and the other one will not take any action (as it is pointed out by the fact that its time is equal to 0). Both ways require specified agent to carry out the action represented by the \(r1\) rule. The same matrix also gives us information that with one action there is a possibility to reach s2 state if both agents engage in cooperative execution of \(r2\) rule. Finally, the \(\mathbb{M}_{K}^{2}\) points out two walks in the state space that lead to the s2 state. Both involve performing the action associated with \(r1\) rule two times (each time by a different agent). It is worth pointing out that in a software implementation of the above algorithm labels should denote specific transition functions rather than reaction rules. While for this particular example it was sufficient to indicate what \"kind\" of changes (i.e., reaction rules) need to occur in the system for automated generation of behavior policies it is necessary to distinguish exactly what transformation (including who participated in a specific transformation) is required.