### Behavior Policy  
We define a behavior policy as a schedule of actions for each object from the beginning of a mission to its end that meets all the requirements listed in Section 2.1. \begin{table}
\begin{tabular}{c c} \hline \hline
**Function** & **Function Definition** \\ \hline \(f_{1}\) & \(f_{1}(c,t)=\left\{\begin{array}{cl}[\langle(b,y),(a,x+2)\rangle,H^{\prime} \cup\{r1_{t+1}\}]&:&c=[\langle(a,x),(b,y)\rangle,H^{\prime}]\\ 0&:&c=0\\ \end{array}\right.\) \\ \(f_{2}\) & \(f_{2}(c,t)=\left\{\begin{array}{cl}[\langle(a,x),(b,y+2)\rangle,H^{\prime} \cup\{r1_{t+1}\}]&:&c=[\langle(a,x),(b,y)\rangle,H^{\prime}]\\ 0&:&c=0\\ \end{array}\right.\) \\ \(f_{3}\) & \(f_{3}(c,t)=\left\{\begin{array}{cl}[\langle(a,z+1),(b,z+1)\rangle,H^{ \prime}\cup\{r2_{t+1}\}]&:&c=[\langle(a,z),(b,z)\rangle,H^{\prime}]\\ 0&:&c\
eq[\langle(a,z),(b,z)\rangle,H^{\prime}]\\ \end{array}\right.\) \\ \(f_{4}\) & \(f_{4}(c,t)=\left\{\begin{array}{cl}[\langle(b,y),(a,x+2)\rangle,H^{\prime} \cup\{r1_{t+1}\}]&:&c=[\langle(a,x),(b,y)\rangle,H^{\prime}]\\ 0&:&c=0\\ \end{array}\right.\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mission progress function definitions for the state space presented in Figure 2. The action represented by \(r_{1}\) reaction rule is assumed to take 2 units of time while the action \(r_{2}\) takes only 1 unit of time. Figure 2: The state space generated from Tracking Transition System defined in Table 3. Mission progress functions definitions are defined in Table 4.  
Having a state space, we can view a behavior policy as a walk (in graph theory sense) indicating what changes (and who did them) are required in order to reach a desired state. To construct a proper policy behavior based on a state space, we need to define the following elements. Please note that by series we will understand a finite sum of elements. * \(K^{t}_{\mathrm{s}}=c_{1}+\cdots+c_{m}=\sum_{i=1:-m}c_{i}\quad c_{i}\in C,s\in \{0,\cdots,n_{s}-1\},t\in\mathbb{N}\)--a series, where summands are mission courses leading to the state \(s\);
* \(N_{K}(K^{t}_{\mathrm{s}})\in\mathbb{N}\)--a function returning a number of elements in a given series. According to the earlier definition, for any series \(K^{t}_{\mathrm{s}}\) this function returns a value of \(m\) (the greatest index of \(c_{i}\));
* \(F_{i,j}(x,t)=\sum_{k\in T_{ij}}f_{k}(x,t)\quad i,j\in\{0,\cdots,n_{s}-1\},t\in \mathbb{N}\)--a series, whose summands are mission progress functions from the \(i\) to the \(j\) state;
* \(\mathbb{M}^{t}_{K}=\left[K^{t}_{0}\quad\cdots\quad K^{t}_{n_{s}-1}\right],t \in\mathbb{N}\)--a matrix whose elements are series indicating possible walks leading to each state. Index \(t\) denotes a number of steps made in a state space. By a step we understand a transition between vertices (including the situation where traversal does not change the vertex);
* \(\mathbb{M}^{t}_{F}=\left[ \begin{array}{ccc}F_{0,0}(x,t)&\cdots\quad F_{0,n_ {s}-1}(x,t)\\ \cdots\quad\cdots\quad\cdots\\ F_{n_{s},0}(x,t)&\cdots\quad F_{n_{s}-1,n_{s}-1}(x,t)\end{array} \right]\)--a matrix of transitions between states. Furthermore, we define two operations:
* \(K^{t}_{\mathrm{s}}\circ F_{i,j}(x,t)=\sum_{k\in T_{ij}}\sum_{l=1:-N_{K}(K^{t} )}f_{k}(c_{l},t)\)--a convolution of the series defined above;
* \(\mathbb{M}^{t+1}_{K}=\mathbb{M}^{t}_{K}\cdot\mathbb{M}^{t}_{F}\)--a multiplication of the matrices defined above. Elements of the new matrix are defined by the formula: \[K^{t+1}_{\mathrm{s}}=\sum_{k=0}^{n_{s}-1}K^{t}_{k}\circ F_{K,s}(x,t)\]  
In order to generate all walks consisting of a specified number of steps from an initial state to a final state one must define the initial state, as a \(\mathbb{M}^{0}_{K}\) matrix and multiply subsequent results by \(\mathbb{M}^{i}_{F}\) the specified number of times. The result will be a \(\mathbb{M}^{x}_{K}\) matrix, whose summands in the \(i\)th column will indicate all possible walks with \(x\) steps that end in the \(i\)th state of the state space. If the element in the specified column is equal to \(0\), it means there is no such walk.