### State Space  
Having a Tracking Transition System we can transform it into a state space of the modeled system. A state space can be later used to generate a behavior policy for agents (as defined in Section 2.1) in the system. We assume the following about modeled systems:  
1. A number of passive and active objects is constant during whole mission;
2. A system cannot change its state without an explicit action of an agent (alone or in cooperation with other agents);
3. No actions performed by agents are subject to uncertainty;
4. A mission can end for each agent separately in different moments. In other words, agents do not have to finish their part of the mission all at the same time;
5. In case of actions involving multiple objects (whether these are active or passive), it is required of all participants to start cooperation at the same moment. A state space \(SS\) of a system consisting of \(n_{o}\) objects and \(n_{s}\) states is defined as:  
\[\mathcal{SS}=(S,E,L,I,C,T,M_{f})\]  
where:  
* \(S\subset\mathbb{N}\)--a set of states in the state space. It corresponds to bigraphs in the Tracking Transition System;
* \(E\subseteq S\times\mathbb{S}\)--a multiset of ordered pairs of states. Elements in this set are directed edges representing transition relations between states;
* \(L\)--a set of labels of changes in the system. It will usually consist of reaction rule names from the Tracking Transition System the state space originated from. To determine what changes, in what order, have led to a specific state we will additionally introduce set \(H=\{l_{t}|l\in L,t\in\mathbb{N}\}\). Elements of the \(H\) set indicate what action (label) took place in what order (index value). * \(I=\{\mathbb{N}_{1}^{2}\times\cdots\times\mathbb{N}_{n_{s}}^{2}\}\)--a set of possible state-at-time (SAT) configurations. The interpretation of elements in such a set is as follows. The first element in each of inner tuples denotes \(id\) of an object (either passive or active) in the system. The second element in each inner tuple is meant to represent time at which the object specified by the \(id\) is at. For example, for \(n_{o}=2\) the element \(i_{x}=\langle(1,777),(2,123)\rangle\) denotes a situation where the object with id 1 is at the moment 777 while the object with id 2 is at the moment 123. * \(C=(I\times 2^{H})\cup\{0\}\)--a set of possible mission courses. 0 denotes the neutral element, i.e., \(\forall_{x\in C}x+0=0+x=x\). For the rest of the elements of \(C\) set the \(+\) symbol serves only as an associative conjunction operator and does not denote any meaningful operation. In other words for the rest of the elements the following formula is true: \(\forall_{x,y\in C\setminus\{0\}}x+y=y+x\). * \(T=\{f_{i}:C\times\mathbb{N}\rightarrow\mathbb{C}|i\in\mathbb{N}\}\cup\{f_{ null}\}\)--a set of functions defining progress of a mission. The \(f_{null}\) function returns 0 regardless of input. Additionally, we will denote by \(T_{i,j}\subset T\) a set of all mission progress functions from the \(i\) state to the \(j\) state. * \(M_{f}:E\to T\)--a bijective mapping of edges to mission progress functions. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Apl** & **Agt** & **Par** & **Res** \\ \hline \(\langle\text{s}0,r_{1}\rangle\) & s1 & \(\{(0,0),(1,1),(3,2)\}\) & \(\{(0,0),(1,2),(2,3),(3,1)\}\) \\ \(\langle\text{s}0,r_{1}\rangle\) & s1 & \(\{(0,0),(2,1),(3,2)\}\) & \(\{(0,0),(1,1),(2,3),(3,2)\}\) \\ \(\langle\text{s}0,r_{2}\rangle\) & s2 & \(\{(0,0),(1,1),(2,2),(3,3)\}\) & \(\{(0,0),(1,3),(2,1),(3,2)\}\) \\ \(\langle\text{s}1,r_{1}\rangle\) & s2 & \(\{(0,0),(1,1),(2,2)\}\) & \(\{(0,0),(1,2),(2,3),(3,1)\}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The Tracking Transition System for the introductory example. Each row defines a single transition in the system. Going back to our introductory example, we will now convert the Tracking Transition System from Table 3 into a state space of the system. We will not define all of the formal elements and rather focus on the key ones.