## 3 Initial Conditions and Definitions of the Algorithm
### Agent Representation  
**Definition 4**.: _Let \(A=\{A_{1},A_{2},...,A_{f}\}\) be a set of agents, where \(A_{f}\) represents a set of agents with the same type/ability, which can operate in the \(i\) subspace type, \(f\in<1,i>\)._  
**Definition 5**.: _Let \(A_{f}=\{a_{\{f,k,1\}},a_{\{f,k,2\}},...,a_{\{f,k,v\}}\}\) be a set of agents that can operate in the \(f\) subspace type \(f\in<1,i>\), where \(k\) represents a subspace identifier, in which the agents are operating \(k\in<1,n>\) and \(v\) represent a number of available agents for the exploration of this type \(f\) of space._  
We assume that \(\forall i\exists A_{f}\) with minimal \(v=1\) (a number of available agents). This is a necessary condition so that the entire space \(S_{real}\) can be explored by agents.  
**Definition 6**.: _The \(As_{x}=\{a_{\{f,x,1\}},a_{\{f,x,2\}},...,a_{\{f,x,v\}}\}\) is a set of agents operated in a subspace \(s_{\{x,f\}}\), \(x\in<1,n>,f\in<1,i>,o\in<1,\|A_{f}\|>\)._  
Figure 1: A heterogeneous exploration space \(S_{real}\) consisting of three independent subspaces.  
Definition 7: Let \(S_{real}\) have an entrance \(e_{S_{real}}\), \(e_{S_{real}}\subset s_{\{n,i\}}\) that an agent \(a_{\{i,n,v\}}\) may use to enter its subspace \(s_{\{n,i\}}\) and start to fulfill an exploration task.  
Definition 8: The \(a_{\{f,k,v\}}\) agent is represented by the following variables,  
\[\begin{array}{ll}a_{\{f,k,v\}}=\{TRa_{\{f,k,v\}},TCa_{\{f,k,v\}},CCa_{\{f,k,v \}},ta_{\{f,k,v\}},TPea_{\{f,k,v\}},\\ &\quad\quad TRr_{\{f,k,v\}},T\Cr_{\{f,k,v\}},CCr_{\{f,k,v\}},tr_{\{f,k,v\}}, TPer_{\{f,k,v\}}\},\end{array} \tag{1}\]  
where  
* \(TRa_{\{f,k,v\}}(TRr_{\{f,k,v\}})\), respectively, are the time periods used when adding (removing) an agent into (from) the group \(As_{k}\);
* \(TCa_{\{f,k,v\}}(TCr_{\{f,k,v\}})\), respectively, are the thresholds on the number of conflicts required to initiate a request for adding (removing) an agent to (from) the group \(As_{k}\), and the following conditions must be met \(TCa_{\{f,k,v\}}\leq TRa_{\{f,k,v\}}\) (\(TRr_{\{f,k,v\}}\geq T\Cr_{\{f,k,v\}}\));
* \(ta_{\{f,k,v\}}(tr_{\{f,k,v\}})\) ), respectively, are timers of exploration time used when adding (removing) the agents;
* \(CCa_{\{f,k,v\}}(CCr_{\{f,k,v\}})\), respectively, are the conflict counters used when adding (removing) an agent into (from) the group \(As_{k}\); and
* \(TPea_{\{f,k,v\}}(TPer_{\{f,k,v\}})\), respectively, represent the thresholds on the pheromone levels for adding (removing) the agent to (from) the group \(As_{k}\).  
Definition 9: Let the agent \(a_{\{f,k,v\}}\) operating in the subspace \(s_{\{k,f\}}\) be at the entrance \(e_{y_{m}}\) belonging to the subspace \(s_{\{y,u\}}\) only if \(s_{\{k,f\}}\) and \(s_{\{y,u\}}\) are adjacent subspaces \(y\sim k\lor k,y\in<1,n>\) and the type of spaces \(f\sim u,\lor f,u\in<1,i>\) (otherwise, the robot would not need transport). Let us assume that there is an agent \(a_{\{u,y,l\}}\) capable of operating in the subspace \(s_{\{y,u\}}\) and waiting at the entrance \(e_{k_{m}}\) to be transported (an agent \(a_{\{u,y,l\}}\) cannot in \(s_{\{k,f\}}\) space), and the agent \(a_{\{f,k,v\}}\) can transport the agent \(a_{\{u,y,l\}}\) to the entrance \(e_{y_{m}}\) through the subspace \(s_{\{k,f\}}\).