# A Lookahead Behavior Model for Multi-Agent Hybrid Simulation
## 3 Problem Description
### Context Overview  
This section further describes the military context, as well as the need for a hybrid simulation. In the military context, the typical CAS model can represent military commanders [37; 38]. A state-of-the-art military agent model is a perception-based agent model. The commander agent is created to model the ability of command and control, which can be achieved by simulating \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt}} \hline \hline
**Symbol** & **Nomenclature** \\ \hline \(N_{agents}\) & The number of agents in the system \\ \hline \(agent_{k}\) & The \(k\)th agent, \(1\leq k\leq N_{agents}\) \\ \hline \(d_{j}^{k}(t)\) & The act performed by the \(k\)th agent at time \(t\) and this action is the \(j\)th action it performs, \(1\leq k\leq N_{agents}\) \\ \hline \(T\) & The time set \\ \hline \(t_{current}\) & The current simulation time, \(t_{current}\in T\) \\ \hline \(t_{start}(a_{j}^{k})\) & The start time of the \(j\)th action \(a_{j}^{k}\) for the \(k\)th agent \(agent_{k}\), \(t_{start}(a_{j}^{k})\in T\) \\ \hline \(t_{duration}(a_{j}^{k})\) & The duration of the \(j\)th action \(a_{j}^{k}\) for the \(k\)th agent \(agent_{k}\), \(t_{duration}(a_{j}^{k})\in T\) \\ \hline \(s_{t}^{k}\) & The state of the \(k\)th agent at time \(t\) where \(1\leq k\leq N_{agents}\) \\ \hline \(s_{t}^{k}(t_{current})\) & The estimated state of the \(k\)th agent at time \(t\) predicted at time \(t_{current}\) where \(t^{\prime}\geq t_{current}\) \\ \hline \(S^{k}=\cup s_{t}^{k}\) & The set of all the states of the \(k\)th agent \\ \hline \(S^{k}=\cup s_{t}^{k}\) & The set of all the lookahead states of the \(k\)th agent \\ \hline \(\Lambda_{j+1}^{k}=(a_{0}^{k}(t_{0}),a_{1}^{k}(t_{1}),\ldots,a_{j}^{k}(t_{j}))\) & The sequence of all actions performed by the \(k\)th agent before time \(t=t_{j+1}\) \\ \hline \(\overline{\Gamma}=\left\{t_{0},t_{1},\ldots,t_{j}\right\}\) & The set of the beginning time of acts performed by the \(k\)th agent before time \(t=t_{j+1}\) \\ \hline \(\Lambda^{k}=\cup\Lambda_{j}^{k}\) & The set of all the actions of the \(k\)th agent \\ \hline \(\Lambda=\cup\Lambda^{k}\) & The set of the system’s actions \\ \hline \(\sum=\cup S^{k}\) & The set of the system’s states \\ \hline \(\delta_{\text{int}}^{k}:S^{k}\to S^{k}\) & The internal state transition function of the \(k\)th agent; \\ \hline \(\delta_{ext}^{k}:\sum\times T\times\Lambda\to S^{rk}\) & The external state transition function of the \(k\)th agent; it indicates that the state transition of the \(k\)th agent occurred at one specific moment is related to the system’s existing acts and states \\ \hline \(\delta_{pre}^{k}:\sum\times T\times\Lambda\to S^{rk}\) & The lookahead function of the state of the \(k\)th agent; this implies that the state of the \(k\)th agent predicted at one specific moment is related to the system’s existing acts and states \\ \hline \(F_{ext}^{2}=F_{ext}^{2}(t,s_{t}^{1})\) & A compound function of \(t\) and \(s_{t}^{1}\) for the second agent \\ \hline \(\delta_{ext}^{2}:\sum\cdot\delta_{ext}^{n}(t,s_{t}^{3},\ldots,s_{t}^{n},s_{t}^ {1})\) & A compound function of \(t\) and \(s_{t}^{1}\) for the second agent \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Table of symbols. the commanders' decision-making process, sensors, weapons, mobile platforms, and communication facilities to conduct processing, detection, lethality, mobility, and communication, respectively [39; 40]. For example, in the Joint Warfare System (JWARS) [41], a Battle Space Entity (BSE), which is the basic element to represent military forces and systems, is composed of C2, Sensor, Platform, Resource Manager, and Communication Manager. The sensor uses different detection radii and probabilities to represent different detection capabilities. Weapons have distinct killing probabilities and ranges. The maneuver platforms have their own moving speeds.