# Artificial Empathy and Imprecise Communication in a Multi-Agent System
## 3 Proposed Model
### Fuzzy Agent State  
The formal representation of an agent's \(\mathcal{A}\) state, encompassing both its environmental perception and its internal knowledge, is divided into two primary components: \(\mathbf{X}^{\mathcal{A}}\), signifying the emotional state or subjective knowledge, and \(\mathbf{Y}^{\mathcal{A}}\), indicating the cognitive state or objective environmental data. The emotional state, \(\mathbf{X}^{\mathcal{A}}\), could include parameters such as the agent's energy level, current operational status, and signals for assistance. The cognitive state, \(\mathbf{Y}^{\mathcal{A}}\), is concerned with external environmental factors such as the agent's proximity to its objective, the presence of neighboring agents, and the identification of obstacles. We represent such states as:  
\[A=\left(\mathbf{X}^{\mathcal{A}},\mathbf{Y}^{\mathcal{A}}\right). \tag{4}\]  
Notably, the elements within \(\mathbf{X}^{\mathcal{A}}\) and \(\mathbf{Y}^{\mathcal{A}}\) are expected to be independent, as they pertain to distinct aspects of the agent's state. Nonetheless, interdependencies between certain elements of \(\mathbf{X}\) and \(\mathbf{Y}\) may exist, reflecting the inevitable connection between the agent's internal state and its external perception. Three variants of state representation can be considered. In all of them, both components of the agent's state are characterized as various types of fuzzy sets:  
\[\mathbf{X}^{\mathcal{A}}=\left(x_{1},x_{2},\cdots,x_{n}\right)\text{ and }\mathbf{Y}^{\mathcal{A}}=\left(y_{1},y_{2},\cdots,y_{m}\right). \tag{5}\]  
Figure 1: Overview of the decision-making scheme of the proposed model from a learning perspective. In the first variant, we use fuzzy sets, where \(x_{i},y_{j}\in[0,1]\) for \(i=1,\ldots,n\) and \(j=1,\ldots,m\). These values represent the degree of presence or relevance of certain features or parameters in the agent's state. This variant assumes no uncertainty regarding the state of the robot and its environment. However, this approach is not always feasible, as imperfection in knowledge is common. Uncertainty may arise from both ambiguous interpretations of the current situation and imperfections or lack of access to measurement sensors. Furthermore, communication between agents can be a significant source of uncertainty. Even when the state \(\left(\mathbf{X}^{A},\mathbf{Y}^{A}\right)\) is not uncertain, communication between agent A and B can introduce it. For example, the actual state \(\left(\mathbf{X}^{A},\mathbf{Y}^{A}\right)\) can be distorted both at the stage of encoding the message \(\left(\mathbf{X}^{A,A},\mathbf{Y}^{A}\right)\) and then at the stage of transmission/reception of the message \(\left(\mathbf{X}^{\prime\prime,A},\mathbf{Y}^{\prime\prime,A}\right)\). Despite this, the agent must be able to make effective decisions. Therefore, the next two models proposed in this work assume representations of uncertainty already at the level of the internal state of the agent, where both components of the agent's state are characterized as:  
1. Interval-valued Fuzzy Sets [24], where: \[x_{i}=\left[\underline{x}_{i},\overline{x}_{i}\right];\] (6)
2. Hesitant Fuzzy Sets [25], where: \[x_{i}\in\left[0,1\right].\] (7)  
Regardless of the formal framework, these (uncertain) values represent the degree of presence or relevance of certain features or parameters in the agent's state. The state representation \(A\) is not static but dynamic, requiring periodic updates to reflect changes in both the agent's internal state and its interactions with the environment. Thus, the agent's performance can be viewed as a set of decisive actions leading to a goal. Such kinds of processes can be viewed as a set of states resulting from performing those actions, where each action is a result of a decision. This approach is key for managing uncertainty, allowing for flexibility and robustness in the agent's behavior. It ensures the agent's effectiveness in dealing with the unpredictable and varying nature of real-world situations.