# Hierarchical Concept Learning by Fuzzy Semantic Cells
## 2 Fuzzy Semantic Cell  
[PERSON] and [PERSON] [10; 11] introduced a novel cognitive structure \(L=\langle P,d,\delta\rangle\) to model the vague concept having the form \"about P\", \"similar to P\" or \"close to P\". In this paper, we continue the definition in [PERSON] and [PERSON]'s work [15] named Fuzzy Semantic Cell. **Definition 1**. : _A fuzzy semantic cell for a vague concept \(L_{i}\) on the domain \(\Omega\) is a triple representation \(L_{i}=\langle P_{i},d,\delta_{i}\rangle\) where \(P_{i}\) is the prototype of \(L_{i}\), \(d\) is the distance metric which measures the neighborhood size of \(L_{i}\), and \(\delta_{i}\) is the probability density function of the neighborhood size of the vague concept \(L_{i}\)._  
In other words, a fuzzy semantic cell for a vague concept \(L_{i}\) is made up of a semantic cell nucleus represented by the prototype \(P_{i}\) and a semantic cell membrane represented by an uncertain boundary of \(L_{i}\) using \(d\) and \(\delta_{i}\). Hence, the fuzzy semantic cell \(L_{i}=\langle P_{i},d,\delta_{i}\rangle\) is assumed to be the smallest semantic unit of vague concepts. Figure 0(a) shows a fuzzy semantic cell \(L_{i}=\langle P_{i},d,\delta_{i}\rangle\) in two-dimensional space as an illustrative example. **Definition 2**. : _On the domain \(\Omega\), for any fuzzy semantic cell \(L_{i}=\langle P_{i},d,\delta_{i}\rangle\), and neighborhood size \(\epsilon\geq 0\), the \(\epsilon\)-neighborhood \(\mathcal{N}^{\epsilon}_{L_{i}}\) of \(L_{i}\) is defined as follows:_  
\[\mathcal{N}^{\epsilon}_{L_{i}}=\{x\in\Omega:d(x,P_{i})\leq\epsilon\}. \tag{1}\]  
According to Definition 2, the \(\epsilon\)-neighborhood \(\mathcal{N}^{\epsilon}_{L_{i}}\) includes all the elements \(x\in\Omega\) whose distance to the prototype \(P_{i}\) is less than the given neighborhood size \(\epsilon\) of \(L_{i}\), as shown in Figure 0(a). Since \(\epsilon\) is a random variable with the probability density function \(\delta_{i}\), \(\mathcal{N}^{\epsilon}_{L_{i}}\) can be considered as a random set neighborhood of \(L_{i}\). Therefore, for any \(x\in\Omega\), the degree of \(x\) belonging to the fuzzy semantic cell \(L_{i}\) should be equal to the probability that the \(\epsilon\)-neighborhood \(\mathcal{N}^{\epsilon}_{L_{i}}\) of \(L_{i}\) is a set containing \(x\), denoted by \(\text{Prob}(\epsilon\colon x\in\mathcal{N}^{\epsilon}_{L_{i}})\). Then, the neighborhood function of \(L_{i}\) can be defined as follows. **Definition 3**. : _On the domain \(\Omega\), for any fuzzy semantic cell \(L_{i}=\langle P_{i},d,\delta_{i}\rangle\), and \(x\in\Omega\), the neighborhood function \(\mu_{L_{i}}(x)\) of \(L_{i}\) is defined as follows:_  
\[\mu_{L_{i}}(x)=\text{Prob}(\epsilon\colon x\in\mathcal{N}^{\epsilon}_{L_{i}})= \text{Prob}(\epsilon\colon\epsilon\geq d(x,P_{i}))=\int_{d(x,P_{i})}^{+\infty} \delta_{i}(\epsilon)de. \tag{2}\]We note that \(\mu_{L_{i}}(x)\in[0,1]\) and \(\mu_{L_{i}}(x)\) is a decreasing function of the distance \(d(x,P_{i})\). In particular, when \(d(x,P_{i})=0\), \(\mu_{L_{i}}(x)=1\); when \(d(x,P_{i})\rightarrow+\infty\), \(\mu_{L_{i}}(x)=0\). Thus \(\mu_{L_{i}}(x)\) can be the belief of a sample \(x\) being a neighbor of the the fuzzy semantic cell \(L_{i}\). Figure 1b shows an example of the neighborhood function \(\mu_{L_{i}}(x)\) of \(L_{i}\). Since \(\mu_{L_{i}}(x)\) is a decreasing function of the distance \(d(x,P_{i})\), an exponential form of \(\mu_{L_{i}}(x)\) is defined in [24] as follows:  
\[\mu_{L_{i}}(x)=\int_{d(x,P_{i})}^{+\infty}\delta_{i}(\epsilon\mid\sigma_{i})d \epsilon=\exp\left(-\frac{d(x,P_{i})}{2\sigma_{i}^{2}}\right) \tag{3}\]  
where \(\sigma_{i}\) is the parameter of the probability density function \(\delta_{i}\), and it is related to the extent of the distribution.