*3.2. Second Layer*

The second layer comprises fuzzy logic neurons that use fuzzy operators [49]. The most well-known is the *t-norm*, ( *T* : [0, 1] 2 → [0, 1]), which represents the intersection of fuzzy sets, and the *t-conorm* (*S* : [0, 1] 2 → [0.1]), capable of the union of fuzzy sets. In this paper, these operations are represented by [49]:

$$t(\mathbf{x}, y) = \mathbf{x} \times \mathbf{y} \tag{35}$$

$$s(\mathbf{x}, y) = \mathbf{x} + y - \mathbf{x} \times y \tag{36}$$

Neurons that use t-norm and t-conorm are extremely common in constructing evolving fuzzy neural networks; and-neuron and or-neuron stand out. These fuzzy neural structures, proposed first by Hirota and Pedrycz [50], can aggregate the model inputs (in the case of this paper, the Gaussian neurons of the first layer) with their respective weights. They are considered operational units capable of nonlinear multivariate operations in unit hypercubes ([0,1] → [0, 1] *n*) with fuzzy inputs and weights. This process generates a unique value that allows the construction of fuzzy rules in the IF–THEN format. Their interpretability is directly related to the operators applied in the procedures of these fuzzy neuronal structures.

In these neurons with two levels, the first one executes aggregate operations, and the last completes the procedures with fuzzy operators. This operation simplifies the aggregation of the fuzzy input with its respective weight, creating interpretable connectivity between the antecedent connectives [51]. In the case of or-neurons (the neuron structure used in this paper), all or-type connectors are present in the connection of all antecedents of a fuzzy rule generated by this structure [50]. It can be exemplified by:

$$\mathbf{z} = or \textit{neuron} (\mathbf{w}; \mathbf{a}) = \mathrm{S}\_{i=1}^{n} (w\_i \ t \ a\_i) \tag{37}$$

where *S* is a *t-conorms* and *t* is *t-norms*, **a** = [*<sup>a</sup>*1, *a*2, ..., *a*3, ...*aN*] is the neuron's inputs (values of fuzzy relevance (Gaussian neurons)), and **w** = [*<sup>w</sup>*1, *w*2, ..., *w*3, ... *w N*] represents the weights, **a, w** ∈ [0, 1] *n*, defined in the range between 0 and 1. Finally, **z** is the neuron's output, which is also seen as a fuzzy rule. This neuron can generate fuzzy rules as shown below:

$$\begin{aligned} \text{Rule}\_1: \text{ If } \mathbf{x}\_1 \text{ is } A\_1^1 \text{ with } impact \, w\_{11} \dots \\ \text{or } \mathbf{x}\_2 \text{ is } A\_1^2 \text{ with } impact \, w\_{21} \dots \\ \text{or } \mathbf{x}\_n \text{ is } A\_1^n \text{ with } impact \, w\_{n1} \dots \\ \text{Then } \mathbf{y}\_1 \text{ is } \mathbf{v}\_1 \\ \text{Rule}\_L: \text{ If } \mathbf{x}\_1 \text{ is } A\_L^1 \text{ with } impact \, w\_{1L} \dots \\ \text{or } \mathbf{x}\_2 \text{ is } A\_L^2 \text{ with } impact \, w\_{2L} \dots \\ \text{or } \mathbf{x}\_n \text{ is } A\_L^n \text{ with } impact \, w\_{nL} \dots \\ \text{Then } \mathbf{y}\_L \text{ is } \mathbf{v}\_L \end{aligned} \tag{38}$$

with *A*<sup>1</sup> *i* , ... , *An i* fuzzy sets represented as linguistic terms for the *n* inputs appearing in the *i*th rules, and *y*1, ... , *yL* are consequent terms (output); *L* is the number of rules, and **v** represents the correspondence value for the output [17].
