The FNN in this study ( Figure 1 ) is considered to be a private case of NFS to generate fuzzy rules and MFs. Note that the core of the system is multilayered network-based structure [1] . Such a system would generate both fuzzy rules and MFs. The source of exemplar input-output data would be described later.

A more detailed scheme of neuro-fuzzy system is depicted in Figure 2 . For simplicity’s sake we presented only two inputs X₁ and X₂ and one output Z.

The first layer of neurons simply distributes inputs of the system among neurons of the subsequent layer. The second layer consists of several groups of neurons equivalent to the number of inputs (for our case 2).

Neurons in each group represent MFs for fuzzy labels used as values for the input connected with this group. Output of every such neuron is value of membership of the input to the corresponding fuzzy level. This process is called “fuzzification” and these neurons are “fuzzifiers.”

Neurons of the third layer represent fuzzy rules. Number of neurons in this layer can be the same as the number of rules in the logical system IF-THEN.

Neurons of the fourth layer determine MFs of fuzzy labels. Neurons of this layer perform the most complex operation, called Compositional Rule of Inference (CRI). Thus, the output MF is determined.

In the fifth layer the defuzzification procedure is performed. This means determination of crisp value output based on inferred fuzzy value.

Figure 2 shows a detailed structure of neuro-fuzzy system, which is like one for BPNN, mentioned in previous section, and hence allows investigation by the similar methods, but there are some differences. In NFS each neuron is specified not by a set of weight/threshold/universal activation function only, but also by complex processing unit with an individual function and set of parameters. And lastly the neurons between consecutive layers are not fully-connected unlike in case of traditional BPNN [1] .

As it was mentioned above, in this study we first are examining well known [1] approach to improve FRM by using genetic-based learning mechanism. Later we propose our alternative way to build FRM, which has significant precision advantages and does not require any adjustment/learning.

In [1] it was stated that the selection of acceptable MFs is generally a subjective decision, but change in MFs may significantly alter the performance of the fuzzy models. It was claimed that the genetic algorithm (GA) allows to generate an optimal set of parameters for the fuzzy model, based either on their initial subjective selection or on a random selection.

From now on we adopt the following fuzzy conditional statements to describe a particular knowledge-based state [1] :

IF x is A₁ THEN z is B₁

ALSO

IF x is A₂ THEN z is B₂

ALSO

…………… (2.1)

ALSO

IF x is A_q THEN z is B_q

where x and z are linguistic variables, and $A_1,\cdots ,A_q$ and $B_1,\cdots ,A_q$ are fuzzy sets on X and Z, respectively. The fuzzy conditional statements (2.1) can be formalized in the form of the fuzzy relation $R\left(X,Z\right)$

$R\left(X,Z\right)=ALSO\left(R_1,R_2,\cdots ,R_i,\cdots ,R_q\right)$(2.2)

where ALSO represents a sentence connective which combines the $R_i$ into the fuzzy relation, $R\left(X,Z\right)$ and $R_i$ denotes the fuzzy relation between X and Z determined by the i-th fuzzy conditional statement, in which z = B_i corresponds to i-th NNs label. The NN learning goal is to find pairs of fuzzy sets $A_i$ and $B_i$, $i=\overline{1,Q}$ such that the mean square error e² between the fuzzy model output values and experimental output values would be the smallest. The mean square error e² is calculated by formula

$e^2=\frac{{\displaystyle {\sum }_{i=1}^Q{\left(z_i^{\ast }-z_i\right)}^2}}{{\displaystyle {\sum }_{i=1}^Q{z}_i^{\ast 2}}}$(2.3)

where $z_i^{\ast }$ is the experimental output value of the object for some current value i; z_i is the corresponding fuzzy model output value; Q is number of experiments.

In [1] the demand function $z=x\cdot \sin \left(1/x\right)$ was used to generate the set of output values z. Results are presented in Table 1 .

Note that $x\in \left[0.15,0.45\right]$, $z\in \left[-0.21,0.358\right]$.

To compare our results with those from [1] we use the same linguistic descriptions of the relationship between x and z to specify the characteristics of the function:

IF x = small THEN z = zero

ALSO

IF x = bit larger than small THEN z = negative small

ALSO

IF x = larger than small THEN z = negative large

ALSO

IF x = smaller than medium THEN z = negative large

ALSO

IF x = bit smaller than medium THEN z = negative medium

ALSO

IF x = medium THEN z = negative small

ALSO (2.4)

IF x = bit larger than medium THEN z = zero

ALSO

IF x = larger than medium THEN z = positive small

ALSO

IF x = smaller than large THEN z = positive medium

ALSO

IF x = bit smaller than large THEN z = larger than medium

ALSO

IF x = large THEN z = smaller than large

All linguistic terms from (2.4) are defined in the following Table 2 .

In [1] it was assumed that to find the crisp output value corresponding to the input value x = 0.26 one had to successively apply the fuzzification, fuzzy logic inference mechanism and defuzzification. Experimental output value, found by formula

$z=x\cdot \sin \left(1/x\right)$, was $z=0.26\cdot \sin \left(1/0.26\right)=-0.17$.

In [1] membership degrees of values for both input fuzzy set, $A_i\subset U_X$, $\forall i\in \left[1,10\right]$ and output one $B_j\subset U_Z$, $\forall j\in \left[1,7\right]$, were determined by (6.1) from Appendix. From Figure 3 we see that variable x has 11 linguistic values, whereas the variable z has 8 (see Figure 4 ) in Appendix. All linguistic values are presented in Table 2 . The following is simulation results from [1] by (a.1):

μ_X(“0.26”) = 0/0 + 0/1 + 0/2 + 0.33/3 + 0.67/4 + 0/5 + 0/6 + 0/7 + 0/8 + 0/9 + 0/10

It was shown that the knowledge-based inference mechanism was applied. The rule base (2.4), consisting of fuzzy linguistic rules, was used. Consequences of multiple (11) rules resulted in the fuzzy output set (see Figure 5 ), constructed on universe U_Z and bounded by the following MF:

μ_z(“−0.17”) = 0.33/0 + 0.67/1 + 0/2 + 0/3 + 0/4 + 0/5 + 0/6 + 0/7.

Then defuzzification was applied. For this matter, the “center” of gravity defuzzification method (a.2) from Appendix was used (see Figure 6 ).

Output values for given input values were calculated in the same way (see Table 3 ). Note that fuzzy rules and MFs were generated heuristically. In [1] it was mentioned that these rules could not provide the model precision required. To achieve the latter, it is necessary to tune appropriately the rules, as well as the shape and the center of the MF. To this end GA was used.

$x=0.26;y=-0.155;e^2=7.7854\times {10}^{-3}$

In [1] it was shown that 11 fuzzy sets were used in linguistic rules preconditions (see Table 2 ). Consequently, it was encoded 11 × 3 − 2 = 31 points. Each point $u_i,i=\overline{1,31}$ took value from a domain $D=\left[a_i,b_i\right]\subseteq U$. It was supposed that if $u_1=0.15$ was the value from an interval $\left[0.12,0.16\right]\subseteq U$, then $u_2=0.18$ from [0.17, 0.20] etc. Then the processes of encoding and decoding were applied, they were described in both [1] and [3] . The GA algorithm is briefly described in Appendix.

The mean square errors for both original ( $e^2=11.53532\times {10}^{-3}$) and GA-based ( $e^2=6.24290\times {10}^{-3}$) fuzzy models are presented in Table 3 .

For each $i=\overline{1,Q}$, where Q is the number of exemplar input, we represent each FNN input $x^i$ as a fuzzy set, forming linguistic variable, described by a triplet of the form

$X=\left\{\langle x^i,U_X,\tilde{X}\rangle \right\},x^i\in T\left(u_x\right),i=\overline{1,Q}$,

where $T\left(u_x\right)$ is extended term set of the linguistic variable “Input” from Table 2 , $\tilde{X}$ is normal fuzzy set with correspondent MF ${\mu }_x:U_X\to \left[0,1\right]$. To normalize values of $x^{*}$ we use

$x_{norm}=\frac{x^i-x_{\min }}{x_{\max }-x_{\min }},i=\overline{1,Q}$,

We will use the following mapping

$\partial :\tilde{X}\to U_X|u_x=Ent\left[\left(Card{U}_X-1\right)\times x_{norm}\right]$,

where

$\tilde{X}={\displaystyle {\int }_{U_x}{\mu }_x\left(u_x\right)/u_x}$(3.1)

On the other hand, to determine the estimates of the MF in terms of singletons from (3.1) in the form ${\mu }_x\left(u_{x_j}\right)/u_{x_j}|\forall j\in \left[0,Card{U}_X\right]$ we propose the following procedure.

$\begin{array}{l}\forall j\in \left[0,Card{U}_X\right],\\ {\mu }_x\left(u_{x_j}\right)=1-\frac{1}{Card{U}_X-1}\times \left|j-Ent\left[\left(Card{U}_X-1\right)\times x_{norm}\right]\right|\end{array}$(3.2)

MF for an input from (3.2) is shown in Figure 7 .

The conceptual difference between our approach to define MF and the one, traditionally used in fuzzy control systems, is that we define all values of a linguistic variable over entire physical scale of input/output parameters via normalization mechanism and therefore mathematically reject the notion of interval based MFs.

Going forward for each $i=\overline{1,Q}$, where Q is the number of exemplar output, we also represent each FNN output $z^i$ as a fuzzy set, forming linguistic variable, described by a triplet of the form

$Z=\left\{\langle z^i,U_Z,\tilde{Z}\rangle \right\},z^i\in T\left(u_z\right),i=\overline{1,Q}$,

where $T\left(u_z\right)$ is extended term set of the linguistic variable “Output “from Table 2 , $\tilde{Z}$ is normal fuzzy set with correspondent MF ${\mu }_z:U_Z\to \left[0,1\right]$. We use the same normalization procedure

$z_{norm}=\frac{z^i-z_{\min }}{z_{\max }-z_{\min }}$, $i=\overline{1,Q}$,

With the following mapping $\Omega :\tilde{Z}\to U_Z|u_z=Ent\left[\left(Card{U}_Z-1\right)\times z_{norm}\right]$, where

$\tilde{Z}={\displaystyle {\int }_{U_Z}{\mu }_z\left(u_z\right)/u_z}.$(3.3)

On the other hand, similarly to the previous cases, to determine the estimates of the MF in terms of singletons from (3.3) in the form ${\mu }_z\left(u_{z_k}\right)/u_{z_k}|\forall k\in \left[0,Card{U}_Z\right]$ we propose the following procedure.

$\begin{array}{l}\forall k\in \left[0,Card{U}_Z\right],\\ {\mu }_z\left(u_z{}_k\right)=1-\frac{1}{Card{U}_Z-1}\times \left|k-Ent\left[\left(Card{U}_Z-1\right)\times z_{norm}\right]\right|,\end{array}$(3.4)

where MF for an output from (3.4) is shown in Figure 8 .

Given the fact that “Output” linguistic variable is represented by normal MF of the type (3.3) and for a goal of defuzzification we must find the value of index $k^{*}$, which corresponds to the following singleton value from (3.3), given (3.4)

$\exists k^{*}|\mu \left(u_{z_{k^{*}}}\right)/u_{z_{k^{*}}}==1,\forall k\in \left[0,Card{U}_Z\right]$,

and the value of Output $z_{k^{*}}\in \left[z_{\max },z_{\min }\right]$ would be defined as

$z_{k^{*}}=k^{*}\times \frac{z_{\max }-z_{\min }}{Card{U}_Z-1}+z_{\min }$(3.5)

To convert (3.1)-(3.4) into fuzzy logic-based statement and terms from Table 2 we use a Fuzzy Conditional Inference Rule (FCIR), formulated by means of “common sense” as a following conditional clause:

P = “IF ( $\tilde{X}$ is X), THEN ( $\tilde{Z}$ is Z)”(3.6)

In other words, we use fuzzy conditional inference of the following type [5] :

Ant 1: If Input is X then Output is Z

Ant 2: Input is X'

-------------------------------------------- (3.7)

Cons: Output is Z'.

where $X,X^{\prime }\subseteq U_X$ and $Z,Z^{\prime }\subseteq U_Z$.

Note that statements (3.6) and (3.7) represent “modus-ponens” syllogism. Given that we use the following type of implication [1]

$X\to Z=\{\begin{array}{l}\left(1-x\right)\cdot z,x>z,\\ 1,x\le z\end{array}$(3.8)

For practical purposes, described down below, we will use Fuzzy Conditional Rule (FCR) of the following type

$\begin{array}{l}R\left(A_1\left(x\right),A_2\left(z\right)\right)=\left(X\times U_X\to U_Z\times Z\right)\cap \left(\neg X\times U_X\to U_Z\times \neg Z\right)\\ ={\displaystyle {\int }_{U_X\times U_Z}\left({\mu }_x\left(u_x\right)\to {\mu }_z\left(u_z\right)\right)}\wedge \left(\left(1-{\mu }_x\left(u_x\right)\right)\to \left(1-{\mu }_z\left(u_z\right)\right)\right)/\left(u_x,u_z\right)\end{array}$(3.9)

Given (3.8) from (3.9) we are getting

$\begin{array}{c}R\left(A_1\left(x\right),A_2\left(z\right)\right)=\left({\mu }_x\left(u_x\right)\to {\mu }_z\left(u_z\right)\right)\wedge \left(\left(1-{\mu }_x\left(u_x\right)\right)\to \left(1-{\mu }_z\left(u_z\right)\right)\right)\\ =\{\begin{array}{l}\left(1-{\mu }_x\left(u_x\right)\right)\cdot {\mu }_z\left(u_z\right),{\mu }_x\left(u_x\right)<{\mu }_z\left(u_z\right),\\ 1,{\mu }_x\left(u_x\right)={\mu }_z\left(u_z\right),\\ \left(1-{\mu }_z\left(u_z\right)\right)\cdot {\mu }_x\left(u_x\right),{\mu }_z(u_z)<{\mu }_x\left(u_x\right).\end{array}\end{array}$(3.10)

Given a unary relationship $R\left(A_1\left(x^{\prime }\right)\right)=X^{\prime }$ one can obtain the consequence $R\left(A_2\left(z^{\prime }\right)\right)$ by CRI to $R\left(A_1\left(x^{\prime }\right)\right)$ and $R\left(A_1\left(x\right),A_2\left(z\right)\right)$ of type (3.10):

$\begin{array}{l}R\left(A_2\left(z^{\prime }\right)\right)=X^{\prime }\circ R\left(A_1\left(x\right),A_2\left(z\right)\right)\\ ={\displaystyle {\int }_{U_X}{\mu }_{x^{\prime }}\left(u_x\right)/u_x}\circ {\displaystyle {\int }_{U_X\times U_Z}\left({\mu }_x\left(u_x\right)\to {\mu }_z\left(u_z\right)\right)}\wedge \left(\left(1-{\mu }_x\left(u_x\right)\right)\to \left(1-{\mu }_z\left(u_z\right)\right)\right)/\left(u_x,u_z\right)\\ ={\displaystyle {\int }_{U_Z}{\displaystyle {\cup }_{x\in U_X}\left[{\mu }_{x^{\prime }}\left(u_x\right)\wedge \left({\mu }_x\left(u_x\right)\to {\mu }_z\left(u_z\right)\right)\wedge \left(\left(1-{\mu }_x\left(u_x\right)\right)\to \left(1-{\mu }_z\left(u_z\right)\right)\right)\right]/u_z}}\end{array}$(3.11)

Corollary 1.

If fuzzy sets $X\subseteq U_X$ and $Z\subseteq U_Z$ are defined as (3.1) and (3.3) respectively, and are represented by unimodal and normal MFs, and also $Card{U}_X\ne Card{U}_Z$, whereas $R\left(A_1\left(x\right),A_2\left(z\right)\right)$ is defined by (3.10), then the number of singles in matrix (3.10) is less or equal 2.

Proof:

Because of unimodality and normality of MFs from (3.1) and (3.3), given (3.10) and the fact that

$\begin{array}{l}\forall j\in \left[0,Card{U}_X\right],\forall k\in \left[0,Card{U}_Z\right]|Card{U}_X\ne Card{U}_Z\\ \to \frac{1}{Card{U}_X-1}\times \left|j-Ent\left[\left(Card{U}_X-1\right)\times x_{norm}\right]\right|\\ \ne \frac{1}{Card{U}_Z-1}\times \left|k-Ent\left[\left(Card{U}_Z-1\right)\times z_{norm}\right]\right|\end{array}$

the following is taking place.

1) The one single in a matrix is always there, because

$\frac{1}{Card{U}_X-1}\times \left|j-Ent\left[\left(Card{U}_X-1\right)\times x_{norm}\right]\right|=0$

and

$\frac{1}{Card{U}_Z-1}\times \left|k-Ent\left[\left(Card{U}_Z-1\right)\times z_{norm}\right]\right|=0$,

or

$j^{*}=Ent\left[\left(Card{U}_X-1\right)\times x_{norm}\right]$ and $k^{*}=Ent\left[\left(Card{U}_Z-1\right)\times z_{norm}\right]$

Therefore from (3.1) and (3.3)

$\begin{array}{l}\forall j\in \left[0,Card{U}_X\right]|\exists !j^{*}|{\mu }_x\left(u_{x_{j^{*}}}\right)/u_{x_{j^{*}}}=1;\\ |\forall k\in \left[0,Card{U}_Z\right]|\exists !k^{*}|{\mu }_z\left(u_{z_{k^{*}}}\right)/u_{z_{k^{*}}}=1;\\ \to {\mu }_R\left(u_{x_{j^{*}}},u_{z_{k^{*}}}\right)/\left(u_x,u_z\right)=1\end{array}$

2) The only second single in a matrix is when

$\frac{1}{Card{U}_X-1}\times \left|j-Ent\left[\left(Card{U}_X-1\right)\times x_{norm}\right]\right|=1$

and

$\frac{1}{Card{U}_Z-1}\times \left|k-Ent\left[\left(Card{U}_Z-1\right)\times z_{norm}\right]\right|=1$,

or

$\left|j-Ent\left[\left(Card{U}_X-1\right)\times x_{norm}\right]\right|=\left(Card{U}_X-1\right)$

and

$\left|k-Ent\left[\left(Card{U}_Z-1\right)\times z_{norm}\right]\right|=\left(Card{U}_Z-1\right)$,

which means $x_{norm}=1$ and $z_{norm}=1\to \exists !i^{*}\in \left[1,Q\right]|x^{i^{*}}=x_{\max }$, $z^{i^{*}}=z_{\max }$ and j = 0, k = 0. (Q. E. D.).

The aggregation (2.2) of knowledge-based situation (2.1) can be formalized in the form of the fuzzy relation $R\left(X,Z\right)$. We interpret a sentence connective ALSO as a fuzzy set Union

$R\left(X,Z\right)=R_{1}OR{R}_2\cdots OR\cdots R_i\cdots OR\cdots R_q$

In terms of (3.9)-(3.11) we use an aggregation of the following form

$R_{aggr}\left(A_1\left(x\right),A_2\left(z\right)\right)={\displaystyle {\cup }_{i=1}^Q{R}_i\left(A_1\left(x\right),A_2\left(z\right)\right)}$(3.12)

We use an experimental input/output value pairs from Table 1 .

Let us define the following in terms of neuro-fuzzy system. We are using 11 rules from (2.4). For input/output fuzzification we use (3.2) and (3.4) respectively. For FCR we use (3.10). For FCIR we use (3.11). For output defuzzification we use (3.5).

1) Neurons of the second layer (fuzzification) for rule 1:

μ_X(“small”) = μ_X(“0.15”) = 1.000/0 + 0.900/1 + 0.800/2 + 0.700/3 + 0.600/4 + 0.500/5 + 0.400/6 + 0.300/7 + 0.200/8 + 0.100/9 + 0.000/10

μ_Z(“zero”) = μ_Z(“0.056”) = 0.571/0 + 0.714/1 + 0.857/2 + 1.000/3 + 0.857/4 + 0.714/5 + 0.571/6 + 0.429/7

2) Neurons of the third layer (FCR) for rule 1:

R₁(A₁(x), A₂(z)) = (μ_X(“small”) → μ_Z(“zero”)) = (μ_X(“0.15”) → μ_Z(“0.056”)) =

3) Neurons of the second layer (fuzzification) for rule 2:

μ_X(“bit larger than small”) = μ_X(“0.18”) = 0.900/0 + 1.000/1 + 0.900/2 + 0.800/3 + 0.700/4 + 0.600/5 + 0.500/6 + 0.400/7 + 0.300/8 + 0.200/9 + 0.100/10

μ_Z(“negative small”) = μ_Z(“−0.12”) = 0.857/0 + 1.000/1 + 0.857/2 + 0.714/3 + 0.571/4 + 0.429/5 + 0.286/6 + 0.143/7

4) Neurons of the third layer (FCR) for rule 2:

R₂(A₁(x), A₂(z)) = (μ_X(“bit larger than small”) → μ_Z(“negative small”)) = (μ_X(“0.18”) → μ_Z(“−0.12”)) =

5) Neurons of the second layer (fuzzification) for rule 3:

μ_X(“0.21”) = 0.800/0 + 0.900/1 + 1.000/2 + 0.900/3 + 0.800/4 + 0.700/5 + 0.600/6 + 0.500/7 + 0.400/8 + 0.300/9 + 0.200/10

μ_Z(“−0.21”) = 1.000/0 + 0.857/1 + 0.714/2 + 0.571/3 + 0.429/4 + 0.286/5 + 0.143/6 + 0.000/7

6) Neurons of the third layer (FCR) for rule 3:

R₃(A₁(x), A₂(z)) = (μ_X(“larger than small”) → μ_Z(“negative large”)) = (μ_X(“0.21”) → μ_Z(“−0.21”)) =

7) Neurons of the second layer (fuzzification) for rule 4:

μ_X(“0.24”) = 0.700/0 + 0.800/1 + 0.900/2 + 1.000/3 + 0.900/4 + 0.800/5 + 0.700/6 + 0.600/7 + 0.500/8 + 0.400/9 + 0.300/10

μ_Z(“−0.205”) = 1.000/0 + 0.857/1 + 0.714/2 + 0.571/3 + 0.429/4 + 0.286/5 + 0.143/6 + 0.000/7

8) Neurons of the third layer (FCR) for rule 4:

R₄(A₁(x), A₂(z)) = μ_X(“smaller than medium”) → μ_Z(“negative large”) = (μ_X(“0.24”) → μ_Z(“−0.205”)) =

9) Neurons of the second layer (fuzzification) for rule 5:

μ_X(“0.27”) = 0.600/0 + 0.700/1 + 0.800/2 + 0.900/3 + 1.000/4 + 0.900/5 + 0.800/6 + 0.700/7 + 0.600/8 + 0.500/9 + 0.400/10

μ_Z(“−0.14”) = 0.857/0 + 1.000/1 + 0.857/2 + 0.714/3 + 0.571/4 + 0.429/5 + 0.286/6 + 0.143/7

10) Neurons of the third layer (FCR) for rule 5:

R₅(A₁(x), A₂(z)) = μ_X(“bit smaller than medium”) → μ_Z(“negative medium”) = (μ_X(“0.27”) → μ_Z(“−0.14”)) =

11) Neurons of the second layer (fuzzification) for rule 6:

μ_X(“0.3”) = 0.500/0 + 0.600/1 + 0.700/2 + 0.800/3 + 0.900/4 + 1.000/5 + 0.900/6 + 0.800/7 + 0.700/8 + 0.600/9 + 0.500/10

μ_Z(“−0.057”) = 0.714/0 + 0.857/1 + 1.000/2 + 0.857/3 + 0.714/4 + 0.571/5 + 0.429/6 + 0.286/7

12) Neurons of the third layer (FCR) for rule 6:

R₆(A₁(x), A₂(z)) = (μ_X(“medium”) → μ_Z(“negative small”)) = (μ_X(“0.3”) → μ_Z(“−0.057”)) =

13) Neurons of the second layer (fuzzification) for rule 7:

μ_X(“0.33”) = 0.400/0 + 0.500/1 + 0.600/2 + 0.700/3 + 0.800/4 + 0.900/5 + 1.000/6 + 0.900/7 + 0.800/8 + 0.700/9 + 0.600/10

μ_Z(“0.037”) = 0.571/0 + 0.714/1 + 0.857/2 + 1.000/3 + 0.857/4 + 0.714/5 + 0.571/6 + 0.429/7

14) Neurons of the third layer (FCR) for rule 7:

R₇(A₁(x), A₂(z)) = (μ_X(“bit larger than medium”) → μ_Z(“zero”)) = (μ_X(“0.33”) → μ_Z(“0.037”)) =

15) Neurons of the second layer (fuzzification) for rule 8:

μ_X(“0.36”) = 0.300/0 + 0.400/1 + 0.500/2 + 0.600/3 + 0.700/4 + 0.800/5 + 0.900/6 + 1.000/7 + 0.900/8 + 0.800/9 + 0.700/10

μ_Z(“0.128”) = 0.429/0 + 0.571/1 + 0.714/2 + 0.857/3 + 1.000/4 + 0.857/5 + 0.714/6 + 0.571/7

16) Neurons of the third layer (FCR) for rule 8:

R₈(A₁(x), A₂(z)) = (μ_X(“larger than medium”) → μ_Z(“positive small”)) = (μ_X(“0.36”) → μ_Z(“0.128”)) =

17) Neurons of the second layer (fuzzification) for rule 9:

μ_X(“0.39”) = 0.200/0 + 0.300/1 + 0.400/2 + 0.500/3 + 0.600/4 + 0.700/5 + 0.800/6 + 0.900/7 + 1.000/8 + 0.900/9 + 0.800/10

μ_Z(“0.213”) = 0.286/0 + 0.429/1 + 0.571/2 + 0.714/3 + 0.857/4 + 1.000/5 + 0.857/6 + 0.714/7

18) Neurons of the third layer (FCR) for rule 9:

R₉(A₁(x), A₂(z)) = (μ_X(“smaller than large”) → μ_Z(“positive medium”)) = (μ_X(“0.39”) → μ_Z(“0.213”)) =

19) Neurons of the second layer (fuzzification) for rule 10:

μ_X(“0.42”) = 0.100/0 + 0.200/1 + 0.300/2 + 0.400/3 + 0.500/4 + 0.600/5 + 0.700/6 + 0.800/7 + 0.900/8 + 1.000/9 + 0.900/10

μ_Z(“0.29”) = 0.143/0 + 0.286/1 + 0.429/2 + 0.571/3 + 0.714/4 + 0.857/5 + 1.000/6 + 0.857/7

20) Neurons of the third layer (FCR) for rule 10:

R₁₀(A₁(x), A₂(z)) = (μ_X(“bit smaller than large”) → μ_Z(“larger than medium”)) = (μ_X(“0.42”) → μ_Z(“0.29”)) =

21) Neurons of the second layer (fuzzification) for rule 11:

μ_X(“0.45”) = 0.000/0 + 0.100/1 + 0.200/2 + 0.300/3 + 0.400/4 + 0.500/5 + 0.600/6 + 0.700/7 + 0.800/8 + 0.900/9 + 1.000/10

μ_Z(“0.358”) = 0.000/0 + 0.143/1 + 0.286/2 + 0.429/3 + 0.571/4 + 0.714/5 + 0.857/6 + 1.000/7

22) Neurons of the third layer (FCR) for rule 11:

R₁₁(A₁(x), A₂(z)) = (μ_X(“large”) → μ_Z(“smaller than large”)) = (μ_X(“0.45”) → μ_Z(“0.358”)) =

23) Neurons of the third layer (FCR) aggregation:

$R_{aggr}\left(A_1\left(x\right),A_2\left(z\right)\right)={\displaystyle {\cup }_{k=1}^{11}{R}_k\left(A_1\left(x\right),A_2\left(z\right)\right)}=$

24) Neurons of the fourth layer (FCIR) composition for rule 1:

μ_Z'(“zero”) = μ_X'(“small”) ∘ R_aggr(A₁(x), A₂(z)) = 1.000/0 + 0.900/1 + 0.500/2 + 1.000/3 + 0.300/4 + 0.229/5 + 0.229/6 + 0.229/7

25) Neurons of the fifth layer (Defuzzification) for output of rule 1:

Defuzzification of μ_Z'(“zero”) ⇒ 0.03342857142857139

26) Neurons of the fourth layer (FCIR) composition for rule 2:

μ_Z'(“negative medium”) = μ_X'(“bit larger than small”) ∘ R_aggr(A₁(x), A₂(z)) = 0.900/0 + 1.000/1 + 0.600/2 + 0.900/3 + 0.400/4 + 0.300/5 + 0.229/6 + 0.229/7

27) Neurons of the fifth layer (Defuzzification) for output of rule 2:

Defuzzification of μ_Z'(“negative medium”) ⇒ −0.12885714285714284

28) Neurons of the fourth layer (FCIR) composition for rule 3:

μ_Z'(“negative large”) = μ_X'(“larger than small”) ∘ R_aggr(A₁(x), A₂(z)) = 1.000/0 + 0.900/1 + 0.700/2 + 0.800/3 + 0.500/4 + 0.400/5 + 0.300/6 + 0.229/7

29) Neurons of the fifth layer (Defuzzification) for output of rule 3:

Defuzzification of μ_Z'(“negative large”) ⇒ −0.21

30) Neurons of the fourth layer (FCIR) composition for rule 4:

μ_Z'(“negative large”) = μ_X'(“smaller than medium”) ∘ R_aggr(A₁(x), A₂(z)) = 1.000/0 + 0.900/1 + 0.800/2 + 0.700/3 + 0.600/4 + 0.500/5 + 0.400/6 + 0.300/7

31) Neurons of the fifth layer (Defuzzification) for output of rule 4:

Defuzzification of μ_Z'(“negative large”) ⇒ −0.21

32) Neurons of the fourth layer (FCIR) composition for rule 5:

μ_Z'(“negative medium”) = μ_X'(“bit smaller than medium”) ∘ R(A₁(x), A₂(z)) = 0.900/0 + 1.000/1 + 0.900/2 + 0.800/3 + 0.700/4 + 0.600/5 + 0.500/6 + 0.400/7

33) Neurons of the fifth layer (Defuzzification) for output of rule 5:

Defuzzification of μ_Z'(“negative medium”) ⇒ −0.12885714285714284

34) Neurons of the fourth layer (FCIR) composition for rule 6:

μ_Z'(“negative small”) = μ_X'(“medium”) ∘ R_aggr(A₁(x), A₂(z)) = 0.800/0 + 0.900/1 + 1.000/2 + 0.900/3 + 0.800/4 + 0.700/5 + 0.600/6 + 0.500/7

35) Neurons of the fifth layer (Defuzzification) for output of rule 6:

Defuzzification of μ_Z'(“negative small”) ⇒ −0.04771428571428571

36) Neurons of the fourth layer (FCIR) composition for rule 7:

μ_Z'(“zero”) = μ_X'(“bit larger than medium”) ∘ R_aggr(A₁(x), A₂(z)) = 0.700/0 + 0.800/1 + 0.900/2 + 1.000/3 + 0.900/4 + 0.800/5 + 0.700/6 + 0.600/7

37) Neurons of the fifth layer (Defuzzification) for output of rule 7:

Defuzzification of μ_Z'(“zero”) ⇒ 0.03342857142857139

38) Neurons of the fourth layer (FCIR) composition for rule 8:

μ_Z'(“positive small”) = μ_X'(“larger than medium”) ∘ R_aggr(A₁(x), A₂(z)) = 0.600/0 + 0.700/1 + 0.800/2 + 0.900/3 + 1.000/4 + 0.900/5 + 0.800/6 + 0.700/7

39) Neurons of the fifth layer (Defuzzification) for output of rule 8:

Defuzzification of μ_Z'(“positive small “) ⇒ 0.11457142857142857

40) Neurons of the fourth layer (FCIR) composition for rule 9:

μ_Z'(“positive medium”) = μ_X'(“smaller than large”) ∘ R_aggr(A₁(x), A₂(z)) = 0.500/0 + 0.600/1 + 0.700/2 + 0.800/3 + 0.900/4 + 1.000/5 + 0.900/6 + 0.800/7

41) Neurons of the fifth layer (Defuzzification) for output of rule 9:

Defuzzification of μ_Z'(“positive medium”) ⇒ 0.1957142857142857

42) Neurons of the fourth layer (FCIR) composition for rule 10:

μ_Z'(“larger than medium”) = μ_X'(“bit smaller than large”) ∘ R_aggr(A₁(x), A₂(z)) = 0.400/0 + 0.500/1 + 0.600/2 + 0.700/3 + 0.800/4 + 0.900/5 + 1.000/6 + 0.900/7

43) Neurons of the fifth layer (Defuzzification) for output of rule 10:

Defuzzification of μ_Z'(“larger than medium”) ⇒ 0.2768571428571428

44) Neurons of the fourth layer (FCIR) composition for rule 11:

μ_Z'(“smaller than large”) = μ_X'(“large”) ∘ R_aggr(A₁(x), A₂(z)) = 0.300/0 + 0.400/1 + 0.500/2 + 0.600/3 + 0.700/4 + 0.800/5 + 0.900/6 + 1.000/7

45) Neurons of the fifth layer (Defuzzification) for output of rule 11:

Defuzzification of μ_Z'(“smaller than large”) ⇒ 0.358.

The mean square error for fuzzy model based on our t-norm approach $e^2=3.41322\times {10}^{-3}$ is shown in Table 3 . This result statistically is almost twice as accurate, as GA-Generated fuzzy model.

In real world of NN based systems a value of their input/output pairs might be significantly changed in accordance with a set of a new requirements/capabilities. It could be a situation of a new label/class introduction. The latter means that aggregated FCR matrix of a system $R_{aggr}\left(A_1\left(x\right),A_2\left(z\right)\right)$ must be modified, based on an additional label, never used originally. We presume that the value of a new label could situate outside of the scale of normalized output values $z_{\text{norm}}\in \left[z_{\max },z_{\min }\right]$, used initially. At this case one must do the following.

1) Expand original scale or re-scale both labels/potential input pairs like that

${z^{\prime }}_{\text{norm}}\in \left[z_{\max }+\Delta z,z_{\min }-\Delta z\right]$, ${x^{\prime }}_{\text{norm}}\in \left[x_{\max }+\Delta x,x_{\min }-\Delta x\right]$,(3.13)

where

$\Delta z=\{\begin{array}{l}\left|z_{\text{label}}-z_{\max }\right|,z_{\max }<z_{\text{label}}\\ \left|z_{\min }-z_{\text{label}}\right|,z_{\text{label}}<z_{\min }\end{array}$(3.14)

On practice the value of $\Delta z=\Delta z+\epsilon$, when ε is defined empirically. In general terms could be the following linear function $\epsilon =f\left(\Delta z\right)$.

2) Find the input value, which corresponds to the new label/class.

For this matter we would use Generalized Modus Tollens [6] mechanism, the scheme of which is the following

Ant 1: IF x is A THEN z is B

Ant 2: z is B'

------------------------------------------- (3.15)

Cons: x is A'.

The most important thing to mention is that in (3.15) Ant 1, is represented by aggregated FCR matrix of a system $R_{aggr}\left(A_1\left(x\right),A_2\left(z\right)\right)$.

In terms of FCR, given a unary relationship $R\left(A_2\left(z^{\prime }\right)\right)=B^{\prime }$ one can obtain the consequence $R\left(A_1\left(x^{\prime }\right)\right)$ by CRI by applying it to $R\left(A_2\left(z^{\prime }\right)\right)$ and $R_{aggr}\left(A_1\left(x\right),A_2\left(z\right)\right)$ of type (3.10):

$\begin{array}{l}R\left(A_1\left(x^{\prime }\right)\right)=R\left(A_2\left(z^{\prime }\right)\right)\circ R_{aggr}\left(A_1\left(x\right),A_2\left(z\right)\right)\\ ={\displaystyle {\int }_{U_Z}{\mu }_{z^{\prime }}\left(u_z\right)/u_z}\circ {\displaystyle {\int }_{U_X\times U_Z}\left({\mu }_x\left(u_x\right)\to {\mu }_z\left(u_z\right)\right)}\wedge \left(\left(1-{\mu }_x\left(u_x\right)\right)\to \left(1-{\mu }_z\left(u_z\right)\right)\right)/\left(u_x,u_z\right)\\ ={\displaystyle {\int }_{U_X}{\displaystyle {\cup }_{z\in U_Z}\left[{\mu }_{z^{\prime }}\left(u_z\right)\wedge \left({\mu }_x\left(u_x\right)\to {\mu }_z\left(u_z\right)\right)\wedge \left(\left(1-{\mu }_x\left(u_x\right)\right)\to \left(1-{\mu }_z\left(u_z\right)\right)\right)\right]/u_x}}\end{array}$(3.16)

3) Based on CRI (3.16) add neuron of the third layer (FCR) for new rule:

$\begin{array}{l}{R}_{new}\left(A_1\left(x\right),A_2\left(z\right)\right)=\left(X^{\prime }\times U_X\to U_Z\times Z^{\prime }\right)\cap \left(\neg X^{\prime }\times U_X\to U_Z\times \neg Z^{\prime }\right)\\ ={\displaystyle {\int }_{U_X\times U_Z}\left({\mu }_{x^{\prime }}\left(u_x\right)\to {\mu }_{z^{\prime }}\left(u_z\right)\right)}\wedge \left(\left(1-{\mu }_{x^{\prime }}\left(u_x\right)\right)\to \left(1-{\mu }_{z^{\prime }}\left(u_z\right)\right)\right)/\left(u_x,u_z\right)\end{array}$(3.17)

4) Repeat an aggregation of neurons of the third layer (FCR) by using (3.17) and by previously aggregated FCR matrix of a system $R_{aggr}\left(A_1\left(x\right),A_2\left(z\right)\right)$.

${R^{\prime }}_{aggr}\left(A_1\left(x\right),A_2\left(z\right)\right)=R_{new}\left(A_1\left(x\right),A_2\left(z\right)\right)\cup R_{aggr}\left(A_1\left(x\right),A_2\left(z\right)\right)$(3.18)

This way we incorporated new knowledge into our system.

1) Suppose we have the new label $z^{\prime }=0.37$ and let $\Delta x=0.05$, $\Delta z=0.02$. Therefore expand (re-scale) both labels/potential input pairs like that $x^{\prime }\in \left[0.1,0.5\right]$, $z^{\prime }\in \left[-0.23,0.378\right]$.

2) The fuzzified value for $z^{\prime }=0.37$ from (3.13) and (3.4) is $\forall k\in \left[0,Card{U}_Z\right]$, ${\mu }_{z^{\prime }}\left(u_{z_k}\right)=1-\frac{1}{Card{U}_Z-1}\times \left|k-Ent\left[\left(Card{U}_Z-1\right)\times b{z^{\prime }}_{norm}\right]\right|$, i.e.

μ_z'(“0.37”) = 0.000/0 + 0.143/1 + 0.286/2 + 0.429/3 + 0.571/4 + 0.714/5 + 0.857/6 + 1.000/7

3) After application of Generalized Modus Tollens (3.15) and (3.16), i.e.

$\begin{array}{c}R\left(A_1\left(x^{\prime }\right)\right)=Rb\left(A_2\left(z^{\prime }\right)\right)\circ R_{aggr}\left(A_1\left(x\right),A_2\left(z\right)\right)\\ ={\displaystyle {\int }_{U_Z}{\mu }_{z^{\prime }}\left(u_z\right)/u_z}\circ {\displaystyle {\int }_{U_X\times U_Z}\left({\mu }_x\left(u_x\right)\to {\mu }_z\left(u_z\right)\right)}\\ \wedge \left(\left(1-{\mu }_x\left(u_x\right)\right)\to \left(1-{\mu }_z\left(u_z\right)\right)\right)/\left(u_x,u_z\right)\end{array}$