Alex Tserkovny
Applied AI Services, Brookline, USA.
DOI: 10.4236/jsea.2026.197011   PDF    HTML   XML   2 Downloads   33 Views   Citations

Abstract

In this work we revise some less investigated aspects of a type-2 fuzzy logic system (FLS), which can handle rule uncertainties. The specificity of this type-2 FLS usually comprises the operations of fuzzification, inference, and output processing. We focus on “inference”, which in our opinion should be based on an operation of fuzzy implication, rather than on commonly used t-norm one. This paper also investigates some original properties of fuzzy grades under the operations of join $\bigsqcup$ , meet $\sqcap$ and implication ⟶, which are defined by using the extension principal L. A. Zadeh, and shows that convex fuzzy grades form normal convex fuzzy grades, which in their turn form a distributive lattice under $\bigsqcup$ and $\sqcap$ . In this article we present much more effective implementation of such operations. We also explore an important feature of IF-THEN rules for type-2 fuzzy sets. This study introduces the notion of semantic similarity measure of these rules and their importance for logical inference in type-2 FLS. All theoretical concepts presented in this work are thoroughly illustrated by relevant examples.

Keywords

Type-2 Fuzzy Set, Type-2 Fuzzy Logic System, Fuzzy Implication, Fuzzy Conditional Inference, IF-THEN Rules, Semantic Similarity

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1. Introduction

It should be mentioned that the concept of fuzzy sets (FS) of type 2 has been defined by L. A. Zadeh [1]-[3] as an extension of ordinary FS. The further investigation of FS of type 2 was done in [4] and [5]. The FS of type 2 can be characterized by a fuzzy membership function the grade (or fuzzy grade), of which is a fuzzy set in the unit interval [0, 1], rather than a point in [0, 1]. Moreover, the algebraic properties of fuzzy grades under the operations $\bigsqcup$ and $\sqcap$ are slightly different from canonical ones. In this study we revisit operations, algebraic properties, and rule semantics in type-2 systems. We propose normalization of physical values and their mapping to a discrete universe. This article provides formulas for constructing membership functions using singletons and linguistic terms. We prove that fuzzy grades defined via triangular secondary MFs are convex. Note that in [6]-[8] the concept of type-2 fuzzy logic system (FLS) was introduced and thoroughly investigated. This paper studies several underexplored aspects of such type-2 FLS. In particular, this article argues that inference in type-2 FLS should rely on fuzzy implication, not t-norms, because implication better reflects semantic reasoning.

2. Fuzzy Logic Features

2.1. Fuzzy Logic Type-1

Let us consider the set of some kind of physical values $p\in \left[P^{min},P^{max}\right]$ .

To normalize these values of $p$ we utilize the following expression

$p^{norm}=\frac{p-P^{min}}{P^{max}-P^{min}}$ (2.1)

Hence, in our research we always consider to operate on a universe of discourse $U$ , i.e. on a set of integers rather, than on original physical scale of $p\in \left[P^{min},P^{max}\right]$ . For this purpose, we use the following mapping

$\gamma :P\to U|u=Ent\left[CardU\cdot p^{norm}\right]$ (2.2)

Let $P$ be correspondent to $p^{norm}$ from (2.1) type-1 FS and let the relevant membership function (MF) of $u\in U$ in $P$ be ${\mu }_p\left(u\right)$ , which is a crisp number in $\left[0,1\right]$ .

Thus, the fuzzy set is presented as

$P={\displaystyle {\int }_U\frac{{\mu }_p\left(u\right)}{u}}$ (2.3)

On the other hand, to determine the estimates of the MF in terms of singletons from (2.3) in the form ${\mu }_p\left(u_i\right)/u_i|\forall i\in \left[1,CardU\right]$ , given (2.1) and (2.2) we propose the following procedure. $\forall i\in \left[1,Card{U}_{ }\right]$

${\mu }_p\left(u_i\right)={\left\{1-\frac{1}{CardU-1}\cdot \left|i-Ent\left[\left(CardU\right)\cdot p^{norm}\right]\right|\right\}}^{{\alpha }_i}$ (2.4)

where ${\alpha }_i$ is a weight coefficient of the i-th input cluster [9] for $\forall i\in \left[1,CardU\right]$ .

In more details from (2.4) we have

$P=\frac{{\mu }_p\left(u_1\right)}{u_1}+\frac{{\mu }_p\left(u_2\right)}{u_2}+\cdots +\frac{{\mu }_p\left(u_n\right)}{u_n}={\displaystyle \sum }_i\frac{{\mu }_p\left(u_i\right)}{u_i},u_i\in U$ (2.5)

In type-1 fuzzy set $P$ is represented as a triplet of the form

$P=\left\{p,U,\tilde{P}>\right\},p\in T\left(u\right),u_i\in U,i=\overline{1,CardU}$ (2.6)

where $\tilde{P}$ is normal FS with correspondent MF ${\mu }_p:U\to \left[0,1\right]$ . $T\left(u\right)$ could be the term from Table 1.

For each fuzzy set $P$ one could define a linguistic term from the following set

$T\left(u\right)=\left\{low,\cdots ,highest\right\}$ (2.7)

The following Table 1 depicts linguistic terms from (2.7).

Table 1. Linguistic terms.

Value of variable
$T\left(k\right)$	$k\in U$
low	1
higher than low	2
middle	3
higher than middle	4
high	5
highest	6

To determine the estimates of the MF in terms of singletons from (2.5) in the form ${\mu }_p\left(u_i\right)/u_i|\forall i\in \left[1,CardU\right]$ , for each k-th term from Table 1. we use the following procedure.

${\mu }_p\left(u_i{}_{}\right)=\left\{1-\frac{1}{CardU-1}\cdot \left|i-k\right|\right\},\forall i\in \left[1,CardU\right]$ (2.8)

Note, that MF from (2.8) has triangular form, which is defined on entire universe of discourse $U$ .

2.2. Fuzzy Logic Type-2

A type-2 fuzzy set in $X$ is $\tilde{P}$ and the membership grade of $x\in X$ in $\tilde{P}$ is ${\mu }_{\tilde{P}}\left(x\right)$ , which is a type-1 FS in $\left[0,1\right]$ . The elements of the domain of ${\mu }_{\tilde{P}}\left(x\right)$ are called primary memberships of $x$ in $\tilde{P}$ and the memberships of the primary memberships in ${\mu }_{\tilde{P}}\left(x\right)$ are called secondary memberships of $x$ in $\tilde{P}$ ; the latter defines the possibilities for the primary membership, e.g., the type-2 FS, $\tilde{P}\in X$ has fuzzy grades (FS in $J_X\subseteq \left[0,1\right]$ ), ${\mu }_{\tilde{P}}\left(x\right)$ that can be denoted for each $x\in X$ as ${\mu }_{\tilde{P}}:X\to {\left[0,1\right]}^{J_X}$

${\mu }_{\tilde{P}}\left(x\right)=\frac{f\left(v_1\right)}{v_1}+\frac{f\left(v_2\right)}{v_2}+\cdots +\frac{f\left(v_n\right)}{v_n}={\displaystyle \sum }_i\frac{f\left(v_i\right)}{v_i},v_i\in J_X$ (2.9)

And to define $v_i\in J_X$ in (2.7) we propose the following expression

$v_i=\left\{\frac{i}{Card{J}_X-1}\right\},v_i\in J_X,\forall i\in \left[1,Card{J}_X\right]$ (2.10)

Thus, we presume that membership grade ${\mu }_{\tilde{P}}\left(x\right)$ is normal, therefore $\exists !l^{*}\in \left[1,Card{J}_X\right]|f\left(v_{l^{*}}\right)=\underset{\forall i\in \left[1,Card{J}_X\right]}{\max }\left\{f\left(v_i\right)\right\}==1$ and the MF of fuzzy grades $f\left(v_i\right)$ we define as

$f\left(v_i\right)=\left\{1-\frac{1}{Card{J}_X-1}\cdot \left|i-l^{*}\right|\right\},\forall i\in \left[1,Card{J}_X\right]$ (2.11)

Example 1.

Suppose the $Card{J}_X=6$ , then from (2.10) we obtain $J_X=\left\{0,0.2,0.4,0.6,0.8,1\right\}\subseteq \left[0,1\right]$ . And also suppose the $X=\left\{1,2,3,4,5,6\right\}$ is the set of numbers $u$ from (2.2), which is correspondent to the set of normalized values $p^{norm}$ from (2.1) and that $P$ is FS of type-2 of Normalized Variable (Figure 1).

P = Normalized Variable = low/1 + middle/3 + high/5

Figure 1. Type-2 MF of “Normalized Variable” from Ex. 1.

From (2.9) and (2.11) by using terms from Table 1. we find the following membership grades (${\mu }_{\tilde{P}}\left(1\right),{\mu }_{\tilde{P}}\left(3\right)$ and ${\mu }_{\tilde{P}}\left(5\right)$ are shown on Figure 1).

${\mu }_{\tilde{P}}\left(1\right)$ = low = 1/0 + 0.8/0.2 + 0.6/0.4 + 0.4/0.6 + 0.2/0.8 + 0/1

${\mu }_{\tilde{P}}\left(3\right)$ = middle = 0.6/0 + 0.8/0.2 + 1/0.4 + 0.8/0.6 + 0.6/0.8 + 0.4/1

${\mu }_{\tilde{P}}\left(5\right)$ = high = 0/0 + 0.2/0.2 + 0.4/0.4 + 0.6/0.6 + 0.8/0.8 + 1/1

Note, that in Ex. 1 $J_1=J_3=J_5=J_X$ , whereas $J_2=J_4=J_6=\varnothing$ .

2.3. Operations of FS Type-2

Let us deliberate some important feature of fuzzy grades, which could be utilize later in this study.

Lemma 1.

If fuzzy grade ${\mu }_P\left(x\right)={\displaystyle \sum }_i\frac{f\left(v_i\right)}{v_i},v_i\in J_X$

from (2.9) is defined as (2.10), then it is a convex for any integers $i,k\in \left[1,Card{J}_X\right]$ , $\exists j\in \left[1,Card{J}_X\right]|i\le j\le k$ , i.e. $f\left(v_j\right)\ge \text{min}\left\{f\left(v_i\right),f\left(v_k\right)\right\}|i\le j\le k$ .

Proof:

Let us consider $f\left(v_{n-1}\right),f\left(v_n\right),f\left(v_{n+1}\right)$ from (2.11) $\forall n\in \left[1,Card{J}_X{}_{ }\right]$ . We have the following cases

1) $n+1\le l^{*}$ ;

2) $n\le l^{*},n+1>l^{*}$ ;

3) $n-1\le l^{*},n>l^{*}$ ;

4) $n-1>l^{*}$ .

For case 1. from (2.11)

$i=n+1\Rightarrow {\text{Δ}}_{n+1}=\left|i-l^{*}\right|=l^{*}-n-1$ ;

$i=n\Rightarrow {\text{Δ}}_n=\left|i-l^{*}\right|=l^{*}-n$ ;

$i=n-1\Rightarrow {\text{Δ}}_{n-1}=\left|i-l^{*}\right|=l^{*}-n+1$ .

From where we have

${\text{Δ}}_{n-1}>{\text{Δ}}_n>{\text{Δ}}_{n+1}\Rightarrow f\left(v_{n-1}\right)<f\left(v_n\right)<f\left(v_{n+1}\right)$

$\Rightarrow f\left(v_n\right)>\min \left\{f\left(v_{n-1}\right),f\left(v_{n+1}\right)\right\}$

For case 2. from (2.11)

$i=n\Rightarrow {\Delta }_n=\left|i-l^{*}\right|=l^{*}-n$ ;

$i=n-1\Rightarrow {\Delta }_{n-1}=\left|i-l^{*}\right|=l^{*}-n+1$ ;

$i=n+1\Rightarrow {\Delta }_{n+1}=\left|i-l^{*}\right|=n+1-l^{*}$ .

Given that ${\Delta }_{n+1}-{\Delta }_n=2\cdot l^{*}-2\cdot n-1|n\le l^{*}\Rightarrow {\Delta }_{n+1}-{\Delta }_n<0$ we have

${\text{Δ}}_{n-1}>{\text{Δ}}_n>{\text{Δ}}_{n+1}\Rightarrow f\left(v_{n-1}\right)<f\left(v_n\right)<f\left(v_{n+1}\right)$

$\Rightarrow f\left(v_n\right)>\min \left\{f\left(v_{n-1}\right),f\left(v_{n+1}\right)\right\}$

For case 3. from (2.11)

$i=n-1\Rightarrow {\Delta }_{n-1}=\left|i-l^{*}\right|=l^{*}-n+1$ ;

$i=n\Rightarrow {\Delta }_n=\left|i-l^{*}\right|=n-l^{*}$ ;

$i=n+1\Rightarrow {\Delta }_{n+1}=\left|i-l^{*}\right|=n+1-l^{*}$ .

Given that ${\Delta }_n-{\Delta }_{n-1}=2\cdot n-2\cdot l^{*}-1|n>l^{*}\Rightarrow {\Delta }_n-{\Delta }_{n-1}<0$ we have

${\text{Δ}}_{n-1}>{\text{Δ}}_n>{\text{Δ}}_{n+1}\Rightarrow f\left(v_{n-1}\right)<f\left(v_n\right)<f\left(v_{n+1}\right)$

$\Rightarrow f\left(v_n\right)>\min \left\{f\left(v_{n-1}\right),f\left(v_{n+1}\right)\right\}$

For case 4. from (2.11)

$i=n-1\Rightarrow {\Delta }_{n-1}=\left|i-l^{*}\right|=n-1-l^{*}$ ;

$i=n\Rightarrow {\Delta }_n=\left|i-l^{*}\right|=n-l^{*}$ ;

$i=n+1\Rightarrow {\Delta }_{n+1}=\left|i-l^{*}\right|=n+1-l^{*}$ .

From where we have

${\text{Δ}}_{n+1}>{\text{Δ}}_n>{\text{Δ}}_{n-1}\Rightarrow f\left(v_{n+1}\right)<f\left(v_n\right)<f\left(v_{n-1}\right)$

$\Rightarrow f\left(v_n\right)>\min \left\{f\left(v_{n-1}\right),f\left(v_{n+1}\right)\right\}$ . ■

Let ${\mu }_P\left(x\right)$ and ${\mu }_Q\left(x\right)$ be two fuzzy grades that is, FSs in $J_X\subseteq \left[0,1\right]$ of FSs of type 2, $P$ and $Q$ , respectively, represented as

${\mu }_P\left(x\right)=\frac{f\left(v_1\right)}{v_1}+\frac{f\left(v_2\right)}{v_2}+\cdots +\frac{f\left(v_n\right)}{v_n}={\displaystyle \sum }_i\frac{f\left(v_i\right)}{v_i},v_i\in J_X,$ (2.12)

${\mu }_Q\left(x\right)=\frac{g\left(w_1\right)}{w_1}+\frac{g\left(w_2\right)}{w_2}+\cdots +\frac{g\left(w_n\right)}{w_n}={\displaystyle \sum }_j\frac{g\left(w_j\right)}{w_j},w_j\in J_X,$ (2.13)

where the functions $f$ and $g$ are MFs of fuzzy grades (FSs in $J_X\subseteq \left[0,1\right]$ ) ${\mu }_P\left(x\right)$ and ${\mu }_Q\left(x\right)$ , respectively, and the values $f\left(v_i\right)$ and $g\left(w_j\right)$ in [0, 1] denote the grades for $v_i$ and $w_j$ in $J_X$ , respectively. Thus, the operations for FS of type 2 are expressed by the following.

Union.

$\begin{array}{l}P\cup Q\iff {\mu }_{P\cup Q}\left(x\right)={\mu }_p\left(x\right)\bigsqcup {\mu }_Q\left(x\right)={\displaystyle \sum }_i\frac{f\left(v_i\right)}{v_i}\bigsqcup {\displaystyle \sum }_j\frac{g\left(w_j\right)}{w_j}\\ ={\displaystyle \sum }_{i,j}\left(f\left(v_i\right)\wedge g\left(w_j\right)\right)/\left(v_i\vee w_j\right),\end{array}$ (2.14)

where $\wedge$ represent min and $\vee$ denote max, whereas $\bigsqcup$ is fuzzy grades operation join.

Intersection.

$\begin{array}{l}P\cap Q\iff {\mu }_{P\cap Q}\left(x\right)={\mu }_p\left(x\right)\sqcap {\mu }_Q\left(x\right)={\displaystyle \sum }_i\frac{f\left(v_i\right)}{v_i}\sqcap {\displaystyle \sum }_j\frac{g\left(w_j\right)}{w_j}\\ ={\displaystyle \sum }_{i,j}\left(f\left(v_i\right)\wedge g\left(w_j\right)\right)/\left(v_i\wedge w_j\right),\end{array}$ (2.15)

where $\wedge$ represent min, whereas $\sqcap$ is fuzzy grades operation meet.

Remark.

In general, if ${\mu }_P\left(x\right)$ and ${\mu }_Q\left(x\right)$ are two fuzzy grades that is, FSs in $J_X\subseteq \left[0,1\right]$ of FS of type 2, $P$ and $Q$ in $X$ respectively, denoted as (2.12) and (2.13), and $\divideontimes$ is a binary operation defined in $X$ , then the operation $\divideontimes$ can be extended to FS $P$ and $Q$ by the definition relation (extension principle).

$\begin{array}{l}P\divideontimes Q\iff {\mu }_{P\divideontimes Q}\left(x\right)={\mu }_p\left(x\right)\divideontimes {\mu }_Q\left(x\right)={\displaystyle \sum }_i\frac{f\left(v_i\right)}{u_i}\divideontimes {\displaystyle \sum }_j\frac{g\left(w_j\right)}{w_j}\\ ={\displaystyle \sum }_{i,j}\left(f\left(v_i\right)\wedge g\left(w_j\right)\right)/\left(v_i\divideontimes w_j\right)\end{array}$ (2.16)

Example 2.

Let $J_X=\left\{0,0.2,0.4,0.6,0.8,1\right\}$ and let fuzzy grades ${\mu }_P\left(x\right)$ and ${\mu }_Q\left(x\right)$ be given as

${\mu }_P\left(x\right)$ = 0.8/0 + 1/0.2 + 0.8/0.4 + 0.6/0.6 + 0.4/0.8 + 0.2/1

${\mu }_Q\left(x\right)$ = 0.2/0 + 0.4/0.2 + 0.6/0.4 + 0.8/0.6 + 1/0.8 + 0.8/1

Then for Join operation from (2.14) we have

${\mu }_p\left(x\right)\bigsqcup {\mu }_Q\left(x\right)$ = (0.8/0 + 1/0.2 + 0.8/0.4 + 0.6/0.6 + 0.4/0.8 + 0.2/1) $\bigsqcup$ (0.2/0 + 0.4/0.2 + 0.6/0.4 + 0.8/0.6 + 1/0.8 + 0.8/1)

$\text{MATRIX}\left({\mu }_p\left(x\right)\bigsqcup {\mu }_Q\left(x\right)\right)=$

(0.8∧0.2)/(0∨0)	(0.8∧0.4)/(0∨0.2)	(0.8∧0.6)/(0∨0.4)	(0.8∧0.8)/(0∨0.6)	(0.8∧1)/(0∨0.8)	(0.8∧0.8)/(0∨1)
(1∧0.2)/(0.2∨0)	(1∧0.4)/(0.2∨0.2)	(1∧0.6)/(0.2∨0.4)	(1∧0.8)/(0.2∨0.6)	(1∧1)/(0.2∨0.8)	(1∧0.8)/(0.2∨1)
(0.8∧0.2)/(0.4∨0)	(0.8∧0.4)/(0.4∨0.2)	(0.8∧0.6)/(0.4∨0.4)	(0.8∧0.8)/(0.4∨0.6)	(0.8∧1)/(0.4∨0.8)	(0.8∧0.8)/(0.4∨1)
(0.6∧0.2)/(0.6∨0)	(0.6∧0.4)/(0.6∨0.2)	(0.6∧0.6)/(0.6∨0.4)	(0.6∧0.8)/(0.6∨0.6)	(0.6∧1)/(0.6∨0.8)	(0.6∧0.8)/(0.6∨1)
(0.4∧0.2)/(0.8∨0)	(0.4∧0.4)/(0.8∨0.2)	(0.4∧0.6)/(0.8∨0.4)	(0.4∧0.8)/(0.8∨0.6)	(0.4∧1)/(0.8∨0.8)	(0.4∧0.8)/(0.8∨1)
(0.2∧0.2)/(1∨0)	(0.2∧0.4)/(1∨0.2)	(0.2∧0.6)/(1∨0.4)	(0.2∧0.8)/(1∨0.6)	(0.2∧1)/(1∨0.8)	(0.2∧0.8)/(1∨1)

=

0.2/0	0.4/0.2	0.6/0.4	0.8/0.6	0.8/0.8	0.8/1
0.2/0.2	0.4/0.2	0.6/0.4	0.8/0.6	1/0.8	0.8/1
0.2/0.4	0.4/0.4	0.6/0.4	0.8/0.6	0.8/0.8	0.8/1
0.2/0.6	0.4/0.6	0.6/0.6	0.6/0.6	0.6/0.8	0.6/1
0.2/0.8	0.4/0.8	0.4/0.8	0.4/0.8	0.4/0.8	0.4/1
0.2/1	0.2/1	0.2/1	0.2/1	0.2/1	0.2/1

=0.2/0 + (0.2∨0.4∨0.4)/0.2 + (0.2∨0.4∨0.6∨0.6∨0.6)/0.4 + (0.2∨0.4∨0.6∨0.6∨0.8∨0.8∨0.8)/0.6 + (0.2∨0.4∨0.4∨0.4∨0.4∨ 0.6∨0.8∨1∨0.8)/0.8 + (0.2∨0.2∨0.2∨0.2∨0.2∨0.2∨0.4∨0.6∨0.8∨0.8∨0.8)/1 = 0.2/0 + 0.4/0.2 + 0.6/0.4 + 0.8/0.6 + 1/0.8 + 0.8/1

For Meet operation from (2.15)

${\mu }_p\left(x\right)\sqcap {\mu }_Q\left(x\right)$ = (0.8/0 + 1/0.2 + 0.8/0.4 + 0.6/0.6 + 0.4/0.8 + 0.2/1) $\sqcap$ (0.2/0 + 0.4/0.2 + 0.6/0.4 + 0.8/0.6 + 1/0.8 + 0.8/1)

$\text{MATRIX}\left({\mu }_p\left(x\right)\sqcap {\mu }_Q\left(x\right)\right)=$

(0.8∧0.2)/(0∧0)	(0.8∧0.4)/(0∧0.2)	(0.8∧0.6)/(0∧0.4)	(0.8∧0.8)/(0∧0.6)	(0.8∧1)/(0∧0.8)	(0.8∧0.8)/(0∧1)
(1∧0.2)/(0.2∧0)	(1∧0.4)/(0.2∧0.2)	(1∧0.6)/(0.2∧0.4)	(1∧0.8)/(0.2∧0.6)	(1∧1)/(0.2∧0.8)	(1∧0.8)/(0.2∧1)
(0.8∧0.2)/(0.4∧0)	(0.8∧0.4)/(0.4∧0.2)	(0.8∧0.6)/(0.4∧0.4)	(0.8∧0.8)/(0.4∧0.6)	(0.8∧1)/(0.4∧0.8)	(0.8∧0.8)/(0.4∧1)
(0.6∧0.2)/(0.6∧0)	(0.6∧0.4)/(0.6∧0.2)	(0.6∧0.6)/(0.6∧0.4)	(0.6∧0.8)/(0.6∧0.6)	(0.6∧1)/(0.6∧0.8)	(0.6∧0.8)/(0.6∧1)
(0.4∧0.2)/(0.8∧0)	(0.4∧0.4)/(0.8∧0.2)	(0.4∧0.6)/(0.8∧0.4)	(0.4∧0.8)/(0.8∧0.6)	(0.4∧1)/(0.8∧0.8)	(0.4∧0.8)/(0.8∧1)
(0.2∧0.2)/(1∧0)	(0.2∧0.4)/(1∧0.2)	(0.2∧0.6)/(1∧0.4)	(0.2∧0.8)/(1∧0.6)	(0.2∧1)/(1∧0.8)	(0.2∧0.8)/(1∧1)

=

0.2/0	0.4/0	0.6/0	0.8/0	0.8/0	0.8/0
0.2/0	0.4/0.2	0.6/0.2	0.8/0.2	1/0.2	0.8/0.2
0.2/0	0.4/0.2	0.6/0.4	0.8/0.4	0.8/0.4	0.8/0.4
0.2/0	0.4/0.2	0.6/0.4	0.6/0.6	0.6/0.6	0.6/0.6
0.2/0	0.4/0.2	0.4/0.4	0.4/0.6	0.4/0.8	0.4/0.8
0.2/0	0.2/0.2	0.2/0.4	0.2/0.6	0.2/0.8	0.2/1

=(0.2∨0.2∨0.2∨0.2∨0.2∨0.2∨0.4∨0.6∨0.8∨0.8∨0.8)/0 + (0.2∨0.4∨0.4∨0.4∨0.4∨0.6∨0.8∨1∨0.8)/0.2 + (0.2∨0.4∨0.6∨ 0.6∨0.8∨0.8∨0.8)/0.4 + (0.2∨0.4∨0.6∨0.6∨0.6)/0.6 + (0.2∨0.4∨0.4)/0.8 + 0.2/1 = 0.8/0 + 1/0.2 + 0.8/0.4 + 0.6/0.6 + 0.4/0.8 + 0.2/1

Note. For both Join and Meet operations the number $N_1$ of calculations needed was $N_1=3\times Card{J}_X^2$ , In our case for $Card{J}_X=6$ we obtain $N_1=108$ ,

2.4. Fast Operations of FS Type-2

Now we introduce significantly more effective way to operate with type-2 FS.

Lemma 2.

If fuzzy grades

${\mu }_P\left(x\right)={\displaystyle \sum }_i\frac{f\left(v_i\right)}{v_i},v_i\in J_X$

and

${\mu }_Q\left(x\right)={\displaystyle \sum }_j\frac{g\left(w_j\right)}{w_j},w_j\in J_X$

are both normal, i.e.

$\{\begin{array}{l}\exists !k^{*}\in \left[1,Card{J}_X\right]|f\left(v_{k^{*}}\right)=\underset{\forall i\in \left[1,CardJ\right]}{\max }\left\{f\left(v_i\right)\right\}==1;\\ \exists !l^{*}\in \left[1,CardJ\right]|g\left(w_{l^{*}}\right)=\underset{\forall j\in \left[1,CardJ\right]}{\max }\left\{g\left(w_j\right)\right\}==1\end{array}$ (2.17)

then the following features are taking place

$v_{k^{*}},w_{l^{*}}\in J|v_{k^{*}}\le w_{l^{*}}\Rightarrow {\mu }_p\left(x\right)\sqcap {\mu }_Q\left(x\right)={\mu }_p\left(x\right)$ ;

$v_{k^{*}},w_{l^{*}}\in J|v_{k^{*}}>w_{l^{*}}\Rightarrow {\mu }_p\left(x\right)\sqcap {\mu }_Q\left(x\right)={\mu }_Q\left(x\right)$ ;

$v_{k^{*}},w_{l^{*}}\in J|v_{k^{*}}\le w_{l^{*}}\Rightarrow {\mu }_p\left(x\right)\bigsqcup {\mu }_Q\left(x\right)={\mu }_Q\left(x\right)$ ;

$v_{k^{*}},w_{l^{*}}\in J|v_{k^{*}}>w_{l^{*}}\Rightarrow {\mu }_p\left(x\right)\bigsqcup {\mu }_Q\left(x\right)={\mu }_p\left(x\right)$

Proof:

1) $v_{k^{*}},w_{l^{*}}\in J|v_{k^{*}}\le w_{l^{*}}\Rightarrow b^{*}=v_{k^{*}}\wedge w_{l^{*}}{}_{}=v_{k^{*}}$

$l\left(b^{*}\right)=l\left(v_{k^{*}}\right)=f\left(v_{k^{*}}\right)=1$

${\mu }_p\left(x\right)\sqcap {\mu }_Q\left(x\right)=f\left(v_{k^{*}}\right)+{\displaystyle \sum }_{v_i\wedge w_j\ne b^{*}}f\left(v_i\right)\wedge g\left(w_j\right)$

${\mu }_p\left(x\right)\sqcap {\mu }_Q\left(x\right)=1+{\displaystyle \sum }_{v_i\wedge w_j\ne b^{*}}f\left(v_i\right)\wedge g\left(w_j\right)={\mu }_p\left(x\right)$

2) $v_{k^{*}},w_{l^{*}}\in J|v_{k^{*}}>w_{l^{*}}\Rightarrow b^{*}=v_{k^{*}}\wedge w_{l^{*}}{}_{}=w_{l^{*}}$

$l\left(b^{*}\right)=l\left(w_{l^{*}}\right)=g\left(w_{l^{*}}\right)=1$

${\mu }_p\left(x\right)\sqcap {\mu }_Q\left(x\right)=g\left(w_{l^{*}}\right)+{\displaystyle \sum }_{v_i\wedge w_j\ne b^{*}}f\left(v_i\right)\wedge g\left(w_j\right)$

${\mu }_p\left(x\right)\sqcap {\mu }_Q\left(x\right)=1+{\displaystyle \sum }_{v_i\wedge w_j\ne b^{*}}f\left(v_i\right)\wedge g\left(w_j\right)={\mu }_Q\left(x\right)$

3) $v_{k^{*}},w_{l^{*}}\in J|v_{k^{*}}\le w_{l^{*}}\Rightarrow b^{*}=v_{k^{*}}\vee w_{l^{*}}=w_{l^{*}}$

$l\left(b^{*}\right)=l\left(w_{l^{*}}\right)=g\left(w_{l^{*}}\right)=1$

${\mu }_p\left(x\right)\bigsqcup {\mu }_Q\left(x\right)=g\left(w_{l^{*}}\right)+{\displaystyle \sum }_{v_i\vee w_j\ne b^{*}}f\left(v_i\right)\wedge g\left(w_j\right)$

${\mu }_p\left(x\right)\bigsqcup {\mu }_Q\left(x\right)=1+{\displaystyle \sum }_{v_i\vee w_j\ne b^{*}}f\left(v_i\right)\wedge g\left(w_j\right)={\mu }_Q\left(x\right)$

4) $v_{k^{*}},w_{l^{*}}\in J|v_{k^{*}}>w_{l^{*}}\Rightarrow b^{*}=v_{k^{*}}\vee w_{l^{*}}=v_{k^{*}}$

$l\left(b^{*}\right)=l\left(w_{l^{*}}\right)=f\left(v_{k^{*}}\right)=1$

${\mu }_p\left(x\right)\bigsqcup {\mu }_Q\left(x\right)=f\left(v_{k^{*}}\right)+{\displaystyle \sum }_{v_i\vee w_j\ne b^{*}}f\left(v_i\right)\wedge g\left(w_j\right)$

${\mu }_p\left(x\right)\bigsqcup {\mu }_Q\left(x\right)=1+{\displaystyle \sum }_{v_i\vee w_j\ne b^{*}}f\left(v_i\right)\wedge g\left(w_j\right)={\mu }_p\left(x\right)$ . ■

Corollary 1.

From the conditions (2.17) of Lemma 2 we can formulate the following fast and simple implementations of Meet and Join operations for type-2 FS

${\mu }_p\left(x\right)\sqcap {\mu }_Q\left(x\right)=\{\begin{array}{l}{\mu }_p\left(x\right),\text{if}{k}^{*}\le l^{*},\\ {\mu }_Q\left(x\right),\text{othervise}\end{array}$

${\mu }_p\left(x\right)\bigsqcup {\mu }_Q\left(x\right)=\{\begin{array}{l}{\mu }_p\left(x\right),\text{if}{k}^{*}\ge l^{*},\\ {\mu }_Q\left(x\right),\text{othervise}\end{array}$

Note. For both Join and Meet operations the number $N_2$ of calculations needed was $N_2=\left(2\times Card{J}_X\right)+1$ , In our case for $Card{J}_X=6$ we obtain $N_2=13$ , Therefore, this algorithm is more than 8.3 times faster than the original one.

3. Type-2 Fuzzy Logic System (FLS)

3.1. Type-2 FLS: Overview

In this chapter we revisit and reconsider some important features of type-2 FLS, in which the antecedent or consequent membership functions are type-2 FS.

Figure 2. The structure of a type-2 FLS.

Figure 2 shows the structure of a type-2 FLS [6]. It is very similar to the structure of a type-1 FLS. The fuzzifier maps the crisp input into a FS. This fuzzy set can, in general, be a type-2 set. The inference engine combines rules and gives a mapping from input type-2 FS to output type-2 FS. To do this one needs to find unions and intersections of type-2 FS sets, as well as compositions of type-2 relations. In the type-2 case, the output of the inference engine is a type-2 set. Usually, in type-2 FLS extended defuzzification is utilized, which gives a type-1 fuzzy set. Since this operation transforms type-2 output sets to the FLS to a type-1 set, this operation is called type reduction and the type-reduced set so obtained a type-reduced set. In [6]-[8] it was mentioned that in order to get a crisp output from a type-2 FLS, the type-reduced set has to be de-fuzzified.

Hence, in order to develop a type-2 FLS, one needs to be able to:

1) perform the set theoretic operations of union, intersection, and complement on type-2 sets;

2) know properties (e.g., commutativity, associativity, identity laws) of membership grades of type-2 sets;

3) deal with type-2 fuzzy relations and their compositions;

4) perform type reduction and defuzzification to obtain a set-valued or crisp output from the FLS [7] [8].

We will concentrate our attention on 1., 2. and 3.

3.2. Type-2 FLS: Inference

Let us consider a type-2 FLS having K inputs, $x_1\in X_1,x_2\in X_2,\cdots ,x_K\in X_K$ , and one output $y\in Y$ . We assume that this FLS has M rules where the m-th rule has the form

$R^m:\text{IF}{x}_1\text{is}\widetilde{P_1^m}\text{and}{x}_2\text{is}\widetilde{P_2^m}\text{and}\cdots \text{and}{x}_K\text{is}\widetilde{P_K^m}\text{THEN}y\text{is}{\tilde{Q}}^m$(3.1)

This rule represents a type-2 fuzzy relation between the input space $X_1\times X_2\times \cdots \times X_K$ and the output space $Y$ of the FLS. We denote the membership function of this type-2 relation as,

$\forall m\in \left[1,M\right]|{\mu }_{\widetilde{P_1^m}\times \widetilde{P_2^m}\times \cdots \times \widetilde{P_K^m}\to {\tilde{Q}}^m}\left(x,y\right)$ , where $\widetilde{P_1^m}\times \widetilde{P_2^m}\times \cdots \times \widetilde{P_K^m}$ denotes the Cartesian product of $\widetilde{P_1^m},\widetilde{P_2^m},\cdots ,\widetilde{P_K^m}$ and $x=\left\{x_1,x_2,\cdots ,x_K\right\}$ . When an input $x^{\prime }$ is applied, the composition of the fuzzy set $\widetilde{P^{\prime }}$ to which $x^{\prime }$ have its place and the rule $R^m$ is found by using the extended sup-star composition [6]-[8]

${\mu }_{\widetilde{P^{\prime }}\circ \widetilde{P_1^m}\times \widetilde{P_2^m}\times \cdots \times \widetilde{P_K^m}\to {\tilde{Q}}^m}\left(y\right)={\displaystyle \coprod }_{x\in \widetilde{P^{\prime }}}\left[{\mu }_{\widetilde{P^{\prime }}}\left(x\right)\sqcap {\mu }_{\widetilde{P_1^m}\times \widetilde{P_2^m}\times \cdots \times \widetilde{P_K^m}\to {\tilde{Q}}^m}\left(x,y\right)\right]$ (3.2)

Let us take a look at the implication membership function from (3.2) for each m-th rule and denote it in the following form

${\mu }_{\widetilde{P_1^m}\times \widetilde{P_2^m}\times \cdots \times \widetilde{P_K^m}\to {\tilde{Q}}^m}\left(x,y\right)=\left[{\displaystyle \prod }_{\forall i\in \left[1,K\right]}{\mu }_{\widetilde{P_i^m}}\left(x\right)\right]\to {\mu }_{\widetilde{Q^m}}\left(y\right)$ (3.3)

We have to underline the fact that the implication operation in (3.3) is an essential element of a decision-making mechanism, known as Fuzzy Conditional Inference Rule (FCIR) of the (3.1) form.

Thus, we use the following type of implication [10]

$P\to Q=\{\begin{array}{ll}\left(1-p\right)\cdot q,\hfill & p>q,\hfill \\ 1,\hfill & p\le q\hfill \end{array}$ (3.4)

For practical purposes, described down below, we will use Fuzzy Conditional Rule (FCR) [11] of the following type

$\left(P\to Q\right)\cap \left(\neg P\to \neg Q\right)=\{\begin{array}{ll}\left(1-q\right)\cdot p,\hfill & p<q,\hfill \\ 1,\hfill & p=q,\hfill \\ \left(1-p\right)\cdot q,\hfill & p>q.\hfill \end{array}$ (3.5)

From (3.3) the correspondent fuzzy binary relationship would look like that $\forall m\in \left[1,M\right]$

$\begin{array}{c}{\mu }_{\widetilde{P_1^m}\times \widetilde{P_2^m}\times \cdots \times \widetilde{P_K^m}\to {\tilde{Q}}^m}\left(x,y\right)=\{\left[{\displaystyle \prod }_{\forall i\in \left[1,K\right]}{\mu }_{\widetilde{P_i^m}}\left(x\right)\right]\to {\mu }_{\widetilde{Q^m}}\left(y\right)\}\\ \wedge \{\left(1-\left[{\displaystyle \prod }_{\forall i\in \left[1,K\right]}{\mu }_{\widetilde{P_i^m}}\left(x\right)\right]\right)\to \left(1-{\mu }_{\widetilde{Q^m}}\left(y\right)\right)\}\end{array}$ (3.6)

Given (3.5) from (3.6) we are getting $\forall m\in \left[1,M\right]$

$\begin{array}{l}{\mu }_{\widetilde{P_1^m}\times \widetilde{P_2^m}\times \cdots \times \widetilde{P_K^m}\to {\tilde{Q}}^m}\left(x,y\right)\\ =\{\begin{array}{ll}\left(1-\left[{\displaystyle \prod }_{\forall i\in \left[1,K\right]}{\mu }_{\widetilde{P_i^m}}\left(x\right)\right]\right)\cdot {\mu }_{\widetilde{Q^m}}\left(y\right),\hfill & \left[{\displaystyle \prod }_{\forall i\in \left[1,K\right]}{\mu }_{\widetilde{P_i^m}}\left(x\right)\right]<{\mu }_{\widetilde{Q^m}}\left(y\right),\hfill \\ 1,\hfill & \left[{\displaystyle \prod }_{\forall i\in \left[1,K\right]}{\mu }_{\widetilde{P_i^m}}\left(x\right)\right]={\mu }_{\widetilde{Q^m}}\left(y\right),\hfill \\ \left(1-{\mu }_{\widetilde{Q^m}}\left(y\right)\right)\cdot \left[{\displaystyle \prod }_{\forall i\in \left[1,K\right]}{\mu }_{\widetilde{P_i^m}}\left(x\right)\right],\hfill & {\mu }_{\widetilde{Q^m}}\left(y\right)<\left[{\displaystyle \prod }_{\forall i\in \left[1,K\right]}{\mu }_{\widetilde{P_i^m}}\left(x\right)\right].\hfill \end{array}\end{array}$ (3.7)

To further investigate the structure and possible implementation of $\left[{\displaystyle {\prod }_{\forall i\in \left[1,K\right]}{\mu }_{\widetilde{P_i^m}}\left(x\right)}\right]$ in (3.7) we introduce the following.

Definition 1.

The set of values $J_X=\left\{v_i\right\}=\left\{v_1,v_2,\cdots ,v_N\right\}\subseteq \left[0,1\right]$ | is a lattice ($J_X,\le$ ), i.e. $v_1<v_2<\cdots <v_N$ and $\forall a,b\in J_X|a\le b\Rightarrow a\wedge b=a;a\vee b=b$ .

Definition 2.

For a lattice $J_X\subseteq \left[0,1\right]$ and N various normal fuzzy grades in $J_X$ , i.e.

$\forall i\in \left[1,Card{J}_X\right],\forall j\in \left[1,N\right]|{\mu }_{P_j}\left(x\right)={\displaystyle \sum }_i\frac{f_j\left(v_{ij}\right)}{v_{ij}}\Rightarrow$

$\exists !k_j^{*}\in \left[1,Card{J}_X\right]|f_j\left(v_{k_j^{*}}\right)=\underset{\forall i\in \left[1,CardJ\right]}{\max }\left\{f_j\left(v_{ij}\right)\right\}==1$ , we define

$v^{upper}=\underset{\forall j\in \left[1,N\right]}{\text{max}}\left\{v_{k_j^{*}}\right\}$ (3.8)

$v^{lower}=\underset{\forall j\in \left[1,N\right]}{\text{min}}\left\{v_{k_j^{*}}\right\}$ (3.9)

Therefore, the relevant indexes from (2.8) are

$i^{upper}=\left(Card{J}_X\right)\cdot v^{upper }$ (3.10)

$i^{lower}=\left(Card{J}_X\right)\cdot v^{lower}$ (3.11)

From (3.8) and (3.10) we define the upper fuzzy grade

for $\forall i\in \left[1,Card{J}_X\right]$

$\begin{array}{c}{\mu }_P^{upper}\left(x\right)=\frac{f\left(v^{upper}\right)}{v^{upper}}+{\displaystyle \sum }_{i\ne i^{upper}}\frac{f\left(v_i\right)}{v_i}\\ =\frac{1}{v^{upper}}+{\displaystyle \sum }_{i\ne i^{upper}}\frac{f\left(v_i\right)}{v_i}\end{array}$ (3.12)

And from (3.9) and (3.11) we define its lower counterpart

for $\forall i\in \left[1,Card{J}_X\right]$

$\begin{array}{c}{\mu }_P^{lower}\left(x\right)=\frac{f\left(v^{lower}\right)}{v^{lower}}+{\displaystyle \sum }_{i\ne i^{lower}}\frac{f\left(v_i\right)}{v_i}\\ =\frac{1}{v^{lower}}+{\displaystyle \sum }_{i\ne i^{lower}}\frac{f\left(v_i\right)}{v_i}.\end{array}$ (3.13)

Based on Definition1 and Definition2 we shall prove the following.

Lemma 3.

If $J_X\subseteq \left[0,1\right]$ is a lattice, then for N various normal fuzzy grades in $J_X$ , i.e.

${\mu }_{P_j}\left(x\right)={\displaystyle \sum }_i{f}_j\left(v_i\right)/v_i|\forall i\in \left[1,Card{J}_X\right],\forall j\in 1,N$

the following is taking place

$\begin{array}{l}{\displaystyle \prod }_{j=\overline{1,N}}{\mu }_{P_j}\left(x\right)={\displaystyle \prod }_{j=\overline{1,N}}{\displaystyle \sum }_{i=Card{J}_X}\frac{f_j\left(v_i\right)}{v_i}={\mu }_P^{lower}\left(x\right)\\ {\displaystyle \coprod }_{j=\overline{1,N}}{\mu }_{P_j}\left(x\right)={\displaystyle \coprod }_{j=\overline{1,N}}{\displaystyle \sum }_{i=\overline{1,Card{J}_X}}\frac{f_j\left(v_i\right)}{v_i}={\mu }_P^{higher}\left(x\right)\end{array}$ (L3.1)

Proof:

From Definition 2 we have the set $V\subseteq J_X=\left\{v_{k^{*}}\right\}=\left\{v_{k_1^{*}},v_{k_2^{*}},\cdots ,v_{k_M^{*}}\right\}$ | $V\subseteq \left[v^{lower},v^{upper}\right]$ .

Since $v^{lower},v^{upper}\in J_X|v^{lower}<v^{upper}$ , from (3.12) and (3.13) we obtain the following

${\displaystyle \prod }_{j=\overline{1,N}}{\mu }_{P_j}\left(x\right)={\mu }_P^{lower}\left(x\right)\sqcap {\mu }_P^{upper}\left(x\right)={\mu }_P^{lower}\left(x\right),$

${\displaystyle \coprod }_{j=\overline{1,N}}{\mu }_{P_j}\left(x\right)={\mu }_P^{lower}\left(x\right)\bigsqcup {\mu }_P^{upper}\left(x\right)={\mu }_P^{upper}\left(x\right).$■

3.3. Type-2 FLS: Implication

It was mentioned in [6]-[8] that “the distinction between type-1 and type-2 is associated with the nature of the MFs, which is not important while forming ‘IF-THEN’ rules”. In opposite, from our point of view, this aspect needs additional attention. To further elaborate on type-2 FLS inference specificity and on correspondent “IF-THEN” rules formulation, we have to mention that in contrast with type-1 case, these rules are describing not a knowledge about input/output system behavior, but rather between our semantic interpretation/understanding of a nature of things under consideration. For this particular reason we are formulating the following.

Proposition 1.

Let ${\mu }_P\left(x\right)$ and ${\mu }_Q\left(x\right)$ be two fuzzy grades in $J_X\subseteq \left[0,1\right]$ of FS of type 2, $P$ and $Q$ , represented as (2.12) and (2.13) correspondingly. Therefore, by applying an extension Zadeh principle [2.16], for type-2 FS implication operator we obtain the following

$\begin{array}{l}P\to Q\iff {\mu }_{P\to Q}\left(x\right)={\mu }_p\left(x\right)\to {\mu }_Q\left(x\right)\\ ={\displaystyle \sum }_i\frac{f\left(u_i\right)}{u_i}\to {\displaystyle \sum }_j\frac{g\left(w_j\right)}{w_j}\\ ={\displaystyle \sum }_{i,j}\left(f\left(u_i\right)\wedge g\left(w_j\right)\right)/\left(u_i\to w_j\right),\end{array}$ (3.14)

where

$\left(u\to w\right)=\{\begin{array}{ll}\left(1-w\right)\cdot u,\hfill & u<w,\hfill \\ 1,\hfill & u=w,\hfill \\ \left(1-u\right)\cdot w,\hfill & u>w.\hfill \end{array}$ (3.15)

Let us represent the implication operation on type-2 FS from (3.14) and (3.15), as

$\begin{array}{l}{\mu }_p\left(x\right)\to {\mu }_Q\left(x\right)=R\left(u,w\right)={\displaystyle \sum }_{i,j}\left(f\left(u_i\right)\wedge g\left(w_j\right)\right)/\left(u_i\to w_j\right)\\ ={\displaystyle \sum }_{u_i<w_j}\frac{f\left(u_i\right)\wedge g\left(w_j\right)}{\left(1-w_j\right)\cdot u_i}+{\displaystyle \sum }_{u_i=w_j}\frac{f\left(u_i\right)\wedge g\left(w_j\right)}{1}\\ +{\displaystyle \sum }_{u_i>w_j}\frac{f\left(u_i\right)\wedge g\left(w_j\right)}{\left(1-u_i\right)\cdot w_j}\end{array}$ (3.16)

We have to mark the fact that the implication operator (3.15). utilized in (3.14), indicates a degree of closeness between $\forall u,w\in J_X$ values. That closeness, by the very nature of type-2 FS linguistic scales in use, might be interpreted as a degree of semantic similarity.

3.4. Type-2 FLS “IF-THEN” Rules Semantic Dissimilarity

Let us denote a fuzzy binary relationship $R\left(u,w\right)$ from (3.16) like a sum of three subsets, i.e. for $\forall i,j\in \left[1,Card{J}_X\right]$ we have

$R\left(u,w\right)=R_{u_i<w_j}\left(u_i,w_j\right)+R_{u_i=w_j}\left(u_i,w_j\right)+R_{u_i>w_j}\left(u_i,w_j\right)$ (3.17)

Based on that let us first consider two subsets of $R\left(u,w\right)$ from (3.17) $R_{u_i<w_j}\left(u_i,w_j\right)$ and $R_{u_i>w_j}\left(u_i,w_j\right)$ for $\forall i,j\in \left[1,Card{J}_X\right]$ , in which

$\left(u\to w\right)=\{\begin{array}{l}\left(1-w\right)\cdot u,u<w,\\ \left(1-u\right)\cdot w,u>w.\end{array}$

In fact, these subsets, when $R_{u_i\ne w_j}\left(u_i,w_j\right)$ for $\forall i,j\in \left[1,Card{J}_X\right]$ indicate the degree of semantic dissimilarity between $\forall u,w\in J_X$ values. In connection with that we have to investigate the following feature of values within single interval $a\in \left[0,1\right]$ . For this reason, let us formulate the following.

Lemma 4.

If $f\left(a\right)=a\cdot \left(1-a\right),\forall a\in \left[0,1\right]$ , then the $\underset{\forall a\in \left[0,1\right]}{\max }\left[f\left(a\right)\right]=0.25$

Proof:

We have $f\left(a\right)=a\cdot \left(1-a\right)=a-a^2$ . To get the value of $a$ , which corresponds to $\underset{\forall a\in \left[0,1\right]}{\max }\left[a\cdot \left(1-a\right)\right]$ we use some basic principle of an optimization theory by taking first derivative of $f\left(a\right)$ and equalize it to zero, i.e. $f^{\prime }\left(a\right)=0$ . Since $f^{\prime }\left(a\right)=1-2\cdot a=0$ , then we obtain $a=0.5$ . Therefore, we are getting the following $\underset{\forall a\in \left[0,1\right]}{\max }\left[f\left(a\right)\right]=0.25$ . ■

Corollary 2.

From the result of Lemma 4 and from (3.15) we obtain the following $\forall a,b\in \left[0,1\right]$ , $f\left(a,b\right)=\{\begin{array}{l}a\cdot \left(1-b\right),a<b\\ \left(1-a\right)\cdot b,a>b\end{array}$ . And since $a\ne b\Rightarrow a\cdot \left(1-b\right)<\underset{\forall a\in \left[0,1\right]}{\max }\left[a\cdot \left(1-a\right)\right]$ and $\left(1-a\right)\cdot b<\underset{\forall a\in \left[0,1\right]}{\max }\left[a\cdot \left(1-a\right)\right]$ we have $a\cdot \left(1-b\right)<0.25$ and $\left(1-a\right)\cdot b<0.25$ .

Example 4.

Assume the $Card{J}_X=6$ , then again from (2.10) we have $J_X=\left\{0,0.2,0.4,0.6,0.8,1\right\}\subseteq \left[0,1\right]$ . Given the fact that $\forall u,w\in J_X$ , then the closest value for both $\left(1-w\right)\cdot u$ , $u<w$ and $\left(1-u\right)\cdot w$ , $u>w$ expressions to max possible value of 0.25 is 0.16 ($u=0.4$ ; $w=0.6$ ) and ($u=0.6$ ; $w=0.4$ ).

3.5. Type-2 FLS: “IF-THEN” Rules Semantic Similarity

Now let us consider the issue of semantic similarity between $\forall u,w_{}\in J_X$ values, i.e. the subset $R_{u_i=w_j}\left(u_i,w_j\right)$ for $\forall i,j\in \left[1,Card{J}_X\right]$ from (3.17), in which $\left(u\to w\right)=1$ , $u=w$ .

From (3.16) and (3.17) the subset $R_{u_i=w_j}\left(u_i,w_j\right)$ for $\forall i,j\in \left[1,Card{J}_X\right]$ is in fact, the major diagonal of a matrix $R\left(u,w\right)$ . And it looks like that $\forall i,j\in \left[1,Card{J}_X\right]$ ,

$R_{u_i=w_j}\left(u_i,w_j\right)={\displaystyle \sum }_{u_i=w_j}\frac{f\left(u_i\right)\wedge g\left(w_j\right)}{1}$ (3.18)

Lemma 5.

If MFs $f\left(u\right)$ and $g\left(w\right)$ of fuzzy grades

${\mu }_P\left(x\right)={\displaystyle \sum }_i\frac{f\left(u_i\right)}{u_i},u_i\in J_X\text{and}{\mu }_Q\left(x\right)={\displaystyle \sum }_j\frac{f\left(w_j\right)}{w_j},w_j\in J_X$ (L5.1)

are both normal (2.17), i.e.

$\{\begin{array}{l}\exists !k^{\text{*}}\in \left[1,Card{J}_X\right]|f\left(u_{k^{\text{*}}}\right)=\underset{\forall i\in \left[1,Card{J}_X\right]}{\max }\left\{f\left(u_i\right)\right\}==1;\\ \exists !l^{\text{*}}\in \left[1,Card{J}_X\right]|g\left(w_{l^{\text{*}}}\right)=\underset{\forall j\in \left[1,Card{J}_X\right]}{\max }\left\{g\left(w_j\right)\right\}==1\end{array}$ (L5.2)

and if MF $h\left(v\right)|\forall v\in J_X$ , is an intersection of MFs $f\left(u\right)$ and $g\left(w\right)$ , $h\left(v\right)=f\left(u\right)\wedge g\left(w\right)$ from (3.18), and it is presented as

$h\left(v\right)={\displaystyle \sum }_{\forall m\in \left[1,Card{J}_X\right]}h\left(v_m\right)={\displaystyle \sum }_{u_i\to w_j=1}f\left(u_i\right)\wedge g\left(w_j\right),$ (L5.3)

then it is normal and unimodal iff $\forall i,j\in \left[1,Card{J}_X\right]|u_i\to w_j=1\Rightarrow f\left(u_i\right)==g\left(w_j\right)$ .

Proof:

By definition of a logical conjunction operation, we have the following $\forall a,b\in \left[0,1\right]$ , $F\left(a,b\right)=a\wedge b=1\Rightarrow a=b=1$ . Therefore, from normality of MFs $f\left(u\right)$ and $g\left(w\right)$ from (2.17) $\exists !k^{\text{*}},l^{\text{*}}\in \left[1,Card{J}_X\right]\Rightarrow \exists !m^{\text{*}}|h\left(v_{m^{\text{*}}}\right)=f\left(u_{k^{\text{*}}}\right)\wedge g\left(w_{l^{\text{*}}}\right)=1$ . In other words, we obtain the case when

$m^{\text{*}}=k^{\text{*}}=l^{\text{*}}|h\left(v\right)=f\left(u\right)=g\left(w\right)\Rightarrow h\left(v\right)$ is normal and unimodal. ■

Example 5.

1) The case when fuzzy grades for both input and output are “low”

${\mu }_P\left(x\right)$ = low = 1/0 + 0.8/.2 + 0.6/.4 + 0.4/.6 + 0.2/.8 + 0/1

${\mu }_Q\left(x\right)$ = low = 1/0 + 0.8/.2 + 0.6/.4 + 0.4/.6 + 0.2/.8 + 0/1

$R\left(u,w\right)=\left({\mu }_P\left(x\right)\to {\mu }_Q\left(x\right)\right)=$

1/1	0.8/0	0.6/0	0.4/0	0.2/0	0/0
0.8/0	0.8/1	0.6/0.12	0.4/08	0.2/04	0/0
0.6/0	0.6/0.12	0.6/1	0.4/0.16	0.2/08	0/0
0.4/0	0.4/08	0.4/0.16	0.4/1	0.2/0.12	0/0
0.2/0	0.2/04	0.2/08	0.2/0.12	0.2/1	0/0
0/0	0/0	0/0	0/0	0/0	0/1

The major diagonal of a matrix $R\left(u,w\right)$ is $\forall i,j\in \left[1,Card{J}_X\right]$

$R_{u_i=w_j}\left(u_i,w_j\right)$ = 1/1 + 0.8/1 + 0.6/1 + 0.4/1 + 0.2/1 + 0/1

2) The case when fuzzy grades for both input and output are “high”

${\mu }_P\left(x\right)$ = high = 0/0 + 0.2/0.2 + 0.4/0.4 + 0.6/0.6 +0.8/0.8 + 1/1

${\mu }_Q\left(x\right)$ = high = 0/0 +0.2/0.2 +0.4/0.4 +0.6/0.6 +0.8/0.8 + 1/1

$R\left(u,w\right)=\left({\mu }_P\left(x\right)\to {\mu }_Q\left(x\right)\right)=$

0/1	0/0	0/0	0/0	0/0	0/0
0/0	0.2/1	0.2/0.12	0.2/08	0.2/04	0.2/0
0/0	0.2/0.12	0.4/1	0.4/0.16	0.4/08	0.4/0
0/0	0.2/08	0.4/0.16	0.6/1	0.6/0.12	0.6/0
0/0	0.2/04	0.4/08	0.6/0.12	0.8/1	0.8/0
0/0	0.2/0	0.4/0	0.6/0	0.8/0	1/1

The major diagonal of a matrix $R\left(u,w\right)$ is $\forall i,j\in \left[1,Card{J}_X\right]$

$R_{u_i=w_j}\left(u_i,w_j\right)$ = 0/1 + 0.2/1 + 0.4/1 + 0.6/1 + 0.8/1 + 1/1

3) The case when fuzzy grades for both input and output are “middle”

${\mu }_P\left(x\right)$ = middle = 0.6/0 + 0.8/0.2 + 1/.4 +0.8/0.6 +0.6/0.8 +0.4/1

${\mu }_Q\left(x\right)$ = middle = 0.6/0 +0.8/0.2 +1/.4 +0.8/0.6 +0.6/0.8 +0.4/1

$R\left(u,w\right)=\left({\mu }_P\left(x\right)\to {\mu }_Q\left(x\right)\right)=$

0.6/1	0.6/0	0.6/0	0.6/0	0.6/0	0.4/0
0.6/0	0.8/1	0.8/0.12	0.8/08	0.6/04	0.4/0
0.6/0	0.8/0.12	1/1	0.8/0.16	0.6/08	0.4/0
0.6/0	0.8/08	0.8/0.16	0.8/1	0.6/0.12	0.4/0
0.6/0	0.6/04	0.6/08	0.6/0.12	0.6/1	0.4/0
0.4/0	0.4/0	0.4/0	0.4/0	0.4/0	0.4/1

The major diagonal of a matrix $R\left(u,w\right)$ is $\forall i,j\in \left[1,Card{J}_X\right]$

$R_{u_i=w_j}\left(u_i,w_j\right)$ = 0.6/1 + 0.8/1 + 1/1 + 0.8/1 + 0.6/1 + 0.4/1

Corollary 3.

If MFs $f\left(u\right)$ and $g\left(w\right)$ of fuzzy grades of an input ${\mu }_P\left(x\right)$ and an output ${\mu }_Q\left(x\right)$ correspondingly, satisfy conditions (L5.1) - (L5.3) from Lemma 5, but $\forall i,j\in \left[1,Card{J}_{}{}_X\right]$ |$u_i\to w_j=1\Rightarrow f\left(u_i\right)\ne g\left(w_j\right)$ , then MF $h\left(v\right)$ from (L5.3) is subnormal.

Proof:

From (L5.1) and (L5.2) and $f\left(u_i\right)\ne g\left(w_j\right)$ ,| $\forall i,j\in \left[1,Card{J}_X\right]|u_i\to w_j=1$ $\Rightarrow$ $k^{\text{*}}\ne l^{\text{*}}$ , therefore $\forall m\in \left[1,Card{J}_X\right]$ , $h\left(v_m\right)=f\left(u_i\right)\wedge g\left(w_j\right)<f\left(u_{k^{\text{*}}}\right)==1$ or $h\left(v_m\right)=f\left(u_i\right)\wedge g\left(w_j\right)<g\left(w_{l^{\text{*}}}\right)==1$ . Hence $\exists !m^{\text{*}}|h\left(v_{m^{\text{*}}}\right)==\underset{\forall m\in \left[1,Card{J}_X\right]}{\max }\left\{h\left(v_m\right)\right\}<1$ . ■

Definition 3.

If ${\mu }_P\left(x\right)$ and ${\mu }_Q\left(x\right)$ are two fuzzy grades in $J_X\subseteq \left[0,1\right]$ of FS of type 2, $P$ and $Q$ , respectively, represented as

${\mu }_P\left(x\right)={\displaystyle \sum }_i\frac{f\left(v_i\right)}{v_i}\text{and}{\mu }_Q\left(x\right)={\displaystyle \sum }_i\frac{g\left(v_i\right)}{v_i},v_i\in J_X,$ (3.19)

then we call FS of type 2 $Q$ from (3.19) Semantically Inversive in relation to FS of type 2 $P$ if $g\left(v_i\right)=1-f\left(v_i\right)$ , $v_i\in J_X$ . In other words we have the following

${\mu }_Q\left(x\right)={\displaystyle \sum }_i\frac{1-f\left(v_i\right)}{v_i},v_i\in J_X$ (3.20)

Lemma 6.

If $\forall a\in \left[0,1\right]|f\left(a\right)=a\wedge \left(1-a\right)=\text{min}\left(a,1-a\right)$ , then $f\left(a\right)\le 0.5$

Proof:

From algebraic definition of a min of two numbers we obtain the following

$f\left(a\right)=\min \left(a,1-a\right)=\frac{a+\left(1-a\right)-\left|a-\left(1-a\right)\right|}{2}.$

Let present $f\left(a\right)$ in the following way $f\left(a\right)=\{\begin{array}{ll}{f}_1\left(a\right),\hfill & a\ge \left(1-a\right)\hfill \\ f_2\left(a\right),\hfill & a<\left(1-a\right)\hfill \end{array}$ .

Thus, we have

$f_1\left(a\right)=\frac{1-a+\left(1-a\right)}{2}=\frac{2-2a}{2}=1-a.$

And

$f_2\left(a\right)=\frac{1-1+2a}{2}=\frac{2a}{2}=a.$

Finally

$\begin{array}{c}f\left(a\right)=\{\begin{array}{ll}1-a,\hfill & a\ge \left(1-a\right)\hfill \\ a,\hfill & a<\left(1-a\right)\hfill \end{array}=\{\begin{array}{ll}1-a,\hfill & 2a\ge 1\hfill \\ a,\hfill & 2a<1\hfill \end{array}\\ =\{\begin{array}{ll}1-a,\hfill & a\ge 0.5\hfill \\ a,\hfill & a<0.5\hfill \end{array}=\{\begin{array}{ll}1-a,\hfill & 1-a\le 0.5\hfill \\ a,\hfill & a<0.5\hfill \end{array}\end{array}$

$\Rightarrow f\left(a\right)\le \mathrm{0.5.}$ ■

Corollary 4.

Hence, from Lemma 6, if MF $h\left(v\right)|\forall v\in J_X$ is an intersection of MFs $f\left(u\right)$ and $g\left(w\right)$ , $h\left(v\right)=f\left(u\right)\wedge g\left(w\right)$ from (3.19) and (3.2), and if it is presented as (L5.3), then it is subnormal and

$\underset{\forall m\in \left[1,Card{J}_X\right]}{\max }\left\{h\left(v_m\right)\right\}\le 0.5$ . (3.21)

Example 6.

Here is a case of Semantically Inversive fuzzy grades “low” and “high” in use.

4) The case when input fuzzy grade is “low” and output one is “high”

${\mu }_P\left(x\right)$ = low = 1/0 + 0.8/0.2 + 0.6/0.4 + 0.4/0.6 + 0.2/0.8 + 0/1

${\mu }_Q\left(x\right)$ = high = 0/0 + 0.2/0.2 + 0.4/0.4 + 0.6/0.6 + 0.8/0.8 + 1/1

$R\left(u,w\right)=\left({\mu }_P\left(x\right)\to {\mu }_Q\left(x\right)\right)=$

0/1	0.2/0	0.4/0	0.6/0	0.8/0	1/0
0/0	0.2/1	0.4/0.12	0.6/08	0.8/04	0.8/0
0/0	0.2/0.12	0.4/1	0.6/0.16	0.6/08	0.6/0
0/0	0.2/08	0.4/0.16	0.4/1	0.4/0.12	0.4/0
0/0	0.2/04	0.2/08	0.2/0.12	0.2/1	0.2/0
0/0	0/0	0/0	0/0	0/0	0/1

The major diagonal of a matrix $R\left(u,w\right)$ is $\forall i,j\in \left[1,Card{J}_X\right]$

$R_{u_i=w_j}\left(u_i,w_j\right)$ = 0/1 + 0.2/1 + 0.4/1 + 0.4/1 + 0.2/1 + 0/1

3.6. Type-2 FLS: Improved Sup-Star Composition

Let, as usual ${\mu }_P\left(x\right)$ and ${\mu }_Q\left(x\right)$ be two fuzzy grades in $J_X\subseteq \left[0,1\right]$ of FS of type 2, $P$ and $Q$ , represented as a system input (2.12) and an output (2.13) correspondingly. And let an input/output fuzzy relationship matrix be defined as $R\left(u,w\right)={\mu }_p\left(x\right)\to {\mu }_Q\left(x\right)$ from (3.16). Hence, we use a Fuzzy Conditional Inference Rule (FCIR), formulated by means of “common sense” as a following conditional clause:

R = “IF (p is P), THEN (q is Q)” (3.22)

In other words, we use fuzzy conditional inference of the following type [11]:

Ant 1: If Input is P, then Output is Q

Ant2: Input is P'

-------------------------------------------- (3.23)

Cons: Output is Q'

where $P,P^{\prime }\subseteq U_{ }$ and $Q,Q^{\prime }\subseteq W$ . Note, that ${\mu }_{P^{\prime }}\left(x\right)$ and ${\mu }_{Q^{\prime }}\left(x\right)$ are two fuzzy grades in $J_X\subseteq \left[0,1\right]$ of FS of type 2, $P^{\prime }$ and $Q^{\prime }$ from (3.23), represented as a system current input of type

$\begin{array}{l}{\mu }_{P^{\prime }}\left(x\right)={\displaystyle \sum }_i\frac{f^{\prime }\left(v_i\right)}{v_i},v_i\in J_X\\ {\mu }_{Q^{\prime }}\left(x\right)={\displaystyle \sum }_i\frac{g^{\prime }\left(w_i\right)}{w_i},w_i\in J_X.\end{array}$ (3.24)

Given a unary relationship $R\left(u\right)=P^{\prime }$ one can obtain the consequence $R\left(w\right)$ by Sup-Star Composition to $R\left(u\right)$ and $R\left(u,w\right)$ of type (3.16):

$\begin{array}{c}R\left(w\right)=R\left(u\right)\circ R\left(u,w\right)={\displaystyle \sum }_i\frac{f^{\prime }\left(u_i\right)}{u_i}\circ {\displaystyle \sum }_{i,j}\left(f\left(u_i\right)\wedge g\left(w_j\right)\right)/\left(u_i\to w_j\right)\\ ={\displaystyle \sum }_{i,j}{\displaystyle \coprod }_i[\frac{f^{\prime }\left(u_i\right)}{u_i}\sqcap \frac{\left(f\left(u_i\right)\wedge g\left(w_j\right)\right)}{(u_i\to w_j)},\forall i,j\in \left[1,Card{J}_X\right]\end{array}$ (3.25)

The value of an output fuzzy set $Q^{\prime }$ could be consider as

$Q^{\prime }=R\left(w\right)$ (3.26)

Lemma 7.

Assume that ${\mu }_P\left(x\right),{\mu }_{P^{\prime }}\left(x\right)$ and ${\mu }_Q\left(x\right)$ are fuzzy grades in $J_X\subseteq \left[0,1\right]$ of FS of type 2; $P$ and $Q$ represent a system input (2.12) and an output (2.13), correspondingly and $P^{\prime }$ represents a system current input (3.24). If an input/output fuzzy relationship matrix is defined as $R\left(u,w\right)={\mu }_p\left(x\right)\to {\mu }_Q\left(x\right)$ from (3.16), then Sup-Star Composition (3.25) is reduced to the following $R\left(w\right)=R\left(u\right)\circ R\left(u,w\right)=R\left(u\right)\sqcap R_{u_i=w_j}\left(u_i,w_j\right)$ for $\forall i,j\in \left[1,Card{J}_X\right]$ . Where $R_{u_i=w_j}\left(u_i,w_j\right)$ for $\forall i,j\in \left[1,Card{J}_X\right]$ from (3.18) is the major diagonal of a matrix $R\left(u,w\right)$ .

Proof:

Since $R\left(u,w\right)$ from (3.17) consists of three subsets $R_{u_i<w_j}\left(u_i,w_j\right)$ , $R_{u_i=w_j}\left(u_i,w_j\right)$ and $R_{u_i>w_j}\left(u_i,w_j\right)$ , then from (3.25) we obtain

$\begin{array}{l}R\left(w\right)={\displaystyle \sum }_{u_i<w_j}{\displaystyle \coprod }_i[\frac{f^{\prime }\left(u_i\right)}{u_i}\sqcap \frac{\left(f\left(u_i\right)\wedge g\left(w_j\right)\right)}{\left(1-w_j\right)\cdot u_i}\\ +{\displaystyle \sum }_{u_i=w_j}{\displaystyle \coprod }_i[\frac{f^{\prime }\left(u_i\right)}{u_i}\sqcap \frac{\left(f\left(u_i\right)\wedge g\left(w_j\right)\right)}{1}\\ +{\displaystyle \sum }_{u_i>w_j}{\displaystyle \coprod }_i[\frac{f^{\prime }\left(u_i\right)}{u_i}\sqcap \frac{\left(f\left(u_i\right)\wedge g\left(w_j\right)\right)}{\left(1-u_i\right)\cdot w_j},\forall i,j\in \left[1,Card{J}_X\right]\end{array}$ (3.27)

From (3.27) for meet operator $\sqcap$ we consider the following three cases for $\forall i,j\in \left[1,Card{J}_X\right]$

$\{\begin{array}{l}\frac{f^{\prime }\left(u_i\right)}{u_i}\sqcap \frac{\left(f\left(u_i\right)\wedge g\left(w_j\right)\right)}{\left(1-w_j\right)\cdot u_i},u_i<w_j;\\ \frac{f^{\prime }\left(u_i\right)}{u_i}\sqcap \frac{\left(f\left(u_i\right)\wedge g\left(w_j\right)\right)}{1},u_i=w_j;\\ \frac{f^{\prime }\left(u_i\right)}{u_i}\sqcap \frac{\left(f\left(u_i\right)\wedge g\left(w_j\right)\right)}{\left(1-u_i\right)\cdot w_j},u_i>w_j;.\end{array}$ (3.28)

Note, that for all (3.28) cases the meet operation is realized for involved MFs like $h\left(u_i,w_j\right)=f^{\text{'}}\left(u_i\right)\wedge f\left(u_i\right)\wedge g\left(w_j\right)$ for $\forall i,j\in \left[1,Card{J}_X\right]$ , whereas for values $\forall u_i,w_j\in J_X$ we have the following

$\{\begin{array}{ll}{v}_{ij}=u_i\wedge \left(1-w_j\right)\cdot u_i,\hfill & u_i<w_j;\hfill \\ v_{ij}=u_i\wedge 1,\hfill & u_i=w_j;\hfill \\ v_{ij}=u_i\wedge \left(1-u_i\right)\cdot w_j,\hfill & u_i>w_j\hfill \end{array}$ (3.29)

From (3.29), given the result of Corollary 2, $\left(1-w_j\right)\cdot u_i<0.25|u_i<w_j$ and $\left(1-u_i\right)\cdot w_j<0.25|u_i>w_j$ . Hence, $v_{ij}<0.25\Rightarrow v_{ij}\notin J_X|u_i\ne w_j$ , whereas $v_{ij}=u_i\Rightarrow v_{ij}\in J_X|u_i=w_j$ and therefore we can rewrite (3.27) as follows $\forall i,j\in \left[1,Card{J}_{}{}_X\right]$

$R\left(w\right)={\displaystyle \sum }_{u_i=w_j}{\displaystyle \coprod }_i[\frac{f^{\prime }\left(u_i\right)}{u_i}\sqcap \frac{\left(f\left(u_i\right)\wedge g\left(w_j\right)\right)}{1}=R\left(u\right)\sqcap R_{u_i=w_j}\left(u_i,w_j\right)$ ■

Example 7.

Let $P$ and $Q$ represent a system input (2.12) and an output (2.13).

Correspondingly, with the following fuzzy grades

${\mu }_P\left(x\right)$ = in between low and mid = 0.8/0 + 1/0.2 + 0.8/0.4 + 0.6/0.6 + 0.4/0.8 + 0.2/1

${\mu }_Q\left(x\right)$ = in between low and mid = 0.8/0 + 1/0.2 + 0.8/0.4 + 0.6/0.6 + 0.4/0.8 + 0.2/1

An input/output fuzzy relationship matrix is

$R\left(u,w\right)=\left({\mu }_P\left(x\right)\to {\mu }_Q\left(x\right)\right)=$

0.8/1	0.8/0	0.8/0	0.6/0	0.4/0	0.2/0
0.8/0	1/1	0.8/0.12	0.6/0.08	0.4/0.04	0.2/0
0.8/0	0.8/0.12	0.8/1	0.6/0.16	0.4/0.08	0.2/0
0.6/0	0.6/0.08	0.6/0.16	0.6/1	0.4/0.12	0.2/0
0.4/0	0.4/0.04	0.4/0.08	0.4/0.12	0.4/1	0.2/0
0.2/0	0.2/0	0.2/0	0.2/0	0.2/0	0.2/1

The major diagonal of a matrix $R\left(u,w\right)$ is $\forall i,j\in \left[1,Card{J}_X\right]$

$R_{u_i=w_j}\left(u_i,w_j\right)$ = 0.8/1 + 1/1 + 0.8/1 + 0.6/1 + 0.4/1 + 0.2/1

And let a system current input $P^{\prime }$ from (3.24) to be

${\mu }_{P^{\prime }}\left(x\right)$ = middle = 0.6/0 + 0.8/0.2 + 1/0.4 + 0.8/0.6 + 0.6/0.8 + 0.4/1

The proposed version of Sup-Star Composition would be

$R\left(w\right)=R\left(u\right)\sqcap R_{u_i=w_j}\left(u_i,w_j\right)={\mu }_{P^{\prime }}\left(x\right)\sqcap R_{u_i=w_j}\left(u_i,w_j\right)=$

(0.6∧0.8)/(0∧1)	(0.6∧1)/(0∧1)	(0.6∧0.8)/(0∧1)	(0.6∧0.6)/(0∧1)	(0.6∧0.4)/(0∧1)	(0.6∧0.2)/(0∧1)
(0.8∧0.8)/(0.2∧1)	(0.8∧1)/(0.2∧1)	(0.8∧0.8)/(0.2∧1)	(0.8∧0.6)/(0.2∧1)	(0.8∧0.4)/(0.2∧1)	(0.8∧0.2)/(0.2∧1)
(1∧0.8)/(0.4∧1)	(1∧1)/(0.4∧1)	(1∧0.8)/(0.4∧1)	(1∧0.6)/(0.4∧1)	(1∧0.4)/(0.4∧1)	(1∧0.2)/(0.4∧1)
(0.8∧0.8)/(0.6∧1)	(0.8∧1)/(0.6∧1)	(0.8∧0.8)/(0.6∧1)	(0.8∧0.6)/(0.6∧1)	(0.8∧0.4)/(0.6∧1)	(0.8∧0.2)/(0.6∧1)
(0.6∧0.8)/(0.8∧1)	(0.6∧1)/(0.8∧1)	(0.6∧0.8)/(0.8∧1)	(0.6∧0.6)/(0.8∧1)	(0.6∧0.4)/(0.8∧1)	(0.6∧0.2)/(0.8∧1)
(0.4∧0.8)/(1∧1)	(0.4∧1)/(1∧1)	(0.4∧0.8)/(1∧1)	(0.4∧0.6)/(1∧1)	(0.4∧0.4)/(1∧1)	(0.4∧0.2)/(1∧1)

=

0.6/0	0.6/0	0.6/0	0.6/0	0.4/0	0.2/0
0.8/0.2	0.8/0.2	0.8/0.2	0.6/0.2	0.4/0.2	0.2/0.2
0.8/0.4	1/0.4	0.8/0.4	0.6/0.4	0.4/0.4	0.2/0.4
0.8/0.6	0.8/0.6	0.8/0.6	0.6/0.6	0.4/0.6	0.2/0.6
0.6/0.8	0.6/0.8	0.6/0.8	0.6/0.8	0.4/0.8	0.2/0.8
0.4/1	0.4/1	0.4/1	0.4/1	0.4/1	0.2/1

Thus, a system current output

$Q^{\prime }$ = 0.6/0 + 0.8/0.2 + 1/0.4 + 0.8/0.6 + 0.6/0.8 + 0.4/1$\Rightarrow {\mu }_{Q^{\prime }}\left(x\right)$ = middle

Example 8.

Here is a case of Semantically Inversive fuzzy grades for input “high” and output “low”.

${\mu }_P\left(x\right)$ = high = 0/0 + 0.2/0.2 + 0.4/0.4 + 0.6/0.6 + 0.8/0.8 + 1/1

${\mu }_Q\left(x\right)$ = low = 1/0 + 0.8/.2 + 0.6/0.4 +0.4/0.6 +0.2/0.8 +0/1

$R\left(u,w\right)=\left({\mu }_P\left(x\right)\to {\mu }_Q\left(x\right)\right)=$

0/1	0/0	0/0	0/0	0/0	0/0
0.2/0	0.2/1	0.2/0.12	0.2/08	0.2/04	0/0
0.4/0	0.4/0.12	0.4/1	0.4/0.16	0.2/08	0/0
0.6/0	0.6/08	0.6/0.16	0.4/1	0.2/0.12	0/0
0.8/0	0.8/04	0.6/08	0.4/0.12	0.2/1	0/0
1/0	0.8/0	0.6/0	0.4/0	0.2/0	0/1

The major diagonal of a matrix $R\left(u,w\right)$ is $\forall i,j\in \left[1,Card{J}_X\right]$

$R_{u_i=w_j}\left(u_i,w_j\right)$ = 0/1 + 0.2/1 + 0.4/1 + 0.4/1+ 0.2/1 + 0/1

And let a system current input $P^{\prime }$ from (3.24) to be

${\mu }_{P^{\prime }}\left(x\right)$ = middle = 0.6/0 + 0.8/0.2 + 1/0.4 + 0.8/0.6 + 0.6/0.8 + 0.4/1

Again, the proposed version of Sup-Star Composition would be

$R\left(w\right)=R\left(u\right)\sqcap R_{u_i=w_j}\left(u_i,w_j\right)={\mu }_{P^{\prime }}\left(x\right)\sqcap R_{u_i=w_j}\left(u_i,w_j\right)=$

(0.6∧0)/(0∧1)	(0.6∧0.2)/(0∧1)	(0.6∧0.4)/(0∧1)	(0.6∧0.4)/(0∧1)	(0.6∧0.2)/(0∧1)	(0.6∧0)/(0∧1)
(0.8∧0)/(0.2∧1)	(0.8∧0.2)/(0.2∧1)	(0.8∧0.4)/(0.2∧1)	(0.8∧0.4)/(0.2∧1)	(0.8∧0.2)/(0.2∧1)	(0.8∧0)/(0.2∧1)
(1∧0)/(0.4∧1)	(1∧0.2)/(0.4∧1)	(1∧0.4)/(0.4∧1)	(1∧0.4)/(0.4∧1)	(1∧0.2)/(0.4∧1)	(1∧0)/(0.4∧1)
(0.8∧0)/(0.6∧1)	(0.8∧0.2)/(0.6∧1)	(0.8∧0.4)/(0.6∧1)	(0.8∧0.4)/(0.6∧1)	(0.8∧0.2)/(0.6∧1)	(0.8∧0)/(0.6∧1)
(0.6∧0)/(0.8∧1)	(0.6∧0.2)/(0.8∧1)	(0.6∧0.4)/(0.8∧1)	(0.6∧0.4)/(0.8∧1)	(0.6∧0.2)/(0.8∧1)	(0.6∧0)/(0.8∧1)
(0.4∧0)/(1∧1)	(0.4∧0.2)/(1∧1)	(0.4∧0.4)/(1∧1)	(0.4∧0.4)/(1∧1)	(0.4∧0.2)/(1∧1)	(0.4∧0)/(1∧1)

=

0/0	0.2/0	0.4/0	0.4/0	0.2/0	0/0
0/0.2	0.2/0.2	0.4/0.2	0.4/0.2	0.2/0.2	0/0.2
0/0.4	0.2/0.4	0.4/0.4	0.4/0.4	0.2/0.4	0/0.4
0/0.6	0.2/0.6	0.4/0.6	0.4/0.6	0.2/0.6	0/0.6
0/0.8	0.2/0.8	0.4/0.8	0.4/0.8	0.2/0.8	0/0.8
0/1	0.2/1	0.4/1	0.4/1	0.2/1	0/1

Hence, a system current output

$Q^{\prime }$ = 0.4/0 + 0.4/0.2 + 0.4/0.4 + 0.4/0.6 + 0.4/0.8 + 0.4/1$\Rightarrow {\mu }_{Q^{\prime }}\left(x\right)$ = unknown!!!

Thus, Semantically Inversive fuzzy grades shouldn’t be used for fuzzy relationship matrix build. Using them leads to fruitless results of type-2 FLS inference.

3.7. Type-2 FLS: Streamlined Inference

3.7.1. Knowledge Base

Remind that we still consider a type-2 FLS with K inputs, $x_1\in X_1,x_2\in X_2,\cdots ,x_K\in X_K$ , and one output $y\in Y$ . Presumably this FLS has M rules, where the m-th rule has the form (3.1). Each rule represents a type-2 fuzzy relation between the input space $X_1\times X_2\times \cdots \times X_K$ and the output space $Y$ of the FLS. And again, when an input $x^{\prime }$ is applied, the composition of the fuzzy set $\widetilde{X^{\prime }}$ , to which $x^{\prime }$ takes place and the rule $R^m$ is found by using the extended sup-star composition (3.2). And also, the implication MF from (3.2) for each m-th rule is presented in in (3.3) form.

Given that in (3.13) we define the lower fuzzy grade ${\mu }_{\widetilde{P^m}}^{lower}\left(x\right)$ , then from (L3.1) of Lemma 3 we obtain for K inputs and m rules of FLS the following

${\displaystyle \prod }_{j=\overline{1,K}}{\mu }_{\widetilde{P_j^m}}\left(x\right)={\displaystyle \prod }_{j=\overline{1,K}}{\displaystyle \sum }_{i=\overline{1,Card{J}_X}}\frac{f_j^m\left(u_i\right)}{u_i}={\mu }_{\widetilde{P^m}}^{lower}\left(x\right),\forall m\in \left[1,M\right]$ (3.30)

From (3.7) given (3.30) we are getting for $\forall m\in \left[1,M\right]$

$\begin{array}{l}{\mu }_{\widetilde{P_1^m}\times \widetilde{P_2^m}\times \cdots \times \widetilde{P_K^m}\to {\tilde{Q}}^m}\left(x,y\right)\\ ={\displaystyle \prod }_{j=\overline{1,K}}{\mu }_{\widetilde{P_j^m}}\left(x\right)\to {\mu }_{\widetilde{Q^m}}\left(y\right)={\mu }_{\widetilde{P^m}}^{lower}\left(x\right)\to {\mu }_{\widetilde{Q^m}}\left(y\right)\end{array}$ (3.31)

Given that

${\mu }_{\widetilde{P^m}}^{lower}\left(x\right)={\displaystyle \sum }_{i=\overline{1,Card{J}_X}}{f}^m\left(u_i^{lower}\right)/u_i^{lower}$

${\mu }_{\widetilde{Q^m}}\left(y\right)={\displaystyle \sum }_{j=\overline{1,Card{J}_X}}{g}^m\left(w_j\right)/w_j$

We obtain $\forall m\in \left[1,M\right]$

$\begin{array}{l}{\mu }_{\widetilde{P^m}}^{lower}\left(x\right)\to {\mu }_{\widetilde{Q^m}}\left(y\right)={\displaystyle \sum }_{i,j}\left(f^m\left(u_i^{lower}\right)\wedge g^m\left(w_j\right)\right)/\left(u_i^{lower}\to w_j\right)\\ ={\displaystyle \sum }_{u_i^{lower}<w_j}\frac{f^m\left(u_i^{lower}\right)\wedge g^m\left(w_j\right)}{\left(1-w_j\right)\cdot u_i^{lower}}\\ +{\displaystyle \sum }_{u_i^{lower}=w_j}\frac{f^m\left(u_i^{lower}\right)\wedge g^m\left(w_j\right)}{1}\\ +{\displaystyle \sum }_{u_i^{lower}>w_j}\frac{f^m\left(u_i^{lower}\right)\wedge g^m\left(w_j\right)}{\left(1-u_i^{lower}\right)\cdot w_j}.\end{array}$ (3.32)

From (3.17) and (3.32) for each m rule

$\begin{array}{l}{R}^m\left(u,w\right)={\mu }_{\widetilde{P^m}}^{lower}\left(x\right)\\ \to {\mu }_{\widetilde{Q^m}}\left(y\right)=R_{u_i^{lower}<w_j}^m\left(u_i^{lower},w_j\right)+R_{u_i^{lower}=w_j}^m\left(u_i^{lower},w_j\right)\\ +R_{u_i^{lower}>w_j}^m\left(u_i^{lower},w_j\right)\end{array}$ (3.33)

Now from (3.32) for each m rule and $\forall i,j\in \left[1,Card{J}_X\right]$ we introduce

$h^m\left(v_{ij}\right)=f^m\left(u_i^{lower}\right)\wedge g^m\left(w_j\right)$ (3.34)

From (3.33) and (3.34) we obtain for each m rule

$R^m\left(u,w\right)={\displaystyle \sum }_{\forall i,j\in \left[1,Card{J}_X\right]}{h}^m\left(v_{ij}\right)/v_{ij}$ (3.35)

Hence, for $\forall i,j\in \left[1,Card{J}_X\right]$ , whereas for values $u_i^{lower},w_j\in J_X$ from (3.35) we have the following

$\{\begin{array}{ll}{v}_{ij}=\left[\left(1-w_j\right)\cdot u_i^{lower}\right],\hfill & u_i^{lower}<w_j;\hfill \\ v_{ij}=1,\hfill & u_i^{lower}=w_j;\hfill \\ v_{ij}=\left[\left(1-u_i^{lower}\right)\cdot w_j\right],\hfill & u_i^{lower}>w_j.\hfill \end{array}$ (3.36)

From (3.36) given the result of Corollary 2

$\{\begin{array}{ll}{v}_{ij}\le 0.25,\hfill & u_i^{lower}<w_j;\hfill \\ v_{ij}=1,\hfill & u_i^{lower}=w_j;\hfill \\ v_{ij}\le 0.25,\hfill & u_i^{lower}>w_j.\hfill \end{array}$ (3.37)

Given (3.37) and from the results of Lemma 7 we obtain for each m rule its fuzzy relationship matrix major diagonals

$R^m\left(u,w\right)={\displaystyle \sum }_{\forall k\in \left[1,Card{J}_X\right]}{h}^m\left(v_k\right)/1$ (3.38)

The knowledge base for m rules of FLS has to be represented as a superposition of all single rules (3.38).

$\begin{array}{c}{R}^{sup}\left(u,w\right)={\displaystyle \coprod }_{\forall m\in \left[1,M\right]}{R}^m\left(u,w\right)={\displaystyle \coprod }_{\forall m\in \left[1,M\right]}{\displaystyle \sum }_{\forall k\in \left[1,Card{J}_X\right]}{h}^m\left(v_k\right)/1\\ =\underset{\forall m\in \left[1,M\right]}{{\displaystyle V}}{\displaystyle \sum }_{\forall k\in \left[1,Card{J}_X\right]}{h}^m\left(v_k\right)/1={\displaystyle \sum }_{\forall k\in \left[1,Card{J}_X\right]}{h}^{sup}\left(v_k\right)/1\end{array}$ (3.39)

Note. The usability of FS $R^{sup}\left(u,w\right)$ for it use in sup-max composition depends on the question of whether FSs for each m-th rule $h^m\left(v_k\right)$ are normal, unimodal and more or less semantically similar.

3.7.2. Decision Making

When an input $x^{\prime }$ is applied with K inputs, $x_1\in {X^{\prime }}_1,x_2\in {X^{\prime }}_2,\cdots ,x_K\in {X^{\prime }}_K$ . The input space ${X^{\prime }}_1\times {X^{\prime }}_2\times \cdots \times {X^{\prime }}_K$ is presented with the fuzzy grade

${\mu }_{\widetilde{P^{\prime }}}\left(x\right)={\mu }_{\widetilde{{P^{\prime }}_1}\times \widetilde{{P^{\prime }}_2}\times \cdots \times \widetilde{{P^{\prime }}_K}}\left(x\right)={\displaystyle \prod }_{j=\overline{1,K}}{\mu }_{\widetilde{{P^{\prime }}_j}}\left(x\right)={\mu }_{\widetilde{P^{\prime }}}^{lower}\left(x\right)$ (3.40)

Given (3.39) and (3.40), the correspondent output space $Y^{\prime }$ will be obtained by using streamlined sup-star composition as the following fuzzy grade

$\begin{array}{c}{\mu }_{\widetilde{Q^{\prime }}}\left(y\right)={\mu }_{\widetilde{P^{\prime }}}^{lower}\left(x\right)\sqcap R^{sup}\left(u,w\right)\\ ={\displaystyle \sum }_{\forall i\in \left[1,Card{J}_X\right]}{f}^{\prime }\left(u_i^{lower}\right)/u_i^{lower}\sqcap {\displaystyle \sum }_{\forall k\in \left[1,Card{J}_X\right]}{h}^{sup}\left(v_k\right)/1\end{array}$

4. Conclusions

The paper argues that inference in type-2 FLS should rely on fuzzy implication, not t-norms, because implication better reflects semantic reasoning. The work revisits operations, algebraic properties, and rule semantics in type-2 systems. It defines normalization of physical values and mapping to a discrete universe. It provides formulas for constructing membership functions using singletons and linguistic terms. It proves that fuzzy grades defined via triangular secondary MFs are convex. The paper revises several underexplored aspects of type-2 fuzzy sets and type-2 fuzzy logic systems (FLS), proposing:

• a shift from t-norm-based inference to fuzzy implication-based inference. The paper argues that t-norm-based inference is semantically weak for type-2 rules.

• dramatically faster method for join, meet, operations on type-2 fuzzy grades (Speedup: 8.3× faster), by providing full matrix-based definitions using Zadeh’s extension principle and by showing that these operations form a distributive lattice for convex fuzzy grades.

• a new framework for analyzing semantic similarity and dissimilarity of type-2 IF-THEN rules,

• an improved formulation of sup-star composition for inference. It uses a conditional implication operator and a more symmetric Fuzzy Conditional Rule (FCR).

The study demonstrates that standard sup-star composition is computationally heavy. The paper proves that, under the proposed implication operator, only the diagonal of the implication matrix matters. This greatly reduces computational cost in inference. Numerous examples illustrating the theory.