Summary of mathematics doctoral thesis: Study of real-world semantics-based interpretability of fuzzy system

GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY
VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

NGUYEN THU ANH

Study of real-world semantics-based
interpretability of fuzzy system

Major: MATHEMATICAL BASIS FOR INFORMATICS
Code: 62.46.01.10

SUMMARY OF MATHEMATICS DOCTORAL THESIS

SCIENTIFIC INSTRUCTOR:
Ph.D. Tran Thai Son

Hanoi 2019


1

INTRODUCTION
In some areas, we expect machinery to be able to simulate behavior,
reasoning ability like human and give human reliable suggestions in the
decision-making process. A prominent feature of human is the ability to reason
on the basis of knowledge formed from life and expressed in natural language.
Because the language characteristic is fuzzy, the first problem that needs to be
solved is how to mathematically formalize the problems of linguistic semantic
and handle semantic language that human often uses in daily life.
In response to those requirements, in 1965, Lotfi A. Zadeh was the first
person to lay the foundation for fuzzy set theory. Based on fuzzy set theory,

Fuzzy Rule Based System (FRBS) has been developed and become one of the
tools of simulating reasoning method and making decisions of human in the
most closely manner. FRBS has been successfully applied in solving practical
problems such as control problem, classification problem, regression problem,
language extraction problem, etc...
When building FRBSs, we need to achieve two goals: accuracy and
interpretability. The thesis will focus on the study of interpretability.
In [1]1 Gacto finds that there are currently two main approaches to
interpretability. The first approach is based on complexity and the second
approach is based on semantics. Another approach proposed by Mencar et. al. in
[2]2, called similar measure function-based approach to assess the
interpretability of semantics-based fuzzy rules. The interpretability of fuzzy
rules is measured by the similarity between knowledge represented by fuzzy set
expression and linguistic expression in natural language.
In 2017, a new approach to the interpretability of fuzzy system is Realworld-semantics-based approach – RWS-approach, has been first-time proposed
and initially surveyed in [3]3. This approach is based on real-world semantics of
words and relations between semantics of fuzzy system components and
corresponding component structures in the real world.
Derived from the recognition that fuzzy set expressions, especially fuzzy
rules of fuzzy systems have no relationship based on methodology with real
world semantics and, therefore, there are no formal basis to study the nature of
interpretability, his thesis chooses the real-world-semantics-based approach
proposed in [3] to study the interpretability of fuzzy systems.

1

M.J. Gacto, R. Alcalá, F. Herrera (2011), Interpretability of Linguistic Fuzzy Rule-Based
Systems: An Overview of Interpretability Measures. Inform. Sci., 181:20 pp. 4340–4360.
2
C. Mencar, C. Castiello, R. Cannone, A.M. Fanelli (2011), Interpretability assessment of fuzzy

knowledge bases: a cointension based approach, Int. J. Approx. Reason. 52 pp. 501–518.
3
Cat Ho Nguyen, Jose M. Alonso (2017), “Looking for a real-world-semantics-based approach to
the interpretability of fuzzy systems”. FUZZ-IEEE 2017 Technical Program Committee and
Technical Chairs, Italy, July 9-12.


2

At the same time, at present, methods of building FRBS from data in
fuzzy set theory-based approach lack a full formal link between fuzzy sets
representing the assumed semantics of a word and its inherent semantics. The
words used in FRBS are only considered as labels or symbols assigned to
corresponding fuzzy sets, are very difficult to fully convey underlying semantics
compared with natural linguistic words. Therefore, this thesis wishes to further
study the interpretability of linguistic fuzzy systems in the semantic approach
based on the hedge algebra proposed by Nguyen and Wechler [4] 4 [5]5. In this
approach, the computational semantics of words shall be defined based on the
inherent order semantics of the words and word domains of the variables that
establish an order-based structure that are rich enough to solve the problems in
fact.
This thesis has achieved some following results:
 Research and analysis of interpretability are as a study of the relationship
between RWS of linguistic expressions and computational semantics of
computational expressions assigned to linguistic expressions. The schema
proposal solves the problem of interpretability of the computational
representation of liguistic frame of cognitive (LFoC).
 The study proposing constraints on interpretation operations is built to
convey, preserve the desired semantic aspects of the LFoC for fuzzy systems.
 Application of HA approach solves the problem of interpretability of

computional representation of LFoC by establishing a granular polymorphism
structure of triangular fuzzy sets or trapezoidal fuzzy sets.
 Further clarify RWS interpretation of human natural languages and word
domains of variables and its basic role in checking RWS interpretability of
components of fuzzy system, at the same time, prove that the standard fuzzy set
algebras are not RWS interpretability.
 Propose formalization method to solve RWS interpretation of fuzzy
systems in the second case and n input variable.
CHAPTER I : BASIC KNOWLEDGE
1.1 Fuzzy set
Definition 1.1. [6]6 Let U be the universe of objects. The fuzzy set A on U is
the set of ordered pairs (x, A(x)), with A(x) being the function from U to [0,1]

4

C.H. Nguyen and W. Wechler (1990), “Hedge algebras: an algebraic approach to structures of sets
of linguistic domains of linguistic truth variables”, Fuzzy Sets and Systems, vol 35, no.3, pp. 281293.
5
Cat-Ho Nguyen and W. Wechler (1992),” Extended hedge algebras and their application to Fuzzy
logic”, Fuzzy Sets and Systems, 52, 259-281.
6
L. A. Zadeh, Fuzzy set, Information and control, 8, (1965), pp. 338-353


3

assigned to each element x of U value A(x) reflects the degree of x belong to
fuzzy set A.
If A(x) = 0, then we say x does not belong to A, otherwise if A(x) = 1, then
we say x belongs to A. In Definition 1.1, function  is also called is a

membership function.
1.2 Linguistic variable
Simply as said by Zadeh, a linguistic variable is a variable in which "its
values are words or sentences in natural language or artificial language".
1.3 Fuzzy rule based system
1.3.1. The components of the fuzzy system
A fuzzy rule based system consists of the following main components:
Database, Fuzzy Rule-based - FRB and Inference System.
- Database is sets of 𝔏j including linguistic label Tj corresponding to fuzzy
sets used to reference domain fuzzy partition UjR (real number set) of variable
𝔛j, (j=1,..,n+1) of problem n input 1 output.
- Fuzzy rule base is a set of fuzzy rules if-then.
- Reasoning system performs an approximate reasoning based on rules and
input values to produce the predicted output value. Some approximate reasoning
directions are as follows:
+ Approximate reasoning based on fuzzy relationship
+ Approximate reasoning by linear interpolation on fuzzy set
+ Reasoning based on the rule burning
1.3.2. Objectives upon building FRBS
 Evaluation of the effectiveness (accuracy) of FRBS
For the objective of the effectiveness of FRBS, we have mathematical
formulas to evaluate how an FRBS is effective.
 Problem of interpretability of FRBS
Interpretability is a complex and abstract problem, it involves many factors.
In [1] Gacto finds that there are currently two main approaches to the
interpretability:
- Interpretability is based on complexity:
 Rule basis level: The less the number of rules of the rule system is, the
shorter the length of the rule is.
 Fuzzy partition level: number of attributes or number of variables,

number of variables used less will increase the interpretability of the rule
system. The number of functions is used in the fuzzy partition, the number of
functions should not be exceeded 7±2 [6].
- Interpretability is based on semantics:
 Semantics at the rule basis level: The rule basis must be consistent, ie. it
does not contain contradictory rules, the rules with the same premise must have


4

the same conclusion, the number of rules burned by an input data is as little as
possible.
 Semantics at fuzzy partition level (word level): The defined domain of
variables must be completely covered by the function of fuzzy sets.
1.4 Hedge algebra
1.4.1. The concept of hedge algebra
Definition 1.2 [7]7: A hedge algebra is denoted as a set of 4 components
denoted by AX = (X, G, H, ) where G is a set of generator, H is a set of hedges,
and “” is a partial ordering relation on X. The assumption in G contains
constants 0, 1, W with the meaning of the smallest element, the largest element
and the neutral element in X. We call each language value xX a term in HA.
If X and H be linearly ordered sets, then AX = (X, G, H, ) is sais a linear
hedge algebra. And if two critical hedges are fitted  and  with semantics being
the right upper bound and right lower bound of the set H(x) when acting on x,
then we get the complete linear HA, denoted by AX* = (X, G, H, , , ). Note
that hn...h1u is called a canonical representation of a term x for u if x = hn...h1u
and hi...h1uhi-1...h1u for i is integer and in. We call the length of a term x is the
number of hedges in its canonical representation for the generated element plus
1, denoted by l(x).
1.4.2. Some properties of linear hedge algebra

Theorem 1.1: [7] Let the sets H- và H+ of a hedge algebra AX = (X, G, H,
) be linearly ordered. Then, the following statements hold:
i) For every uX, H(u) is a linearly ordered set.
ii) If X is a primarily generated hedge algebra and the set G of the primary
generators of X is linearly ordered, then so is the set H(G). Furthermore, if uand u, v are independent, i.e. uH(v) và vH(u), thì H(u) H(v).
The theorem below looks at the comparison of two terms in the linguistic
domain of variable X
Theorem 1.2: [7] Let x = hn…h1u and y = km…k1u be two arbitrary
canonical representations of x and y w.r.t. u. Then there exists an index j ≤
min{n, m} + 1 such that hj' = kj' for all j'hj = I, hj is the unit operator I, for j = n + 1 ≤ m or kj = I for j = m + 1 ≤ n) and
i) xii) x = y iff m = n and hjxj = kjxj.
iii) x and y are not comparable iff hjxj and kjxj are not comparable.

7

C. H. Nguyen and V. L. Nguyen (2007), Fuzziness measure on complete hedges algebras and
quantifying semantics of terms in linear hedge algebras, Fuzzy Sets and Syst., vol.158 pp.452-471.


5

1.4.3. Fuzziness measure of linguistic values
Definition 1.3: [7] Let AX *= (X, G, H, , , ) be a linear ComHA. An
fm: X [0,1] is said to be an fuzziness measure of terms in X provided:
(i) fm is complete, i.e. fm(c-) + fm(c+) =1 và hHfm(hu) = fm(u), uX;
(ii) fm(x) = 0, for all x such that H(x) = {x} and fm(0) = fm(W) = fm(1) = 0;
(iii) x,y X, h H, fm(hx)  fm(hy) , that is this propotion does not

fm( x)
fm( y )
depend on particular elements and, hences, is called the fuzziness measure of
hedge h and is denoted by (h)
We summarize some properties of the fuzziness measure of linguistic term
and hedges in the following proposition:
Proposition 1.1: [7] Let fm và  be defined in Definition 1.3, then:
(i) fm(c-) + fm(c+) = 1 and 
fm(hx)  fm( x) ;
hH


(iii) 
(ii)

1
j  q

xX k

 (h j )   ,



p
j 1

 (h j )   , for ,> 0 and  + = 1;

fm( x)  1 , where Xk is the set of all term in X = H(G) of length k;

(iv) fm(hx) = (h).fm(x), and xX, fm(x) = fm(x) = 0;
(v) Given fm(c-), fm(c+) and (h), hH, the for x = hn...h1c, c {c-, c+},
one can easily comput fm(x) như sau: fm(x) = (hn)...(h1)fm(c).
1.4.4. Fuzziness interval
Definition 1.4 [7]: Fuzziness interval of terms xX, denoted by fm(x), is a
subset of paragraph [0, 1], fm(x)  Itv([0, 1]), has the length equal to the fuzzy
measure, |fm(x)| = fm(x).
1.4.5. Quantifying semantics of linguistic values.
Definition 1.5 [7]: Let AX*= (X, G, H, ) be a linear HA, we define:
1) Function sign(k, h) ∈ {-1, 1} is said to be relative sign function of k for h
if sign(k, h) = 1((x≤ hx) hx ≤ khx)(x≥hx) hx≥khx)), and
sign(k, h) = -1  ((x ≤ hx) hx≥ khx ≥ x)  (x ≥ hx) hx≤ khx≤ x))
2) Function Sign: X {-1, 0, 1} is said to be sign function of words x if hn
… h1c, c∈G, is a formal representation, i.e. hjhj-1 … h1c ≠ hj-1 … h1c, for every j
= 1, …, n and h0 = Id, identity, i.e. h0c = c, then:
Sign(x)=Sign(hnhn-1…h1c) = sign(hn,hn-1) × … × sign(h2,h1) × sign(h1)
×sign(c).
Based on the sign function definition, we have the standard to compare hx
and x.
Proposition 1.2 [7]. For any h and x, if Sign(hx) = +1 then hx>x; if Sign(hx)
= -1 then hxFrom the above proposition we have:
0≤ H(x) ≤ 1 and H(x) ≤ H(y), x, y, i.e. xH(x) and yH(y)
(1.2)
Sgn(hpx) = +1 H(h-qx) ≤…≤ H(h-1x) ≤ x ≤ H(h1x) ≤…≤ H(hpx) (1.3)


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Sgn(hpx) = 1 H(h-qx) ≥… ≥ H(h-1x) ≥ x ≥ H(h1x) ≥…≥ H(hpx) (1.4)
Definition 1.6 [7]: Let AX be a free linear ComHA and fm be a fuzziness
measure on X . Then, a mapping

: X [0, 1] is said to be included by fm ,if it

is defined recursively as follows:
(i)

(W)= =fm(c-),

(c-)=– fm(c-) = .fm(c-),

(c+) =  +fm(c+);

i  sign( j )

(x)+ Sign(h x)
( j )  (hi ) fm( x)   (h j x) (h j x) fm( x) , (1.5)
j isign


for j, –qjp và j 0,
1
 (h j x)  1  Sign(h j x) Sign(hp h j x)(   )  ,   ;
2
With this definition, it has been proven that it satisfies the requirements of a
semantic quantitative function and assures its discretion with the word classes of
AX in paragraph [0, 1].
1.5 Conclusion of chapter 1

In this chapter, we summarizes the basic knowledge that serves as a basis
for research. It includes fuzzy set theory, fuzzy system based on rules,
applications, theory of HA.

(ii)

(hjx)=

CHAPTER 2. INTERPRETABILITY OF LINGUISTIC COGNITIVE
FRAMEWORK IN LINGUISTIC FUZZY SYSTEMS
In this chapter, we will show the schema that solves the interpretability
problem of the computational representation of the linguistic cognitive
framework, propose additional semantic constraints on interpretative maps. The
next section will survey the representation of the granular polymorphism
structure generated from the semantics of the word domain and show that these
representions meet the relevant constraints. The results of this chapter are
presented based on the work [2] in the List of scientific works of the author
related to the thesis.
2.1. The interpretability of LRBSs on the word level
Nguyen and colleagues [8]8, proposed a new approach to the interpretability
of LRBSs which leads to the investigation of the order-based semantics of the
LRBS components. The basis of the new approach is that the word-domain of a
variable 𝒳, denoted by Dom(𝒳), is modeled by an order-based structure induced
by the inherent meaning of the word, called hedge algebras(HAs).

8

C.H. Nguyen, V.Th. Hoang, V.L. Nguyen (2015), “A discussion on interpretability of linguistic
rule base systems and its application to solve regression problems”, Knowledge-Based Syst., vol. 88,
pp. 107-133.

7

The essence of computational interpretation is that the interpretation of the
semantics of words which cannot be calculated, needs to be converted to
computable objects, but the transformation must "preserve the semantics" of the
words. This requires us to investigate to propose the necessary constraints on
semantic interpretation.
We use the concept of LFoCs of variables, interpreting as word
vocabularies used to describe real world entities. So, the study of the
interpretability of a comput-representation of an LFoC is just to examine how
much semantic information of the words of the LFoC a desired interpretation
can convey or represent.
2.1.1. Scheme to solve the problem of interpretability of calculation
representation of linguistic frame of cognitive
Syntactical expressions of
an LFoC and its formal
properties
The low level (word level):
- - Words (syntactical strings)
- - Formalized LFoC (a set of
formalized words) and their
relationship structure
(semantic order-based
relation of words,
generality-specificity
relation etc.)

The HA AX modeling the

word-domain D
containing the LFoC
The HA of the worddomain:
I1 - - HA-expressions: string
representations of words
in D
- LFoCs and their
relationship structure

I = I2 o

The desired
computational objects
of a comput.
math. structure

I2

Comput.
structure:
(number, fuzzy set,
interval, ...)
-The
objects
of
comput. structure CS
and the relationships
between them.
-Set of comput-objects
representing LFoC

I
1
Figure 2.1. A schema of a computational interpretation
I of an LFoC

oC
In the study, for easily understandable we first schematize the process of
solving the interpretability of the comput-representation of the LFoCs of
LRBSs, as represented in Fig. 2.1, in which I1 is an interpretation assigning an
appropriate HA-element of 𝒜𝒳 to every word and I2 assigns an object of a
comput-structure 𝔖 to an HA-element of AX.
2.1.2. General constraints on the computational interpretation of the
words of variables
The authors in [8] proposed the initial constraints applied to the
interpretations described in Figure 2.1 for linguistic frame of cognitive LFoC to
maintain the semantics of LFoCs in the context of the entire word domain
instead of constraints imposed only on fuzzy sets.
Constraint 2.1 [8] (Essential role of the word): The inherent semantics of
words of a variable appearing in a f-rule base (FRB) must, in principle, be
explicit-ly taken into account or, must create a formalized basis to determine the
comput-semantics of the words, including the fuzzy set based semantics, to
handle the comput-semantics of the FRB.


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Constraint 2.2 [8] (Formalization of word quantification): The computsemantics of words, including f-sets semantics, should be produced based on an
adequate formal formalization of the word-domains of variables. Moreover, they
can be produced by a procedure developed based on this formalization system

that can then perform computational semantics of words automatically.
Constraint 2.3 [8] (Interval-interpretation of the words and G-S relation):
Let be given variable 𝒳, whose word-domain is Dom(𝒳), and denote by Intv the
set of all intervals of U(𝒳), an interval-interpretation 𝒜: Dom(𝒳) → Intv,
declared to be an interval-semantics of 𝒳, should preserve the G-S relationships
between the words, i.e. for any two words x and hx of 𝒳, where h is a hedge, we
should have 𝒜 (hx)  𝒜 (x).
Constraint 2.4 [8] (Interpretation as order isomorphism): To study the
order-based semantics of ling-rules, the comput-interpretation of words of 𝒳, ℑ:
Dom(𝒳) → C(𝒳), must preserve the word semantics, i.e.x,yDom(𝒳), xy &
x≤ y  ℑ(x) ℑ(y) & ℑ(x)≼ ℑ(y), where ≼ is an order-relation on ℑ(Dom(𝒳)).
That is, ℑ should be an order isomorphism.
2.1.3. Additional constraints on the computational representations of
linguistic frames of cognition
To study the LRBS interpretability at the low level, we propose the
following additional constraint on semantic core of the words of the LFoCs used
for the designed LRBSs.
Definition 2.1. An LFoC 𝔉 of a variable 𝒳 (in a user natural language) with
the set H of all hedges of 𝒳 is a collection of words satisfying the following:
(i) {0, c, W, c+, 1}  𝔉;
(ii) hx𝔉  (h’H)(h’x𝔉) (all words hx, h’H, together belong to 𝔉);
(iii) x𝔉 & x=hx’& hH  x’𝔉 (closed with respect to taking ancestors).
Denote by k the greatest length of the words present in 𝔉, it is called to be of
specificity k.
Note that the hedges in (ii) and (iii) make the word to be more specific. A
demonstrations of the important role of the G-S relation of words in the HAapproach.
Any word in an LFoC should be considered or selected in the context of the
whole LFoC. That is, the semantics of the words of an LFoC is dependent on
each other and these dependences lead to certain constraints imposed on the
comput-interpretations described in Fig. 2.1.

First, we examine the so-called semantics core of a word x introduced in


9

[11]9 to make a constraint related to the core of words. The semantics of core(x)
was analyzed to establish (2.1), for x, yDom(𝒳):
xand x and core(x) are not comparable.
Intuitively, the semantic core is in the semantics of the word, so we need to
give constraints to the computational semantics of the semantic core.
Therefore, for each variable 𝒳, with U𝒳 is its numerical domain, we use the
int(U𝒳) to denote the set of intervals of U𝒳, including the degenerate range [a,
a].
By methodology, we can consider mapping the interval value, denoted by
ℐint, ℐint : Dom(𝒳) → int(U𝒳). In order for ℐint to be interpreted as an
interpretation of the interval value of a variable with core semantics, we provide
the following constraint:
Constraint 2.5 (On the interval-interpretation, ℐint, of words and their
cores): ℐint is said to be an interval-interpretation of the words of 𝒳 and their
cores, if it must be satisfied by the condition: for x of 𝒳, C-core(x) = ℐint(h0x)
 ℐint(x).
We now examine the interpretation of words when using semantics of
triangle fuzzy set or trapezoidal fuzzy set. As mentioned above, we can use a
triple-interpretation for both types of fuzzy set.
Consider a triple-interpretation ℐtrp : Dom(𝒳) → {(a, b, d) : a, d∈U𝒳,
b∈int(U𝒳)}, called the tripe-semantics of the words x and core(x) ∈Dom(𝒳),
where a, b and d are included in U(𝒳). Every ℐint(core(x)) = ℐint(h0x) consists of
the values of U(𝒳) that are most compatible with x and, hence, they semantically
cannot belong to the interval-semantics of the others. So, the interval

ℐint(core(x)) = (b, c) can be rewritten as a triple (b, b, c), where b = (b,c) and, for
xy, ℐint(core(x)) ℐint(core(y)) = . This with (2.1) suggest a constraint to
maintain the order-based semantics as follows:
Constraint 2.6. For a desired order relation ≼ on the triples, the tripleinterpretation ℐtrp(x), the interval-interpretation ℐint(x) of the words of 𝒳 should
maintain the semantics of the words and there cores, i.e.:
(i) ℐtrp(core(x)) = ℐint(core(x))
(ii) For any two words x and y:
x
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C.H. Nguyen, T. S. Tran, D.P. Pham (2014), Modeling of a semantics core of linguistic terms
based on an extension of hedge algebrasemantics and its application, Knowl-Based Syst., Vol. 67 pp.
244-262.


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2.2. Comput-interpretation of LFoCs with triangle/trapezoid fuzzy sets
Let be given an LFoC 𝔉 of 𝒳 of specificity κ. For simplicity, assume that
the set H consisting of two hedges, L(little) and V(very), and an artificial one h0.
We start with the order-based structure of 𝔉 and their G-S relation of the words
in 𝔉: hx is more specific than x or, equivalently, x is more general than hx, for
any hx𝔉.
1) Construction of
W
0
1
multi-levels
of

specificity
of
𝔉
:
Lev
e
Decompose 𝔉 into
c
1
0
l c
specificity-levels so that
+
 Lev
1
1
the words on each level
0e
have the same G-S
l
V
Lc
L
Vc+ 1 degree or, equivalently,
0
Lev
c
c
1
2

2
they have
the same
e
+

l
length. Denote by 𝔉k
LLc
VLc
VL
0VV LVc
LL
LV VV 1 the set of the words of
Lev


2 c
3c
3
c
c length
𝔉cof the
k, k = 0,
e +

+
+
+
1, …, κ, where 𝔉0 = {0,

l
W, 1}, 𝔉1 = {01, c, c+,
Figure 2.3: The multi-granularity with tri/trap3 fuzzy sets of
11}, 𝔉2 = {02,Vc, Lc,
words in LFoC 𝔉
Lc+, Vc+, 12}, 𝔉3 =
{03,VVc, LVc, LLc, VLc, VLc+, LLc+, LVc+, VVc+, 13} … The pre-sence of the
artificial words 0k and 1k comes from the requirement that in the f-partition of 𝔉j
must be complete. Moreover, they makes the set 𝔉 richer.
2) Construction of fuzzy multi-granularity representtation of 𝔉: Every
specificity vevel 𝔉k is represented by tri/trap f-set partition as represented in Fig.
2.3, in which there are three f-partitions. Such a structure of f-sets is called
multi-granularity. It can easily be verified that it maintains the G-S relationship
of the words of 𝔉: the support of the f-set of hx is included in the support of the
f-sets of x.
3) Interpretations of 𝔉 defined by the constructed f-multi-granularity structure:
For a multi-granularity structure, such as the structure given in Figure 2.3.
From the multi-granularity structure, we can define interpretations as follows:
(I1) F-set interpretation of 𝔉: It is the interpretation, denoted by ℐfuz, which
assigns every word x in 𝔉 to the tri/trap f-set whose core is ℑ(h0x).
(I2) The interval-interpretation ℐint of 𝔉: The ℐint is defined simply as follows:
for x𝔉, (i) ℐint(x) is the support of the tri/trap fuzzy set ℐfuz(x); (ii) ℐint(h0x) =
ℑ(h0x), the core of the f-set of x.
(I2) The triple-interpretation ℐtrp of 𝔉: For x𝔉, ℐtrp(x) = (a, b, d), where (a, d)
is the support of the f-set ℐfuz(x) and b = ℐint(h0x) = ℑ(h0x).


11

Theorem 2.1. The interpretations ℐfuz and ℐtrp of 𝔉, which are associated

with the interval interpretation ℐint, defined by the f-multi-granularity structure
constructed as above satisfy all Const’s 2.1 – 2.6.
Proved that the interpretability defined above satisfies the above constraints
2.3. Conclusion of chapter 2
In this chapter, we has researched and solved some of the following
problems:
 Research and analysis of interpretability are as a study of the relationship
between RWS of linguistic expressions and computational semantics of
computational expressions assigned to linguistic expressions. The schema
proposal solves the problem of interpretability of the computational
representation of linguistic cognitive frameworks.
 Propose constraints on interpretation operations that are built to convey,
preserve the desired semantic aspects of the LFoC for fuzzy systems.
 Application of HA approach solves the problem of interpretability of
computional representation of LFoC by establishing a granular polymorphism
structure of triangular fuzzy sets or trapezoidal fuzzy sets.
CHAPTER 3. INTERPRETABILITY BASED ON REAL WORLD
SEMANTICS OF LINGUISTIC EXPRESSIONS
In essence, each fuzzy system is a fuzzy set expression manipulated based
on a computational basis in fuzzy set theory, in which each fuzzy set is assigned
to a linguistic label. Therefore, each fuzzy set expression is corresponding to a
human readable and comprehensible linguistic expression and it is considered a
fuzzy set representation of that linguistic expression. Therefore, the
interpretability problem of a fuzzy set expression consists of at least 02
problems: (1) Do fuzzy sets in a given fuzzy set express semantics correctly of
the linguistic label? (2) Is its linguistic expression easy to understand for
human?
The objective of this chapter is to study the interpretability based on RWS
of theoretical foundation to develop methodology or algorithm. At the same
time, the study surveying the interpretability based on RWS of the theory of HA

and on that basis, the study of interpretability based on RWS of the components
of fuzzy systems. The results of this chapter are presented based on the work
[1,3,4] in the List of scientific works of the author related to the thesis.
3.1. RWS-interpretability of variable word-domains & its crucial
3.1.1. The novel concept of RWS-interpretability of any formalized
theories
Methodologically, human beings cognize the reality around their daily lives
by using symbolic languages associated with implicit semantic interpretation
assignments, by which their elementary symbolic elements convey real-world
semantics, such as natural languages of human communities, mathematical


12

languages, physical languages…So, it is necessary to put the study of the FSyst
interpretability in relationships between human beings, the RW and natural
languages, as it is exhibited by a scheme given in Fig. 3.1.

Structures of the
real-world

Real world
models
of formal
theories
Applications/algorithms
designed based on certain
formal theories interacting
with their RW-counterparts

Formal theories
developed based on
their axioms

Figure 3.1. Relationships between formalized theories, their models and
applications and their RW counterparts

3.1.1.1. The concept of RWS-interpretability of formalized theories
Therefore, the study [3] introduce the following definition:
Definition 3.1 [3] A formalized method/theory T formulated in its
formalized language to simulate a real-world structure, denoted by WT, is said to
be RWS-interpretable if there exists an interpretation mapping RT: WT → T,
which assigns real-world objects of W to elementary formalized elements of T
that can convey or preserve the essential properties of WT. In this case, T is
called an RWS-model of WT or WT is interpretable in T. Such a formalized
method T is called RWS-interpretable. Note that, the structure WT is a subjective
concept as it depends on the observation/perception of a human user. In this
sense, most of classical mathematical theories are RWS-interpretable.
Based on the concept of the interpretability defined in Def. 3.1 and the
successful applications of math-theories in reality, we adopt the following
hypothesis:
Hypothesis 3.1. The developments of math-theories based on axiomatic
methods ensure their RWS-interpretability, that is, a math-theory with its axioms
whose semantics is justified to represent key structural relationships between
entities of the RW-counterpart of the theory is RWS-interpretable.
3.1.1.2. Proposal of a scheme to solve a RWS-interpretablity problem
In math-logics, the inference mechanisms of predicate logics guarantee that
a conclusion derived from valid statements is also valid. However, in the
fuzzy/uncertain environment with inexact statements, one has no strict
mechanism that permits to derive valid statements from given valid ones, a

similar assertion is even more difficult to prove. Thus, in a fuzzy environment, it
is necessary to introduce a scheme to solve a given RWS-interpretability
problem as shown in Fig. 3.2, where the RWS-interpretability of a formalized
fuzzy expression depends on which a structure of its RW-counterpart can be
discovered, including expressions representing Approximate Reasoning Methods


13

(ARMs).
Detection
Given W in its own
structure, denoted SW, and
𝔼 is expressed by the
expert, try to find the
essential relation in 𝔼
(relation among variables
of
SW;
among
the
linguistic elements of each
variable)

Construction of
computational space
(i)
Construct
a
computational

space
CSW;
(ii) Define the essential
concepts
and
key
relations in CSW
to
modeling
the
corresponding structures
found in SW

Construction of
interpretation
assignment
+ Build a map f to
convert SW to CSW.
+ Check if f can
preserve the key
relations of SW or
not.

Figure 3.2. The interpretable problem-solving schema RWS
3.1.2. The RWS-interpretability of human natural languages and of HAs of
variables
3.1.2.1. The RWS-interpretability of natural languages of any human
communities
Every language of a human community considered as a whole cannot be
formalized as a structure and, therefore, we cannot show that it is RWSinterpretable in the sense of Def. 3.1. However, the history of fighting for the

human existence and development, in which its language has been being used
throughout the length of its history to communicate with each other or to make
decisions in daily lives, proves that its language must be RWS-interpretable,
otherwise the human community cannot exist.
Hypothesis 3.2. Any human natural languages are RWS-interpretable.
3.1.2.2. The RWS-interpretability of HAs–Mathematical models of worddomains of variables
Straight lines in reality and the human calculation demand based on their
structures motivate the development the theory of the real numbers. The
successful applications of this theory in reality demonstrate that it is RWSinterpretable. So, methodologically, the RWS-interpretability of the theory of
real numbers can be ensured by the following two facts:
 The RWS-interpretability of its axioms: It is well known that straight
lines are RW-models of the theory of the real numbers and, hence, its axioms
that formulate the properties of the operations on the real numbers are
compatible with the respective measurements made by human being on
corresponding segments of a straight line. These demonstrate that the theory of
the real numbers has a close connection with its RW-counterpart. Thus, the
RWS-interpretability of its axioms are justified by RW-semantics representing
distance relationships between real points on the straight lines.


14

 The development of the theory that is based on the inference rules of a
formal logics does maintain the RWS-interpretability of the whole theory. The
(logical) validity of a sentence is verified based on the RW-events. For instance,
the validity of the sentence “If we have free time on Sunday AND this day is
sunshine, then we will visit your family” is verified based on the respective
events actually happen or do not happen in that day and, hence, it is RWSinterpretability. Since all inference rules preserve the validity of the sentences, it
follows the validity of the mentioned above fact.
Based on these facts, we can show that the theory of HAs including their

quantification are RWS-interpretable. Since the logical rules are the sameas in
the field of the mathematics, it is sufficient to show that the axioms of the HAs
and their quantification are RWS-interpretable.
On one hands, human societies including their natural languages are
evidently considered as a part of the RW. On other hand, as previously argued,
natural languages are RWS-interpretable and, hence, the word-domains of the
variables, which are ordered based on their inherent meaning and viewed as
their sub-RW-parts, can also be considered as RW-parts. Thus, to model the
word-domains of variables, similarly as for the theory of the real numbers, the
theory of HAs is developed in an axiomatic manner and their axioms are formal
formulations of key essential properties of words and hedges of their respective
word-domains [5,4, 1010,7], viewed as RW-counterparts. As examined in [7], the
quantification of HAs is also developed in axiomatic way and its axioms are
established based on the structure of HAs. This ensures the RWS-interpretability
of the quantification axioms.
In such a way, their RWS-interpretability is guaranteed by the hypothesis
formulated above. So, similarly as for any classical math-theory, we have the
following proposition:
Proposition 3.1. Any HA and its quantification theory are RWSinterpretable.
3.2. RWS-interpretability of the fuzzy systems components
A Fsyst can be viewed as comprising a fuzzy knowledge base, which
includes its linguistic frames of cognition (LFoC) and its linguistic fuzzy rule
base (LRB), and a fuzzy inference engine, which is constructed mainly based on
an approximate reasoning method (ARM).
Here, we will examine the RWS interpretability of these components.

10

C. H. Nguyen and N.V. Huynh (2002), An algebraic approach to linguistic hedges in Zadeh's
fuzzy logic, Fuzzy Sets and Syst., vol.129 pp.229-254.

15

3.2.1. The RWS-interpretability of LFoCs
In the RWS-approach, an RWS-interpretable computational representation
of a given LFoC 𝔉 is constructed based on the scheme given in Fig. 2 described
as follows, where 𝔉=X(k)={x∈X: the set of the words of a variable whose length
is not greater than k> 0}.
3.2.1.1. Try to discover structural relationships between words of 𝔉 being
considered as an 𝕃E:
As previously discussed, 𝔉 can be regarded as a RW-counterpart (or, its words
can properly convey their RW-semantics) and we try to find out key structural
features of 𝔉. It can be seen that on this set there exist two relations, denoted by
≤ and GS(x, y), which are still not considered in the fuzzy set framework.
∘ 𝔉 is linearly orderedset induced by the word meaning. Its structure is
denoted as (𝔉, ≤).
∘ GS(x, y) is a generality-specificity relation on 𝔉. For example, for the
variable AGE, “old” is more general than “very_old”,and “rather very young” is
more specific than“young”, if they all are in 𝔉. It can be verified that G(x, y)
have the following properties:
- Anti-symmetry: GS(x, y) &GS(y, x) =>x = y.
- Transitivity: GS(x, y) &GS(y, z) =>GS(x, z).
These relations are taken as constraints imposed on two respective
interpretation mappings of sound computational representations of LFoCs,
where a sound computational representation of an LFoC
Definition 3.2. A fuzzy set presentation of a given LFoC 𝔉, FR(𝔉)={F(x):
x∈𝔉}, where F(x) denotes the fuzzy set assigned to x, is said to be RWSinterpretable if the following conditions hold:
(i) On FR(𝔉) can be defined two relations: the first one is denoted by ≤*,
which is reflexive, anti-symmetric and transitive, and the second one is denoted

by GS*, which is anti-symmetric and transitive.
(ii) There exists an interpretation assignment I and IGS that both map 𝔉 into
FR(𝔉), such that they respectively preserve the relations ≤ and GS on 𝔉, i.e., for
all x, y in 𝔉, we have x ≤ y =>I(x) ≤*I(y) and GS(x, y) =>GS*(IGS(x), IGS(y)).
3.2.1.2. Try to construct a computational space that can properly represent
the semantics of 𝔉:
The study [3] argued that the topology of multi-granularity structure of
fuzzy sets, as shown in Fig. 3.3 can meet the both above discovered constraints,
where the fuzzy sets are arranged in three level of their generality. Here, the
fuzzy setson the k th-level represent the semantics of the words of the generality
of degree k. For k = 0, the words on this level are the most general and the fuzzy
setsare either all triangles or all trapezoids, whore core are therefore represented
by bolded points in the figure and whose supports are uniquely defined by the
fuzziness intervals of the respective words. Representing triangles/trapezoids by


16

triples of the form (a, b, d), where b’s are the cores of triangles/trapezoids, the
order between the fuzzy setsare defined as follows: (a, b, d) ≤* (a’, b’, d’) {b
≤ b’& there is at least one of the remaining components of the triples, say the
first one, satisfying the
W
00
10
inequality a ≤ a’}. The
relation GS* between the
triples are defined as
01
young

11
old
follows: GS*((a, b, d), (a’,
b’, d’))  [a, d] [a’,
d’].Obviously,the triples on
02 Vyoung
Ryoung
Rold Vold
12
the top level are of the most
g
generality
which
is
compatible
with
the
Figure 3.3 RWS-Interpretable triangle/trapezoid
semantics of the words on
multi-granular representation of XAGE,(2)
this level.
3.2.2. The RWS-interpretability ofcomputational representation of
LRBs and ARMs
In [3], it is argued that one may acquire a piece of knowledge about a
numeric dependentrelationship between two variables only when it is observed
that they are monotonically dependent oneach other on a certain interval of
each variable, otherwise their dependence is chaotic. Compatibly with this, as
discussed in that study, it is interesting that the semantics of a multi-variable
linguistic rule does also expressmonotonic dependent relationships betweenits
unique output variable and one of its input variables.

Consider a linguistic rule with one output and m input variables written in
the following form:
(r) IF 𝒳1L is x1 & … &𝒳mL is xm, THEN 𝒳m+1,L is xm+1
(1)
in which each expression “𝒳jL is xj” is a linguistic predicate, for j = 1 to m + 1.
Similar as for analyzing a classical multi-variable function, for every rule r, one
may consider m dependent relations ‘IF 𝒳jL is xj, THEN 𝒳m + 1,L is xm+ 1’, j = 1 to
m + 1, and therefore, r denotes m monotonic dependences between variables 𝒳m
+ 1, L and 𝒳jL on certain interval of each respective RW-variable of the RWcounterpart.
To further analyze, we assume that a LRB ℛℬ is full of conditions, that
is, all m variables 𝒳jL, j = 1, …,m, are explicitly present in each rule (as it
happens with the well-known Wang and Mendel [11]11). In this case, it is simply
called a full LRB. It is natural to require ℛℬto beconsistent, i.e. if the

11

L.-X. Wang and J. M. Mendel (1992), “Generating fuzzy rules by learning from examples,” IEEE
Transactions on Systems, Man and Cybernetics, vol. 22, no. 6, pp. 1414–1427.


17

antecedents of any two rules of ℛℬ are identical, then so are their conclusions.
Then, a full and consistent LRB ℛℬ may be considered as representing a
linguistic functional dependence of 𝒳m+1,L on 𝒳jL’s, j = 1 to m. By the RWSinterpretability of natural languages and the RW-semantics of linguistic rules
and LRBs, ℛℬ represents a real-world functional dependence of 𝒳m+1 on 𝒳j’s,
for j = 1 to m.
Definition 3.3. Given comput-space 𝒮 = (𝒞, ≤𝒮) defined on Cartesian product
oforder-based structures CSj’s. A CRep-method ℳ with its interpretations I𝒳j’s,
I𝒳j: Dom(𝒳j) → CSj, is said to be RWS-interpretable in 𝒮 provided that

1) The interpretations I𝒳j’s are order isomorphisms.
2) For a given LRB 𝔹, ℳ preserves the discovered monotonicity, if any, of
𝔹. That is, if 𝔹 is increasing (or decreasing) and for a = (xi1, …,xim) ≤a’ = (xi’1,
…,xi’m), where a and a’are any two linguistic vectors formed by m words
appearing in, respectively, some two rules ra and ra’of 𝔹, then ℳ(ra)|𝒳m+1 ≤
ℳ(ra’)|𝒳m+1 (or, ℳ(ra)|𝒳m+1 ≥ ℳ(ra’)|𝒳m+1).
It can be observed in general that any method that represents any LRB as an
(m+1)-dimensional fuzzy relation cannot be RWS-interpretable, as the order of
words and their fuzzy sets is ignored.
ARMds eveloped to solve application problems plays an important role to
build FSysts and therefore, its interpretability is essential to ensure their
performance in solving application problems, due to in the opposite case we
have no formal basis to ensure that the outputs of their ARMd are compatible
with the results expected by human designer. This question strongly depends on
the RWS-interpretability of the constructed computational representation
method, ℳ, to produce computational representations of LRBs as well as of
ARMds running on. Any ARMd, say ℝ, needs to be developed to be able to
work on the computational representation of ℛℬ and this implies that its realworld-semantics interpretability depends heavily on ℳ. Therefore, the RWSinterpretability of an ARMds should be defined based on the computational
representation method associated with it. In [13], the authors introduced the
following definition, in which a = (a1, ..., am) is the input vector and ℝ(a)
denotes the numerical output of the vector a produced by ℝ.
Definition 3.4. Assume that an ARMd ℝ is developed to work on
computational representations of LRBs produced by a computational
representation method ℳ. Then, ℝ is said to be RWS-interpretable if for any
give LRB ℛℬ being increasingly monotonic to all individual input variables of
ℛℬ, ℝ must satisfy the following condition:
(a, a’){[a ≼ a’  ℝℳ(𝔹)(a)  ℝℳ(𝔹)(a’)]
and [a  a’  ℝℳ(𝔹)(a)  ℝℳ(𝔹)(a’)]}
(2)

18

3.3. The
RWS-interpretability
concept
of
linguistic
fuzzy
expressions/theories
As discussed in previous sections, the novel RWS-interpretability of any
fuzzy theories, in general, and of any FSysts, in particular, seems to be very
essential and practical. So, a natural question that arises is that whether the fuzzy
sets theory or its expressions are RWS-interpretable and if it is not, whether
there exist methodologies to develop RWS-interpretable FSysts?
3.3.1. Examination of the RWS-interpretability of some fuzzy
expressions of the fuzzy set theory
The RWS-interpretability of the fuzzy set theory is a too big problem and,
therefore, in this section it is restricted to examine the RWS-interpretability of
the standard fuzzy set algebras.
3.3.1.1. An analysis of the RWS-interpretability of the standard fuzzy set algebra
Let us consider a universe U and denote by F(U) the set of all fuzzy sets of
U, F(U) = { : ∈ [0, 1]U}, where [0, 1]U is the set of all (membership)
functions from U into [0, 1] and the fuzzy sets and their membership functions
can be viewed as to be identical. It is well-known that the union (), intersection
(), complementation () can be defined in F(U) as a generalization of the
respective operations on the crisp sets of U. They are pointwise defined on the
membership functions of the fuzzy sets in the whole F(U). Then, we have a
standard fuzzy set algebra that can be denoted by F𝔸= (F(U), , , ).
To examine the RWS-interpretability of F𝔸, we have to find out which is

the RW-semantics of individual fuzzy sets and the RW-semantics of the
operations ,  and  when they act on fuzzy sets. To answer these questions,
we should come back to the ultimate aim of the development of the fuzzy sets
theory: to simulate human capabilities in handing words. This is why in
applications the operations ,  and  are usually interpreted as representing
the computational semantics of the respective logical connectives in natural
languages, AND, OR and NOT. So, we will examine the RWS-interpretability
of the operations of the standard fuzzy set algebra based on the real-world
semantics of the connectives AND, OR and NOT.
There are two main reasons that show that F𝔸 is not be able RWSinterpretable.
1) A methodological reason present in the fuzzy set framework. Let us
consider the variable HIGH of people of a community and the meaning of the
sentence “he is ‘Tall OR Rather_tall’”. The semantics of this sentence must be
considered in the context of the word-expressions of the word-domain of HIGH,
LDom(HIGH). Assume that the standard fuzzy set algebra defined on the
universe U of HIGH is F𝔸= (F(U), , , ).
Proposition 3.2. As LDom(HIGH) is finite, while F(U) is innumerable
and even of continuum power, there does not exists an interpretation mapping ℑ


19

from LDom(HIGH) into F(U) that can maintains the relationships that
characterize the structure of LDom(HIGH), noting that the operations of F𝔸 are
defined in the whole F(U).
The above clause is demonstrated and illustrated by Figure 3.4.
2) A methodological reason on standpoint of the RWS-approach.
First, we adopt an assumption
AB
that we deal withonly variables

with numeric linear universe and
A
B
hence the word-domains of their
linguistic variables are linearly
Figure 3.4. The union of the two given fuzzy
ordered. So, their respective HAs
sets of variable HIGH
are also linear.
In Section 1, it is shown by
Proposition 1 that the HA AXHIGH is RWS-interpretable. That is there exists an
interpretation mapping ℑHIGH from LDom(HIGH) into the underlying set of
AXHIGH, which implies that ℑHIGH(wA AND wB)=ℑHIGH(wA) ℑHIGH(wB), where 
is join operation defined in the order-based structure AXHIGH. Since AXHIGH is
RWS-interpretable, ℑ preserves the relationships between words of
LDom(HIGH), the expression ℑHIGH(wA) ℑHIGH(wB) represents the semantics of
the word-expression “wA AND wB”. As AXHIGH is linear, ℑHIGH(wA)ℑHIGH(wB) =
max{ℑHIGH(wA), ℑHIGH(wB)} and it represents the RW-semantics of the
expression “wA AND wB”. Since as mentioned above, the fuzzy sets A and B
associated with respectively the word wA and wB, but AB{A, B}, which is not
compatible with ℑHIGH(wA) ℑHIGH(wB) = max{ℑHIGH(wA), ℑHIGH(wB)} which
represents the RW-semantics of “wA AND wB”. This asserts that the standard
fuzzy set algebra F𝔸 is not RWS-interpretable.
3.3.1.2. A discussion of the RWS-interpretability of Mamdani fuzzy
reasoning method
Table. 3.1. Simplified FRB for
In Mamdani fuzzy reasoning
the actuator on the 1th-storey
method which is denotedby ARMMmd,
ẋ2

NS
Z
PS
its fuzzy rule base (FRB), 𝔹, consists of
x2
n rules of a similar form as in (1), but at
NS
NM
NS
Z
the positions of the words xjk’s are
Z
NS
Z
PS
fuzzy setf(xjk)’s assigned to the words
denoted also by xjk’s:
(𝔹) IF 𝒳1L is f(x1,k)& … &𝒳mL is xmi,k,
THEN 𝒳(m+1)L is x(m+1),k, for k = 1 to n

Z

PS
(3)

PS

PM

20

This thesis has proved with monotonous language rule base (LRB) 𝔅,
given in Table 3.1, simplified from the LRB given in the study [12]12 to include
only 9 fuzzy rule, a Mamdani fuzzy approximation method ARMMmd based on
rule burning is not RWS-interpretable because it does not satisfy condition (2) of
Definition 3.4.
3.3.2. Discuss the RWS-interpretability of graphic representation of
LRBs and of the interpolation reasoning HA-based method
A question arising is whether there exists an RWS-interpretable ARM? In
this section, we will follow the HA approach to the inherent order-based
semantics of words and the inherent semantic structures of word-domains of
variables. Since this approach establishes a formalism to immediately handle the
variable words, we use the terminology linguistic rules (or,linguistic rule bases
(LRBs)) instead of fuzzy rules (or, fuzzy rule bases (FRBs) to emphasize this
linguistic characteristic.
There are three basic quantitative semantics of the words of each variable
𝒳, defined in close relation to each other: fuzzy measure, fuzzy interval
(considered as interval semantics) and semantically quantifying mapping (SQM)
of the words of variables. They are uniquely defined when the numerical values
of the independent fuzzy parameters of variables are provided. The SQM-values
of words are called the numerical semantics of words. In this section, however,
we utilize only SQMs which are characterized by two properties that they are
order isomorphisms, i.e. they must preserve the order relations among words and
the images of linguistic domains of variables under these isomorphisms are
dense in the reference domains of the corresponding variables (similar as the
countable set of the rational numbers is dense in the real line).
On the mathematical point of view, when word-domains are formalized to
become math-structures, everylinguistic rule of the form given in (3) can be
considered as a point in the respective linguistic(m+1)-dimensional Cartesian

space. So, every given LRB of the form (3) can be considered as modeling a
linguistic function of m variables going through these n points, called graphic
representation of (3).
Construction method a graphic representation of 𝔅: Given a linguistic
rule base 𝔅 describes the function relationship of XL,(m+1) into variables XL,j, j =
1, .., m. Then, the steps to build the function fN represent the language law base
calculation 𝔅 including the following steps
Step 1) Determine the qualitative and quantitative semantics of linguistic
variables:

12

R. Guclu, H. Yazici (2008), Vibration control of a structure with ATMD against earthquake using
fuzzy logic controllers. Journal of Sound and Vibration, 318, 36–49.


21

(1.1) Building hedge algebra of domain from Dom(Xj), j = 1, …, m +1
(1.2) Selecting values of fuzzy parameters of Xj’s variables. This greatly
affects quantitative semantics of the words of the variable.
Step 2) Represent the graph of 𝔅 by defining the grid of points in space [0,
1]m+1 where the function fN,𝔅 , qua goes through:
(2.1) For each variable Xj, j = 1, …, m +1, list all the words of the variable
present in 𝔅, denoted by xjk, k = 1, …, Kj. SQMj notation is a quantitative
mapping of Xj determined by the values of the fuzzy parameters of Xj and the
numerical semantic values of the words xjk, SQMj(xjk), k = 1, …, Kj.
(2.2) Set the approximate graph grid of function fN,𝔅 as a calculation
representation of the linguistic rule base 𝔅 as follows:
- For each language rule ri in 𝔅 there is the form (1), which is of the form:

ri : IF X1L is x1,i & … &XmL is xm,i, THEN Xm+1,L is xm+1,i, i = 1, …, n,
We denote ri|Xj = xj,i, j = 1, …, m + 1, and set the following points:
(SQM1(ri|X1), …, SQMm+1(ri|Xm+1)) ∈ [0, 1]m+1.
- Set grid in space [0, 1]m+1:
Grid(𝔅) = {( SQM1(ri|X1), …, SQMm+1(ri|Xm+1) : i = 1, …, n }.
Because the quantitative maps SQMj are all isomorphisms that preserve the
order of language words of the variables Xj, it is easy to verify the correctness of
the following theorem:
Theorem 3.2. The graphic representation of LRBs is RWS-interpretable.
Proof: Suppose for 𝔅 is monotonous, meaning that if we use the above
notation and we have ri|Xj ≤ ri’|Xj with all j = 1, …, m, then we also have ri|Xm+1 ≤
ri’|Xm+1. Because the quantitative mappings SQMj are all isomorphic preserving
the order of the linguistic words of Dom(Xj), Grid (𝔅) defines a function fN on
the domain of Grid(𝔅) which is monotonous , that is something to prove.
3.3.3. Approximate reasoning method on graphical representations of
LRBs
3.3.3.1. Interpolative approximate reasoning method
Approximate reasoning problem: Give a numerical vector ain = (ain,1, …,
ain,m) ∈ U𝒳1 …  U𝒳m and a linguistic rule base ℛℬ, calculate a numerical
semantic of the output corresponding to the input ain, denoted by Outℛℬ(ain),
based on the knowledge given by ℛℬ.
This problem can be solved in this study by an interpolative method in
Euclide an space as follows:
Interpolative method on LRB ℛℬ: Let be given values of the fuzzy
parameters of the variables present in ℛℬ and a graphical representation method
𝕄Graph. Then, ℳGraph(ℛℬ) defines a grid of a surface Sℛℬ in Euclidean space [0,
1]m+1. So, every (numerical) interpolative method INTMd on the surface Sℛℬ can
be apply to define a ARMd to solve the approximate reasoning problem for the
given linguistic rule knowledge base ℛℬ.

22

For a give an INTMd ℳInter, it is clear that, for each input vector ain,
Outℛℬ(ain) can be calculated by applying ℳInter on the surface Sℛℬ, denored by
ℳInter(Sℛℬ), and obtain Outℛℬ(ain) = ℳInter(Sℛℬ)(ain), i.e. it is the value calculated
by ℳInter on Sℛℬ in the Euclidean space [0, 1]m+1.
3.3.3.2.RWS-interpretability of interpolative approximate reasoning
methods
1) Linear interpolative approximate reasoning methods: In case that
the LRB has two inputs, we have a linear interpolative approximate reasoning
method on surface in [0, 1]3. For example, the LRB ℛℬ given in Table 3.1 with
9 linguistic rules defines a surface Sℛℬ as represented in Figure 3.6.
 Interpolative approximate reasoning method Li
This interpolative method is called the Li-method, which is extended from
the method studied in the work [13]13, but it RWS-interpretability is still not
examined, and is described as follows:
- For each input vector ain = (a1, a2), define the smallest rectangle, whose
three vertices are denoted by Pk, k = 1, 2, 3, in the coordinate plane x  y
containing point (a1, a2)
- Establish
the
section whose projection
l (0.73)
on the coordinate plane x
Ll(0.67
 y is the above defined
)
triangle:
Denote

by
Sℛℬ(Pk), k = 1, 2, 3, the
W(0.40)
points in [0,1]3 lying on
Ls(0.30)
the surface Sℛℬ whose
s(0.18)
projections on the plane x
W (0.40)
s (0.18)
l (0.73)
 y are the points Pk, k =
s(0.18)
1, 2, 3 and establish the
plane equation going
W
through these points,
(
denoted by z = EQ(Sℛℬ(P1), l (0.73)
0
x
.
(x,
y).
Sℛℬ(P2), Sℛℬ(P3))
4 graphical representation of LRB
- Calculate
the Figure 3.6. numerical
passing
0 through 9 points

output
by
equality
)
Out(ain)=EQ(Sℛℬ(P1), Sℛℬ(P2), Sℛℬ(P3))(a1, a2).
We can easily demonstrate the correctness of the following theorem:

13

M. Antonelli, P. Ducange, B. Lazzerini, F. Marcelloni (2011), Learning concurrently data and
rule bases of Mamdani fuzzy rule-based systems by exploiting a novel interpretability index. Soft
Comput., 15 pp. 1981–1998.


23

Theorem 3.3. The linearly interpolative Li-method, denoted by -ℳ, is
RWS-interpretable.
Proof: Assuming that LRB ℛℬ describes an increasing linguistic function,
as this equation is linear it is easy to prove that the inequality (a1, b1) ≤ (a2, b2)
implies that -ℳ(Sℛℬ)(a1, b1) ≤ -ℳ(Sℛℬ)(a2, b2).
2) In case m > 2
There are many interpolative methods with the number of dimensions n >
3 but they are in general very complicated when n is large. In this case, we can
use an aggregation operator usually used in fuzzy set theory to convert
approximate reasoning problems in m + 1 dimensional space to two-dimensional
one.
Theorem 3.4. Let be given a LRB ℛℬ and assume that the aggregation
operator used is a weighted average with weight vector w = (w1, …, wm)
corresponding to m antecedent variables of ℛℬ, denoted by 𝔤w. Then, the linear

interpolation using 𝔤w, denoted by Li_IntM2,w is RWS-interpretable.
Proof: Assume that ℛℬ is a LRB represented by the graph 𝒩Gph𝕀∘𝕗 (ℛℬ)
with the grid
Grid2(ℛℬ) = {(𝔤w[SQM1(x1,i),…,SQMm(xm,i)], SQMm+1(xm+1,i)): i =1,…, n }.
Due to ℛℬ is increasingly monotonic and assume that there are two rules ri
and ri’ in form (*) whose linguistic vectors created by the words in their
antecedent parts, denoted by x(ri) = (x1,i, …, xm,i) and x(ri’) = (x1,i’, …, xm,i’),
satisfy the condition that x(ri) ≤ x(ri’), i.e. xj,i ≤ xj,i’, for j = 1, …, m, implies
ri|𝒳m+1 = xi,m+1 ≤ ri’|𝒳m+1 = xi’,m+1. As SQMj are order isomorphisms, we have
SQMj(xj,i) ≤ SQMj(xj,i’), j = 1, …, m+1, and therefore we obtain 𝔤w(x(ri)) ≤
𝔤w(x(ri’)).
Consider two input vectors ain = (ain,1, …, ain,m) ≤ bin = (bin,1, …, bin,m).
Then, similarly as above, we have 𝔤w(ain,1, …, ain,m) ≤ 𝔤w(bin,1, …, bin,m). There
are two cases:
Case 1: There exists a smallest interval [𝔤w(x(rj1)), 𝔤w(x(rj2))] containing
the both values 𝔤w(ain,1, …, ain,m) and 𝔤w(bin,1, …, bin,m) computed from the two
given inputs.
Case 2: The two values 𝔤w(ain,1, …, ain,m) and 𝔤w(bin,1, …, bin,m) lie on
different intrvals I1 = [𝔤w(x(rj1)), 𝔤w(x(rj1*))] and I2 = [𝔤w(x(rj2)), 𝔤w(x(rj2*))]
created by the adjacent horizon coordinates of the grid Grid 2(ℛℬ) in [0, 1]2.
We have proved that the method of approximating linear interpolation
Li_IntM2, also preserves the monotonicity of the linguistic law base 𝔅 in these
cases. The theorem is proven.
3.4. Conclusion of chapter 3
The purpose of this chapter is to discuss and analyze clearly that the
interpretability of RWS is an essential general concept and not only for FSysts
but also for languages, theories, signing methods. ... The interpretability of RWS


24

of two basic concepts, LRB-Rep method and approximate reasoning are
formalized more accurately.
The fuzzy set theory is proved initially including basic parts failing to
interpret based on RWS. Meanwhile, graph representation of the linguistic rule
basis and approximate reasoning method are developed based on linear
interpolation proved to be interpretable based on RWS.
CONCLUSION OF THE THESIS
The thesis is conducted with the desire to study in depth the interpretability
of the fuzzy linguistic systems in the semantic approach based on the hedge
algebra proposed by Nguyen and Wechler. In this approach, the computational
semantics of a word must be defined based on the inherent order semantics of
the words and word domains of the variables that establish an order-based
structure that is rich enough to solve the problems in fact. The thesis has focused
on studying the problem of interpretability of FRBS in HA-based approach and
proposed some constraints, definitions and theorems in this approach.
At the same time, the thesis also studies the approach based on real-world
interpretability for the problem of interpretability of fuzzy system and deeper
and more practical analysis of the interpretability of formal theories includes
natural human languages in general and formalized fuzzy systems.
From the results achieved in the thesis, we can draw some following
conclusions:
• The interpretability of a computional expression depends on the idea that
the interpretability needs to preserve discovered relationships based on the
semantics of linguistic expression, and the proposal of constraints for
interpretation operations are different from the concept of interpretability of
fuzzy systems studied in the fuzzy set theory.
• The HA-based approach is similar to the idea of interpretability of formal
programming languages. In which, the semantics of syntactic expressions
defined by interpretation operations assign them certain mathematical objects

that are considered to be their semantics. This interpretation aims to preserve the
validity of their semantics on the basis of preserving axioms for the
mathematical structure assigned to syntactic expressions.
• As shown in this study, the fuzzy set theory, more specifically the
standard fuzzy set algebra may not be interpretable of real world semantics.
Methodologically, it is an essential shortcoming of fuzzy set theory based on the
perspective of a real-world semantic approach. However, despite this, fuzzy set
theory is still one of the great theories because it is very flexible in applications,
this omission can be overcome by testing experimental studies until realizing the
objective.