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

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).

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

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 world-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

Real-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.

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

281-293.

5 Cat-Ho Nguyen and W Wechler (1992),” Extended hedge algebras and their application to Fuzzy

logic”, Fuzzy Sets and Systems, 52, 259-281

6L A Zadeh, Fuzzy set, Information and control, 8, (1965), pp 338-353

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

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 = h n h1u and hi h1uh i-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 u<v,

and 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 = k m …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'<j (here if j = min {n, m} + 1 then either

h j = I, hj is the unit operator I, for j = n + 1 ≤ m or kj = I for j = m + 1 ≤ n) and i) x<y iff h j x j<kj x j, where xj = hj-1 h1u

ii) x = y iff m = n and h j x j = kj x j

iii) x and y are not comparable iff h j x j and kj x j 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

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H fm(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,

)()()()(

y fm hy fm x fm hx

fm  , that is this propotion does not 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:

(v) Given fm(c - ), fm(c +) and (h), hH, the for x = h n 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 hj h j-1 … h1c ≠ h j-1 … h1c, for every j

= 1, …, n and h0 = Id, identity, i.e h0c = c, then:

Sign(x)=Sign(h n h n-1 …h1c) = sign(h n ,h n-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 hx<x and if Sign(hx) = 0 then hx = x

From 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(h p x) = +1 H(h -q x) ≤…≤ H(h-1x) ≤ x ≤ H(h1x) ≤…≤ H(h p x) (1.3)

j sign i j sign i

x fm x j h x j h x fm i h x

j h

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.

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

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

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

oC

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

relat ion etc.)

The HA AX modeling the word-domain D

containing the LFoC

The HA of the domain:

word word HAword expressions: string representations of words

in D

- LFoCs and their relationship structure

The desired computational objects

of a comput math structure

Constraint 2.2 [8] (Formalization of word quantification): The

comput-semantics of words, including f-sets comput-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 HA-approach

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

[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):

x<y  core(x)<y & x<core(y) & core(x)<core(y) (2.1)

and 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 interpretation ℐtrp(x), the interval-interpretation ℐint(x) of the words of 𝒳 should

triple-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<yℐtrp(core(x))≼ℐtrp(y) &ℐtrp(x)≼ℐtrp(core(y))

9 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

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 multi-levels of specificity of 𝔉: Decompose 𝔉 into

specificity-levels so that

the words on each level have the same G-S degree or, equivalently, they have the same length Denote by 𝔉k the set of the words of

{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)

0

Lev

e l

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

 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

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

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

Structures of the

real-world

Formal theories developed based on their axioms

Real world models

of formal theories

Applications/algorithms designed based on certain formal theories interacting with their RW-counterparts

Figure 3.1 Relationships between formalized theories, their models and

applications and their RW counterparts