REPRESENTATION THEOREM FOR FUZZY FUNCTIONS

Graded Form

Martina Daˇnkov´a

Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22, Ostrava, Czech Republic

Keywords:

Fuzzy function, Extensionality, Functionality, Fuzzy rules, Approximate reasoning.

Abstract:

In this contribution, we will extend results relating to representability of a fuzzy function using a crisp function.

And additionally, we show for which functions there exist fuzzy function of a speciﬁc form. Our notion of

fuzzy function has a graded character. More precisely, any fuzzy relation has a property of being a fuzzy

function that is expressed by a truth degree. And it consists of two natural properties: extensionality and

functionality. We will also provide a separate study of these two properties.

1 INTRODUCTION

Historically, fuzzy functions took many forms by

their deﬁnition and they live contemporaneously with

their applications. As an example, we put here two

mostly knowninterpretations of this notion: 1’st – it is

any mapping that assign a fuzzy set to a fuzzy set, see

e.g., (Nov´ak et al., 1999); 2’nd – it is a fuzzy relation

speciﬁed by various properties (see e.g., (Demirci,

1999a; Demirci, 2001; Demirci and Recasens, 2004)),

which gave rise to notions such as partial, perfect,

strong fuzzy function etc.

In this work, we will turn our attention to the

second class of the interpretation, i.e., we will ex-

plore fuzzy relations that meet some special require-

ments. Wide overview together with applications can

be found in the following exemplary sources (Kla-

wonn, 2000; Demirci, 2001; Bˇelohl´avek, 2002). As

noted in (Demirci, 2000), not all notions of the fuzzy

function do coincide with the classical notions for

crisp functions. In this paper, we will try to avoid

this problem and all the subsequent deﬁnitions will

be consistent with the classical notions whenever ap-

plied on crisp input. As a basis for our work we will

take Demirci’s deﬁnition of fuzzy function (Demirci,

2001) adjusted to our framework.

Our framework and methodology stems from

(Bˇehounek and Cintula, 2006) and it can be charac-

terized by the following items:

1. We will work inside a speciﬁc fuzzy logic.

2. Fuzzy sets (relations) will be handled as objects

in formal language deﬁned by a formula without

direct interpretation using truth values (over some

chosen structure).

3. Statements about the objects of the interest will be

in the form of graded theorems, which means that

instead of the usual statement of the problem

If ⊢ ϕ then ⊢ ψ – Classical theorem (1)

we search for more informative and general form

(not equivalent) of this statement

⊢ ϕ

→ ψ – Graded theorem. (2)

Practically, we analyze how many times we need

to incorporate an antecedent ϕ to prove the con-

sequent ψ and we code the result into the degree

Graded theorems may become very difﬁcult for

non-experienced reader therefore, each section will

be equipped by paragraphs that translate the most

important formulae. Translation will be given ﬁrst

into the language of models for some special theories

and second into the special language that mathemati-

cians can use whenever they work over some “fuzzy

logic”

. Those who prefer formal notation may skip

these parts of the text and concentrate only on the

technical aspects of the addressed problems.

The subsequently proposed reading of the graded

theorem will be completely analogous to the classi-

cal case (using classical mathematical logic (CML))

The notion Fuzzy logic does not represent here a wide

range of applications as it is usual in engineers papers, but

it denotes formal logics (of speciﬁed order and type) having

syntax and semantics.

nková M..

REPRESENTATION THEOREM FOR FUZZY FUNCTIONS - Graded Form.

DOI: 10.5220/0003080900560064

In Proceedings of the International Conference on Fuzzy Computation and 2nd International Conference on Neural Computation (ICFC-2010), pages

56-64

ISBN: 978-989-8425-32-4

 2010 SCITEPRESS (Science and Technology Publications, Lda.)

and we will distinguish between them using a special

typeface. For the classical two-valued logic we will

reserve the following typeface:

Language of CML Reading

ϕ ⇒ ψ If ϕ then ψ

ϕ ⇔ ψ ϕ iff ψ (if and only if)

ϕ∧ ψ ϕ and ψ

ϕ∨ ψ ϕ or ψ

And for the chosen fuzzy logic, we set

Language of FL Reading

ϕ → ψ IF ϕ THEN ψ

ϕ ↔ ψ ϕ IFF ψ

ϕ∧ ψ ϕ AND ψ

ϕ∨ ψ ϕ OR ψ

ϕ&ψ ϕ and ψ

ϕ⊕ ψ ϕ or ψ

Indeed, meta-language that we are using when speak-

ing about provable formulae over some fuzzy logic is

the formal language of classical mathematical logic

(obviously by the fact that a given formula is either

provable or not). Hence, using the completeness of

the used background logic, we can carefully rein-

terpret known results proved in an algebraic setting

onto syntactical leveland we obtain classical theorem.

And further, we can try it for graded theorems.

The main difference between readings of a classi-

cal theorem and a graded theorem is as follows. Hav-

ing a classical theorem, we use language of classical

logic to read propositions that include provable for-

mulae. And in the case of graded theorem, we read

directly a formula using the generalized language. We

can say:

“We write classically, but we think in grades.”

As an example of classical theorem (1), we can

assume that ϕ can be interpreted as “relation is exten-

sional” and ψ as “relation is Lipschitz continuous”. In

this case, we can read (1) as

“If a relation is extensional then it is Lipschitz

continuous.”

Here, the given relation must be extensional to the de-

gree 1 to deduce that it is Lipschitz continuous to the

same degree. It can be shown that there is also graded

theorem (2) with n = 1 for this statement that can be

read in the analogous way

“IF a relation is extensional THEN it is Lipschitz

continuous.”

In this case, we have incorporated also additional hid-

den grades for both properties. Hence, a relation is

Lipschitz continuous (e.g. to the degree b) as much as

it is extensional (to the degree a such that a ≤ b).

The paper will be organized in the following way:

- First we introduce our logical framework, basic no-

tions and overview related results.

- The following two sections will be devoted to the

notions of extensionality and functionality, re-

spectively.

- And ﬁnally, Section 4 will provide a deep insight

into the “theory” of fuzzy functions.

2 LOGICAL FRAMEWORK

Let us work in the framework of fuzzy class the-

ory (FCT) (Bˇehounek and Cintula, 2005), which is a

schematic extension of a background logic (that con-

tain crisp equality = and Baaz-delta ∆) by axiom of

comprehension and extensionality axiom. Provability

in FCT will be denoted simply by the same shortcut

in front of ⊢ or it will be explicitly written. The back-

ground logic may be various (so that completeness

theorem is valid for FCT) due to our actual require-

ments. In our case, the weakest background logic

will be a many-sorted ﬁrst order involutive monoidal

t-norm based logic (IMTL) and we will deal only with

this logic throughout the whole text.

The language J consists of the following set of

basic connectives (&, →, ∧), involutive negation

¬.

The quantiﬁer ∀, truth constants ⊤,⊥ and variables of

the speciﬁc sorts.

Standardly, we introduce the following connec-

tives and quantiﬁer:

x∨ y ≡

d f

((x → y) → y) ∧ ((y → x) → x), (3)

x⊕ y ≡

d f

¬(¬x&¬y), (4)

x ↔ y ≡

d f

(x → y)&(y → x), (5)

(∃x)ϕ ≡

d f

¬(∀x)¬ϕ. (6)

IMTL extends MTL by the following schemata of

axioms:

(INV) ¬¬ϕ → ϕ.

Let us summarize properties of MTL and its vari-

ous extensions.

Proposition 1. IMTL proves

ϕ ↔ ¬¬ϕ, (7)

(ϕ → ψ) ↔ (¬ψ → ¬ϕ). (8)

Interpretation of the connectives is given by the

corresponding operations {∗,→

∗

,∧

,∨

,∼}, and

Mostly, ¬ is reserved for residual negation and ∼ is

usual symbol for involutive negation. In this paper, we work

only with involutive negation, hence, there is no danger of

confusion between these two various notations.

REPRESENTATION THEOREM FOR FUZZY FUNCTIONS - Graded Form

the constant ⊥ is interpreted as 0, which together form

an IMTL-algebra denoted by L .

An L -structure for the language J is of the form

M = h(X

)

fors

,(r

)

P−predicate

,(m

)

c−constant

i = 1,2,. . . ,n, where each X

is non-empty set of ob-

jects, r

is an L-fuzzy relation of the respective type

and m

belongs to D

provided that c is of the type s

2.1 Basic Notions

Let F, ≈

,≈

be predicates of the type

),(s

), respectively, i.e. their are in-

terpreted as F

⊂

∼

× X

,≈

≈

⊂

∼

× X

,≈

≈

⊂

∼

× X

;

0 be a constant denoting an atomic individual of the

domain of discourse. For the better orientation, we

will use the same terminology on the syntax as well

as on the semantical level. Moreover, we will omit

the speciﬁcation of sorts, whenever it will be clear

from the concept.

Moreover, let R,S are fuzzy relations or fuzzy sets

of the same type and ¯x includes all free variables of

R,S then we deﬁne the following properties:

• Reﬂexivity:

Reﬂ

≡

d f

(∀x)Rxx

• Symmetry:

Sym

≡

d f

(∀x,y)(Rxy → Ryx)

• Transitivity:

Trans

≡

d f

(∀x,y,z)[(Rxy&Ryz) → Rxz]

• Similarity:

Sim

≡

d f

Reﬂ

&Sym

&Trans

• Subsethood:

R ⊆ S ≡

d f

(∀x,y)(R(x,y) → S(x,y))

• Strong set-similarity:

∼

S ≡

d f

(∀x,y)(R(x,y) ↔ S(x,y))

• Set-similarity:

R ≈ S ≡

d f

(R ⊆ S)&(S ⊆ R)

• Totality:

Tot

≡

d f

(∀x∃y)Rxy

• Surjectivity:

Sur

≡

d f

(∀y∃x)Rxy

• Injectivity:

Inj

≈

R ≡

d f

(∀x,x

′

)[(∃y)(Rxy&Rx

′

y) → (x ≈

′

)]

Moreover, the following set operations can be intro-

duced:

A∪ B =

d f

{x | (x ∈ A) ⊕ (x ∈ B)} strong union

A∩ B =

d f

{x | (x ∈ A)&(x ∈ B)} strong intersec.

A⊔ B =

d f

{x | (x ∈ A) ∨ (x ∈ B)} union

A⊓ B =

d f

{x | (x ∈ A) ∧ (x ∈ B)} intersection

We will additionally deal with relational compo-

sitions deﬁned using a class notation. A systematic

study can be ﬁnd in (Bˇelohl´avek, 2002). We will use

three basic relational compositions.

• sup-T composition:

R◦ S =

d f

{xy | (∃z)(Rxz&Szy)}

• BK-subproduct:

R⊳ S =

d f

{xy | (∀z)(Rxz → Szy)}

• BK-superproduct:

R⊲ S =

d f

{xy | (∀z)(Rxz ← Szy)}

3 EXTENSIONALITY AND ITS

PROPERTIES

The extensionality is one of the most important prop-

erties of fuzzy relations that are used in fuzzy control.

Indeed, only such relations are considered for approx-

imation by fuzzy rules, see (H´ajek, 1998; Daˇnkov´a,

2007). It is deﬁned by the following formula:

Ext

≈

1,2

F ≡

d f

(∀x,x

′

,y,y

′

)

[(x ≈

′

)&(y ≈

′

)&F(x,y) → F(x

′

)].

Using ≈

1(2)

we capture a relationship between ele-

ments of the input (output) space. When assuming

two concrete individuals, the requirement says the

following: closer are the individuals {a,b,c,d} then

more equivalent are the degrees to which the proposi-

tion ﬁres for these individuals, we can read it symbol-

ically as

“IF a ∈ U(b)

≈

and c ∈ U(d)

≈

and F(a, c)

THEN F(b,d)”,

where

U(x)

≈

d f

{y | x ≈ y}

expresses a neighbourhood of the element x.

3.1 Extensionality of Set Operations

and Relational Compositions

Let us summarize properties relating to extensional-

ity.

ICFC 2010 - International Conference on Fuzzy Computation

Proposition 2. FCTproves

Ext

≈

1,2

F&Ext

≈

1,2

E → Ext

≈

1,2

(F ∩E), (9)

Ext

≈

1,2

F&Ext

≈

1,2

E → Ext

≈

1,2

(F ∪E), (10)

Ext

≈

1,2

F ∧Ext

≈

1,2

E → Ext

≈

1,2

(F ⊓E), (11)

Ext

≈

1,2

(F ⊔E) → Ext

≈

1,2

F ∨ Ext

≈

1,2

E. (12)

Readings of the above results:

(9) – “IF F and E are extensional THEN their strong

intersection is extensional.”

(10) – “IF F and E are extensional THEN their strong

union is extensional.”

(11) – “IF F AND E are extensional THEN their inter-

section is extensional.”

(12) – “IF the union of F and E is extensional THEN F

is extensional OR E is extensional.”

In the following, the extensionality of superset and

similar set is studied.

Proposition 3. FCTproves

(F ⊆ E)

→ [Ext

≈

1,2

F → Ext

≈

1,2

E], (13)

(F ≈ E)

→ [Ext

≈

1,2

F ↔ Ext

≈

1,2

E]. (14)

Readings of the results:

(13) – “IF F is a subset of E (we need this requirement

twice) and F is extensional THEN E is extensional.”

(14) – “IF F and E are similar sets (we need this re-

quirement twice) THEN F is extensional IFF E is ex-

tensional.”

The above formulae together with properties of

the relational compositions produces a long list of

consequences.

Corollary 1. Let

≡

d f

⊆ E

)

≡

d f

⊆ F

)

≡

d f

≈ E

)

≡

d f

≈ F

)

Then FCTproves

→[Ext

≈

1,2

(F ◦ E

) → Ext

≈

1,2

(F ◦ E

)],

→[Ext

≈

1,2

(F ⊳E

) → Ext

≈

1,2

(F ⊳E

)],

→[Ext

≈

1,2

(F ⊲E

) → Ext

≈

1,2

(F ⊲E

)],

→[Ext

≈

1,2

⊳ E) → Ext

≈

1,2

⊳ E)],

→[Ext

≈

1,2

⊲ E) → Ext

≈

1,2

⊲ E)],

→[Ext

≈

1,2

(F ◦ E

) ↔ Ext

≈

1,2

(F ◦ E

)],

→[Ext

≈

1,2

(F ⊳E

) ↔ Ext

≈

1,2

(F ⊳E

)],

→[Ext

≈

1,2

(F ⊲E

) ↔ Ext

≈

1,2

(F ⊲E

)],

→[Ext

≈

1,2

⊳ E) ↔ Ext

≈

1,2

⊳ E)],

→[Ext

≈

1,2

⊲ E) ↔ Ext

≈

1,2

⊲ E)],

Intersection:

Ext

≈

1,2



F∈A



◦ E



→ Ext

≈

1,2



F∈A

(F ◦ E)



Ext

≈

1,2



[

F∈A

(F ⊳ E)



→ Ext

≈

1,2



F∈A



⊳ E



Ext

≈

1,2



[

E∈A

(F ⊳E)



→ Ext

≈

1,2



F ⊳

[

E∈A



Union:

Ext

≈

1,2



[

F∈A



◦ E



↔ Ext

≈

1,2



[

F∈A

(F ◦ E)



Ext

≈

1,2



F∈A

(F ⊳ E)



↔ Ext

≈

1,2



[

F∈A



⊳ E



Ext

≈

1,2



E∈A

(F ⊳E)



↔ Ext

≈

1,2



F ⊳

E∈A



3.2 Duality between Extensionality and

1-Lipschitz Continuity

The duality between pseudo-metrics and similari-

ties based on the additive generator of a t-norm has

been used to prove the duality between extensional-

ity (on the model where & is interpreted as continu-

ous archimedean t-norm) and Lipschitz continuity in

the induced pseudo-metric space (Mesiar and Nov´ak,

1999) (later in (Perﬁlieva, 2004)). Below, we provide

a graded form of the result from (Daˇnkov´a, 2010).

By the above observations, the mutual inversion

between extensionality and Lipschitz continuity ob-

viously follows.

Theorem 1. Let

Lipschitz

d,d

′

F ≡

d f

(∀ ¯x, ¯y)[d(F( ¯x),F( ¯y)) → d

′

( ¯x, ¯y)],

and moreover, Deﬁne

′

( ¯x, ¯y) =

d f

¬(x

≈

) ⊕ ¬(x

≈

), (15)

d(x,y) = ¬(x ↔ y). (16)

Then FCTproves

Ext

≈

1,2

F ↔ Lipschitz

d,d

′

Reading of the result:

“F is extensional w.r.t. ≈

1,2

IFF

F is Lipschitz continuous w.r.t. d, d

′

.”

Whenever ≈

1,2

are not similarities, we should

not speak about an analogy with Lipschitz continu-

ity. There, we obtain only one pseudo-metric dual

to equivalence, i.e. d, and a correct interpretation of

Lipschitz

d,d

′

F can be expressed as a domination of

pseudo-metric d applied on values F by relation d

′

REPRESENTATION THEOREM FOR FUZZY FUNCTIONS - Graded Form

4 FUNCTIONALITY AND ITS

PROPERTIES

In this section, we will introduce a property of fuzzy

relation called functionality that is a direct generaliza-

tion of the related crisp notion.

Let the functionality property be given by the

following formula (semantic version in (Demirci,

1999b)):

Func

≈

1,2

F ≡

d f

(∀x,x

′

,y,y

′

)

[(x ≈

′

)&F(x,y)&F(x

′

) → (y ≈

′

)]. (17)

It is not an obvious generalization of the classical def-

inition of functionality axiom:

If F(x,y) and F(x,y

′

) then y = y

′

represented by

F(x,y)&F(x, y

′

) → y = y

′

whenever F is crisp. The classical functionality can

be tested on fuzzy relations as well and it can be ex-

pressed within the fuzzy logic using Baaz-delta ∆ as

follows:

∆F(x,y)&∆F(x, y

′

) → y = y

′

thus for crisp relations, the form of functionality ax-

iom does not differ from its origin.

The most natural generalization stands in remov-

ing all crisp constrains to F and the equality, which is

now replaced by an equivalence relation determining

which elements are indistinguishable on the universe

of discourse

F(x,y)&F(x, y

′

) → y ≈

′

, (18)

and it seems that we are done at that point. However,

taking closer look at the above formula, we uncover

the hidden crisp equality there relating to the variable

x. Since we work in the setting where ≈

possesses

what is distinguishable on the input space, therefore

we should incorporate this fact by modifying left side

of the implication in the functionality formula

x ≈

′

&F(x,y)&F(x

′

) → y ≈

′

Observe that when assuming reﬂexivity of ≈

, we

have (18) as the special instance of the above formula

and so of the formula deﬁning Func

≈

1,2

Example 1. Duality between extensionality and

Let L = h[0, 1],⊗,⇒,∨,∧,0,1i be the standard

Łukasiewicz algebra,

x ∼ y = (x ⇔ y)

= (x ⇔ y) ⊗ (x ⇔ y) ⊗ (x ⇔ y).

The following is the example of a relation that is

functional to the degree 1 w.r.t. =,=: F

(x,y) =

{

0.5

/sin(x)}.

Considering similarity relation ∼, we ﬁnd out that

1. F

(x,y) = y ∼ sin(x) is functional to the degree 1

w.r.t. ∼,∼.

2. F

(x,y) = y ∼ f(x), where

f(x) =



1.7x

, x ∈ [0,0.5];

cos(0.9x) − 0.5, otherwise.

is functional to the degree 1 w.r.t. =,∼.

3. F

(x,y) = F

∨ F

is functional to the degree 0

w.r.t. ∼, ∼. Take x = x

′

= 1 and y = 0.1, y

′

= 0.8

then F(x,y) ⊗ F(x

′

) = 1, but y ∼ y

′

= 0.

Figure 1: Functional relation F

w.r.t. ∼,∼.

Figure 2: Functional relation F

w.r.t. ∼,∼.

Figure 3: Non-functional relation F

w.r.t. ∼,∼.

4.1 Functionality of Set Operations and

Relational Compositions

Let us summarize properties relating to functionality.

ICFC 2010 - International Conference on Fuzzy Computation

Proposition 4. FCTproves

Func

≈

1,2

F&Func

≈

2,3

S → Func

≈

1,3

(F ◦ S), (19)

Func

≈

1,2

F&Func

≈

1,2

S → Func

≈

1,2

(F ∩S), (20)

Func

≈

1,2

F ∧Func

≈

1,2

S → Func

≈

1,2

(F ⊓S), (21)

Func

≈

1,2

(F ⊔ S) → Func

≈

1,2

F ∧Func

≈

1,2

(22)

We can also introduce the following properties of

relations analogous to the classical notions.

Proposition 5. FCTproves

(S ⊆ F)

→ [Func

≈

1,2

F → Func

≈

1,2

S], (23)

(F ≈ S)

→ [Func

≈

1,2

F ↔ Func

≈

1,2

S]. (24)

Similarly as in the previous section, we may generate

a long list of corollaries for the relational composi-

tions.

Note that we have decided to omit the reading of

the above results because it is in a direct analogy with

reading of the results in Section 3.1.

5 FUZZY FUNCTIONS AND

THEIR RELATION TO CRISP

FUNCTIONS AND VICE-VERSA

When having crisp functions, we can always express

this special relational dependency by y = f(x), which

follows from the functionality property throughout

the compatibility property w.r.t. =,= deﬁned by (25).

If we exchange = by ≈

, i.e. we assume that each

x is mapped to a neighbourhood of f(x), which can

be represented using ≈

as y ≈

f(x). This relation is

the fuzzy functions w.r.t. =,≈

, provided that ≈

reﬂexive.

Let us ﬁrst explore how does the relation be-

tween compatibility and functionality/extensionality

look like for a specially chosen relation F

(x,y) :=

y ≈

f(x).

Lemma 1. Let us deﬁne

Comp

≈

,≈

f ≡

d f

(∀x,y)(x ≈

y) → ( f(x) ≈

f(y)), (25)

(x,y) ≡

d f

y ≈

f(x), (26)

C ≡

d f

Sym

≈

&(Trans

≈

)

. (27)

Then FCTproves

Tot f → TotF

, (28)

C →[Comp

≈

,≈

f → Func

≈

,≈

], (29)

C →[Comp

≈

,≈

f → Ext

≈

,≈

], (30)

When deﬁning fuzzy function (property Function

deﬁned by (31) assigned to a relation F), it is usu-

ally assumed the extensionality and functionality to-

gether. Extensionality says that we can substitute

the original inputs (x,y) by the indistinguishable one

′

). The formula representing the functionality is

the exact analogy with the classical deﬁnition, where

we assume that the images of indistinguishable ele-

ments are indistinguishable. Relating to this interpre-

tation, we must still keep on mind that ≈

1,2

repre-

sent the granularity of the input (output) space, which

means that they are coarsest relations in our system

(∀R)(≈

1(2)

⊆ R) enabling us to distinguish elements

of the universe.

Theorem 2. Let us deﬁne

Function

≈

1,2

F ≡

d f

Ext

≈

1,2

F ∧ Func

≈

1,2

F, (31)

Deﬁnition

F, f

≡

d f

(∀x)[F(x, f(x)) ↔ (∃y)F(x,y)], (32)

be deﬁned by F

(x,y) ≡

d f

y ≈

f(x), where f is

some unary functional symbol.

Then FCTproves

C&Comp

≈

,≈

f → Function

≈

,≈

, (33)

Tot f&Deﬁnition

, f

→ ( f

≈ f). (34)

Reading of the results:

(33) – “IF ≈

is symmetric and transitive (we

need transitivity twice) and f is compatible THEN

y ≈

f(x) is fuzzy function.”

(34) – “IF f is compatible and total and f

is so that

for an arbitrary x : [ f

(x) ≈

f(x) IFF there exists

y : y ≈

f(x)] THEN f

is similar to f.”

Now, let us address the reverse problem: consider

a fuzzy relation F and let us ﬁnd a crisp function f

such that it is compatible with (≈

,≈

) and its exten-

sion to fuzzy relation F

is similar to F.

Theorem 3. Let D ≡

d f

TotF&Deﬁnition

F, f

Then FCTproves

D&(TotF)&Func

≈

1,2

F → Comp

, (35)

D&Function

≈

1,2

F&Reﬂ

≈

→ (F

≈ F). (36)

Reading of the results:

(35) – “IF F is functional and total (we need totality

twice) and f

is so that for an arbitrary x : [F(x, f

(x)

IFF there exists y : F(x,y)] THEN f

is compatible

function.”

(36) – “IF F is a total fuzzy function and f

is as

above and ≈

is reﬂexive THEN F

is similar to F.”

REPRESENTATION THEOREM FOR FUZZY FUNCTIONS - Graded Form

Let us recall the original representation theorem

(translated into our framework) to provide a compar-

ison with our graded variant. For the semantical ver-

sion see e.g. (Bˇelohl´avek, 2002).

Theorem 4. Let

FEqual

≡

d f

Sim

&(∀x,y)[∆R(x,y) → (x = y)],

= { Sim

≈

,FEqual

≈

,Comp

≈

,≈

= { Sim

≈

,FEqual

≈

,TotF,Function

≈

1,2

Deﬁnition

F, f

= { Deﬁnition

, f

} ∪ T

Then

FCT∪ T

⊢ Function

≈

1,2

, (37)

FCT∪ T

⊢ Comp

≈

1,2

. (38)

Moreover,

FCT∪ T

⊢ f

= f, (39)

FCT∪ T

⊢ F

= F. (40)

Reading of the above formulae:

(37) – “If ≈

is a similarity and ≈

is a fuzzy equality

and f is compatible w.r.t. ≈

1,2

then F

is the fuzzy

function.”

(39) – “If we additionally assume that

(x, f

(x)) =

y∈Y

(x,y), for an arbitrary x,

then f

is equal to f.”

(38)+(40) – “If ≈

is a similarity and ≈

is a fuzzy

equality and F is a total fuzzy function then f

satisfying F(x, f

(x)) =

y∈Y

F(x,y) is compatible

function and moreover, F

and F are identical.”

We see that the readings do not differ signiﬁcantly.

The main difference is in gradualness of the new re-

sults, which reﬂects also in similarity of f

and F

with their pre-images, respectively.

An intended class of applications of the intro-

duced theory of fuzzy functions is connected with

fuzzy rules, especially, the implicative fuzzy rules.

Originally, implicative fuzzy rules where introduced

as an approximation of fuzzy relations that are func-

tional (to the degree 1). In this case, we obtain an up-

per approximation of the original relation and more-

over, a precision of such approximation can be esti-

mated. But if we have at the disposal only partially

functional relation (to the degree 0 < α < 1) then we

cannot expect that implicative rules will provide an

upper approximation. The graded approach gives at

least an estimation of a residua between the implica-

tive rules and the original relation. The following ex-

ample illustrates an approximation of partially func-

tional fuzzy relations using implicative rules.

Example 2. Let us assume the standard Łukasiewicz

algebra L

as an interpretation for the connectives,

i.e., L

= h[0,1], ⊗,→

⊗

,∧,∨,∼i, where

x⊗ y = max(0, x + y− 1),

x →

⊗

y = max(1, 1 − x+ y),

x∧ y = min(x,y),

x∨ y = max(x,y),

∼ x = 1 − x.

In this case, we have x ↔

⊗

y = max(0,1−|x−y|) and

x ↔

⊗

y = (x ↔

⊗

y)... (x ↔

⊗

| {z }

k−times

= max(0,1− k|x− y|).

In (Mesiar and Nov

ak, 1999), it has been shown

that Lipschitz continuity of a function f : [0, 1] 7→ [0,1]

w.r.t. the standard metric and with the Lipschitz con-

stant k is equivalent with compatibility of f w.r.t.

↔

⊗

,↔

⊗

over Łukasiewicz algebra, i.e. the interpre-

tation of Comp

↔

,↔

f over L

is equal to 1, we

write symbolically ||Comp

↔

,↔

f|| = 1. Formula (29)

says that ||Comp

↔

,↔

f|| ≤ ||Func

↔

,↔

||. More-

over, it can be proved that if ||Func

↔

,↔

|| = 1 then

∀x,y ∈ [0,1] :

(x,y) ≤ Rules

(x,y), where (41)

Rules

(x,y) =

c∈M

[(x ↔

⊗

c) →

⊗

(y ↔

⊗

f(c))],

(42)

and M ⊂ [0,1]. And even more

FCT ⊢ Func

↔

,↔

→ (F

⊆ Rules

), (43)

which gives the following estimation:

||Func

↔

,↔

|| ≤ ||F

⊆ Rules

||. (44)

Let us ﬁx k = 1. Then we obtain the following table of

degrees of compatibility for the speciﬁc functions:

f ||Comp

↔,↔

f||

sin(x) 1

sin(2x) 0.6576 = e

sin(3x) 0.4675 = e

Figures 4–6 depict the sample relations F

, where f

is from the above table. Additionally, Figures 7–9

visualize implicative rules with 11 equidistantly dis-

tributed nodes M = {0,0.1,. . .,0.9,1}.

From (44) and (29), we conclude that

sin(x)

(x,y) ≤ Rules

sin(x)

(x,y),

≤ min

(x,y)∈[0,1]

sin(2x)

(x,y) →

⊗

Rules

sin(2x)

(x,y),

≤ min

(x,y)∈[0,1]

sin(3x)

(x,y) →

⊗

Rules

sin(3x)

(x,y).

Figures 10 and 11 show residua of F

sin(ix)

from

Rules

sin(ix)

, where the visualized z-coordinate is set

to [e

,1] for i = 2, 3.

ICFC 2010 - International Conference on Fuzzy Computation

Figure 4: F

sin(x)

Figure 5: F

sin(2x)

Figure 6: F

sin(3x)

Figure 7: Rules

sin(x)

Figure 8: Rules

sin(2x)

Figure 9: Rules

sin(3x)

Figure 10: F

sin(2x)

→

⊗

Rules

sin(2x)

Figure 11: F

sin(3x)

→

⊗

Rules

sin(3x)

REPRESENTATION THEOREM FOR FUZZY FUNCTIONS - Graded Form

6 CONCLUSIONS

In this contribution, we have generalized the well

known representation theorem for fuzzy function into

the graded form. The main advantage of our approach

is that it incorporates the whole scale for degrees of

truth, while in the original approach the results were

applied to properties valid in the degree 1. More-

over, evaluation of the degrees of the antecedents in

our graded theorems provides an additional informa-

tion about a “precision” of the consequent. E.g., if ≈

is similarity relation then an evaluation of D in (34)

estimates closeness of f

≈ f, or in other words, a

distance (dual to ≈) between f

and f.

Hence, we have shown that graded theorems bring

a new light into the already well established theory of

fuzzy functions. And additionally, the logical frame-

work provides an uniﬁed approach to mathematics of

fuzzy logic that corresponds with the classical one

(also in the notational standards).

ACKNOWLEDGEMENTS

We gratefully acknowledge support of the grant

MSM6198898701 of the M

SMT

CR.

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ICFC 2010 - International Conference on Fuzzy Computation