# Comparison of Different Algorithms of Approximation by Extensional Fuzzy Subsets

TL;DR: Different methods to approximate an arbitrary fuzzy subset by an adequate extensional one are compared in order to understand better the performance and improvement they give.

Abstract: How to approximate an arbitrary fuzzy subset by an adequate extensional one is a key question within the theory of Extensional Fuzzy Subsets. In a recent paper by the authors [19] different methods were provided to find good approximations. In this work these methods are compared in order to understand better the performance and improvement they give.

Topics: Fuzzy logic (56%)

Comparison of Different Algorithms of

Approximation by Extensional Fuzzy Subsets

Gabriel Mattioli and Jordi Recasens

Abstract How to approximate an arbitrary fuzzy subset by an adequate extensional

one is a key question within the theory of Extensional Fuzzy Subsets. In a recent

paper by the authors [19] different methods were provided to ﬁnd good approxi-

mations. In this work these methods are compared in order to understand better the

performance and improvement they give.

1 Introduction

Indistinguishability operators were introduced to fuzzify the concept of crisp equiv-

alence relations. These operators allow to model the idea of ”similarity” between

elements, which is key to understand how we ”identify” objects. The operation of

identiﬁcation is the mainstone to simplify the representation we have of the en-

vironment and understand the information given by our perception. Being able to

identify objects enables us to store less quantity of information if favour of being

able to extract a qualitative analysis of it.

An eye without a mechanism to identify objects is nothing but a sensor of outern

reality. An eye with this mechanism becomes a perceptive system that can ”under-

stand” the environment.

Under an indistinguishability operator the observable fuzzy sets are the exten-

sional ones. These sets correspond to the fuzziﬁcation of classical equivalence

classes. Within the theory of Fuzzy Logic the ﬁrst researcher to point the relevance

of these sets was Zadeh when he discussed the concept of granularity [24].

Department of Mathematics and Computer Science

ETS Arquitectura del Valles

Universitat Polit`ecnica de Catalunya

C/ Pere Serra 1-15. 08190 Sant Cugat del Valls. Spain

e-mail: gabriel.mattioli@upc.edu

e-mail: j.recasens@upc.edu

1

2 Gabriel Mattioli and Jordi Recasens

If we assume that indistinguishability operators are a good model to understand

similarities between objects (and there is evidence to think so), then a very interest-

ing problem is how an arbitrary fuzzy subset can be approximated by an extensional

one with the minimum loss of accuracy.

In a previous work by the authors [19] this problem was faced and 3 methods

were derived for Archimedean t-norms (T = Łukasievicz and T =

Π

product) and

one for the Minimum t-norm.

Restricting to Archimedean t-norms, the ﬁrst method was based on ﬁnding an

adequate mean between two operators that provide the best upper and lower ap-

proximation by extensional fuzzy subsets of a given fuzzy subset

µ

. The second

one computed an adequate power of the lower approximation of

µ

, and the last one

found the solution solving a Quadratic Programming problem.

Big differences can be found between the ﬁrst two and the last method. The QP-

based one guarantees that the solution found is optimal while the ﬁrst two do not.

On the other hand, the last method suffers drastically the curse of dimensionality

and becomes computationally unaffordable for large cardinalities of the universe

of discourse X. The ﬁrst two do not have this problem and work even when X is

non-ﬁnite.

The aim of this work is to compare in depth the mean-based and the power-based

methods. In order to reduce the scope of this comparison we will restrict to the

Łukasievicz t-norm and to ﬁnite sets. This has been done because in [19] explicit

formulas were provided to ﬁnd the best approximations when T = Ł, while the best

approximation for T =

Π

had to be found by numerical methods.

The work is structured as follows:

In Section 2 the Preliminaries to this work are given. In this section the deﬁnition

and main properties of indistinguishability operators and extensional sets will be

recalled.

Section 3 will show how the mean-based method can be built. Natural weighted

means will be introduced ﬁrst and explicit formulas will be provided to ﬁnd the

extensional fuzzy subset that better approximates

µ

following this method.

In Section 4 the power-based method will be given. First of all it will be shown

how powers can be deﬁned with respect to a t-norm T and further how this can be

used to ﬁnd good approximations by extensional fuzzy subsets.

In Section 5 a comparison between these two methods will be provided. Fixed an

indistinguishability operator E we will study the output and error committed by each

of the methods when approximating different fuzzy subsets and some conclusions

will be extracted.

Finally, the Concluding Remarks of this work will be given in Section 6.

Comparison of Different Algorithms of Approximation by Extensional Fuzzy Subsets 3

2 Preliminaries

In this section the main concepts and results used in this work will be given. The

deﬁnition of indistinguishability operator will be recalled as well as the main prop-

erties of the extensional fuzzy subsets related to an indistinguishability operator.

First of all let us recall the well known Ling’s Theorem which introduces the

concept of additive generator t of a continuous Archimedean t-norm. Additive gen-

erators will prove to be very useful further in this work.

Theorem 1 [15] A continuous t-norm T is Archimedean if and only if there exists a

continuous and strictly decreasing function t : [0,1] → [0, ∞] with t(1) = 0 such that

T(x,y) = t

[−1]

(t(x) + t(y))

where t

[−1]

is the pseudo inverse of t deﬁned by

t

[−1]

(x) =

1 if x ≤ 0

t

−1

(x) if 0 ≤ x ≤ t(0)

0 if t(0) ≤ x.

The function t will be called an additive generator of the t-norm and two generators

of the t-norm T differ only by a positive multiplicative constant.

If T = Ł is the Łukasievicz t-norm, then an additive generator is t(x) = 1− x.

If T =

Π

is the Product t-norm, then t(x) = −log(x).

Deﬁnition 1 Let T be a t-norm.

• The residuation

−→

T of T is deﬁned for all x,y ∈ [0,1] by

−→

T (x|y) = sup{

α

∈ [0,1]|T(

α

,x) ≤ y}.

• The birresiduation

←→

T of T is deﬁned for all x,y ∈ [0,1] by

←→

T (x,y) = min{

−→

T (x|y),

−→

T (y|x)} = T(

−→

T (x|y),

−→

T (y|x)).

When the t-norm T is continuous Archimedean, these operations can be rewritten

in terms of the additive generator t.

Proposition 1 Let T be a continuous Archimedean t-norm generated by an additive

generator t. Then:

• T(x,y) = t

[−1]

(t(x) + t(y))

•

−→

T (x|y) = t

[−1]

(t(y) − t(x))

•

←→

T (x,y) = t

[−1]

(|t(x) − t(y)|).

4 Gabriel Mattioli and Jordi Recasens

Indistinguishability operators are the fuzziﬁcation of classical equivalence re-

lations and model the intuitive idea of ”similarity” between objects. For a more

detailed explanation on this operators readers are referred to [4], [21].

Deﬁnition 2 Let T be a t-norm. A fuzzy relation E on a set X is a T-indistinguish-

ability operator if and only if for all x,y,z ∈ X

a) E(x,x) = 1 (Reﬂexivity)

b) E(x,y) = E(y,x) (Symmetry)

c) T(E(x, y),E(y,z)) ≤ E(x,z) (T -transitivity).

Whereas indistinguishability operators represent the fuzziﬁcation of equivalence

relations, extensional fuzzy subsets play the role of fuzzy equivalence classes al-

together with their intersections and unions. Extensional fuzzy subsets are a key

concept in the comprehension of the universe of discourse X under the effect of

an indistinguishability operator E as they correspond with the observable sets or

granules of X.

Deﬁnition 3 Let X be a set and E a T-indistinguishability operator on X. A fuzzy

subset

µ

of X is called extensional with respect to E if and only if:

∀x,y ∈ X T(E(x,y),

µ

(y)) ≤

µ

(x).

We will denote H

E

the set of all extensional fuzzy subsets of X with respect to E.

Extensional fuzzy subsets have been widely studied in the literature [7], [11],

[12].

If the t-norm T is continuous Archimedean then the condition of extensionality

can be rewritten in terms of additive generators. This result will be recalled several

times along this paper.

Lemma 1 Let E be a T -indistinguishability operator on a set X.

µ

∈ H

E

if and only

if ∀x,y ∈ X:

t(E(x,y)) + t(

µ

(y)) ≥ t(

µ

(x)).

Proof.

µ

∈ H

E

⇔ T(E(x,y),

µ

(y)) ≤

µ

(x)

⇔ t

−1

(t(E(x,y)) + t(

µ

(y))) ≤

µ

(x).

And as t is a monotone decreasing function this is equivalent to

t(E(x,y)) + t(

µ

(y)) ≥ t(

µ

(x)).

3 Approximation using Means

In this section we will propose a method to approximate an arbitrary fuzzy subset

by an extensional one. First we will introduce two approximation operators,

φ

E

(

µ

)

Comparison of Different Algorithms of Approximation by Extensional Fuzzy Subsets 5

and

ψ

E

(

µ

), that provide the best upper and lower approximation respectively by

extensional fuzzy subsets of

µ

given an indistinguishability operator E. The method

will consist in computing an adequate weight in order to minimize an error function

between

µ

and the natural weighted mean of

φ

E

(

µ

) and

ψ

E

(

µ

).

Deﬁnition 4 Let X be a set and E a T-indistinguishability operator on X. The maps

φ

E

: [0,1]

X

→ [0,1]

X

and

ψ

E

: [0,1]

X

→ [0,1]

X

are deﬁned ∀x ∈ X by:

φ

E

(

µ

)(x) = sup

y∈X

T(E(x,y),

µ

(y)),

ψ

E

(

µ

)(x) = inf

y∈X

−→

T (E(x, y)|

µ

(y)).

φ

E

(

µ

) is the smallest extensional fuzzy subset greater than or equal to

µ

; hence

it is its best upper approximation by extensional fuzzy subsets. Analogously,

ψ

E

(

µ

)

provides the best approximation by extensional fuzzy subsets smaller than or equal

to

µ

. From a topological viewpoint these operators can be seen as closure and inte-

rior operators on the set [0,1]

X

[11]. It is remarkable that these operators also appear

in a natural way in ﬁelds such as fuzzy rough sets [20], fuzzy modal logic [6], [5],

fuzzy mathematical morphology [8] and fuzzy contexts [3] among many others.

Though

φ

E

(

µ

) and

ψ

E

(

µ

) provide extensional fuzzy subsets that approximate

µ

there is no guarantee in general that there are no better approximations of

µ

by

extensional fuzzy subsets. In [19] the authors faced this problem and provided three

methods to ﬁnd approximations for Archimedean t-norms and one for the Minimum

t-norm. The two methods compared in this paper were introduced there.

Deﬁnition 5 [1] Let t : [0,1] → [−∞,∞] be a non-increasing monotonic map, x,y ∈

[0,1] and r ∈ [0,1]. The weighted quasi-arithmetic mean m

t

of x and y is deﬁned as:

m

r

t

(x,y) = t

−1

(r·t(x) + (1− r) · t(y))

m

t

is continuous if and only if {−∞,∞} * Ran(t).

There is a bijection between the set of continuous Archimedean t-norms and

the set of quasi-arithmetic means by taking as map the additive generator t of the

t-norm [14]. Under this interpretation in the literature quasi-arithmetic means are

sometimes called natural means [17], as we will recall them from now on.

We want to approximate

µ

by m

r

t

(

φ

E

(

µ

),

ψ

E

(

µ

)). Below we prove that this mean

is extensional for any value of r.

Proposition 2 [19] Let X be a set and

µ

,

ν

extensional fuzzy subsets of X with

respect to an indistinguishability operator E on X. Then:

m

r

(

µ

,

ν

) ∈ H

E

.

Corollary 1 Let

µ

be a fuzzy subset on a set X and E an indistinguishability oper-

ator. Then:

m

r

(

φ

E

(

µ

),

ψ

E

(

µ

)) ∈ H

E

.

##### Citations

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17 Mar 2016Abstract: This thesis can be summarized in the following sentence: All scientific models, for instance structural atlases of brain MRI, can be read in terms of fuzzy extensional sets. The work done in this doctoral thesis backs up and explains how this interpretation can be performed and provides future lines of application of the results developed in the field of automatized analysis of brain MRI neuroimages. Does this make any sense? Why should a framework be developed in order to apply fuzzy techniques in brain analysis? The first part of this work asks itself precisely this question, the why and for what of the thesis proposed. Uncertainty, as a multidimensional phenomenon, is studied and tried to discriminate the different sources and natures of vagueness, imprecision, partial truth and knowledge? 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15 Jul 2014

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30 Jun 2015

TL;DR: This paper considers dierent types of monotonicity and shows that obtained results allow us to describe approximate system based on the constructed operators of upper and lower general aggregation operators.

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### Cites background or methods from "Comparison of Different Algorithms ..."

...Fuzzy sets φE(μ) and ψE(μ) were introduced to provide upper and lower approximation of a fuzzy set μ with respect to fuzzy equivalence relation E [11]....

[...]

...Extensional fuzzy subsets have been widely studied in the literature [2], [8], [9], [11]....

[...]

##### References

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•

03 Mar 1993TL;DR: The book is a solid reference for professionals as well as a useful text for students in the fields of operations research, management science, industrial engineering, applied mathematics, and also in engineering disciplines that deal with analytical optimization techniques.

Abstract: COMPREHENSIVE COVERAGE OF NONLINEAR PROGRAMMING THEORY AND ALGORITHMS, THOROUGHLY REVISED AND EXPANDED"Nonlinear Programming: Theory and Algorithms"--now in an extensively updated Third Edition--addresses the problem of optimizing an objective function in the presence of equality and inequality constraints. Many realistic problems cannot be adequately represented as a linear program owing to the nature of the nonlinearity of the objective function and/or the nonlinearity of any constraints. The "Third Edition" begins with a general introduction to nonlinear programming with illustrative examples and guidelines for model construction.Concentration on the three major parts of nonlinear programming is provided: Convex analysis with discussion of topological properties of convex sets, separation and support of convex sets, polyhedral sets, extreme points and extreme directions of polyhedral sets, and linear programmingOptimality conditions and duality with coverage of the nature, interpretation, and value of the classical Fritz John (FJ) and the Karush-Kuhn-Tucker (KKT) optimality conditions; the interrelationships between various proposed constraint qualifications; and Lagrangian duality and saddle point optimality conditionsAlgorithms and their convergence, with a presentation of algorithms for solving both unconstrained and constrained nonlinear programming problemsImportant features of the "Third Edition" include: New topics such as second interior point methods, nonconvex optimization, nondifferentiable optimization, and moreUpdated discussion and new applications in each chapterDetailed numerical examples and graphical illustrationsEssential coverage of modeling and formulating nonlinear programsSimple numerical problemsAdvanced theoretical exercisesThe book is a solid reference for professionals as well as a useful text for students in the fields of operations research, management science, industrial engineering, applied mathematics, and also in engineering disciplines that deal with analytical optimization techniques. The logical and self-contained format uniquely covers nonlinear programming techniques with a great depth of information and an abundance of valuable examples and illustrations that showcase the most current advances in nonlinear problems.

6,258 citations

•

01 Feb 20062,434 citations

### Additional excerpts

...Definition 5 [1] Let t : [0,1]→ [−∞,∞] be a non-increasing monotonic map, x ,y∈ [0,1] and r∈ [0,1]....

[...]

••

TL;DR: M Modes of information granulation (IG) in which the granules are crisp (c-granular) play important roles in a wide variety of methods, approaches and techniques, but this does not reflect the fact that in almost all of human reasoning and concept formation thegranules are fuzzy (f- Granular).

Abstract: There are three basic concepts that underlie human cognition: granulation, organization and causation. Informally, granulation involves decomposition of whole into parts; organization involves integration of parts into whole; and causation involves association of causes with effects. Granulation of an object A leads to a collection of granules of A, with a granule being a clump of points (objects) drawn together by indistinguishability, similarity, proximity or functionality. For example, the granules of a human head are the forehead, nose, cheeks, ears, eyes, etc. In general, granulation is hierarchical in nature. A familiar example is the granulation of time into years, months, days, hours, minutes, etc. Modes of information granulation (IG) in which the granules are crisp (c-granular) play important roles in a wide variety of methods, approaches and techniques. Crisp IG, however, does not reflect the fact that in almost all of human reasoning and concept formation the granules are fuzzy (f-granular). The granules of a human head, for example, are fuzzy in the sense that the boundaries between cheeks, nose, forehead, ears, etc. are not sharply defined. Furthermore, the attributes of fuzzy granules, e.g., length of nose, are fuzzy, as are their values: long, short, very long, etc. The fuzziness of granules, their attributes and their values is characteristic of ways in which humans granulate and manipulate information.

2,402 citations

••

TL;DR: An extended version of Szpilrajn's theorem is proved and various properties of similarity relations and fuzzy orderings are investigated and, as an illustration, a fuzzy preordering is investigated which is reflexive and antisymmetric.

Abstract: The notion of ''similarity'' as defined in this paper is essentially a generalization of the notion of equivalence. In the same vein, a fuzzy ordering is a generalization of the concept of ordering. For example, the relation x @? y (x is much larger than y) is a fuzzy linear ordering in the set of real numbers. More concretely, a similarity relation, S, is a fuzzy relation which is reflexive, symmetric, and transitive. Thus, let x, y be elements of a set X and @m"s(x,y) denote the grade of membership of the ordered pair (x,y) in S. Then S is a similarity relation in X if and only if, for all x, y, z in X, @m"s(x,x) = 1 (reflexivity), @m"s(x,y) = @m"s(y,x) (symmetry), and @m"s(x,z) >= @? (@m"s(x,y) A @m"s(y,z)) (transitivity), where @? and A denote max and min, respectively. ^y A fuzzy ordering is a fuzzy relation which is transitive. In particular, a fuzzy partial ordering, P, is a fuzzy ordering which is reflexive and antisymmetric, that is, (@m"P(x,y) > 0 and x y) @? @m"P(y,x) = 0. A fuzzy linear ordering is a fuzzy partial ordering in which x y @? @m"s(x,y) > 0 or @m"s(y,x) > 0. A fuzzy preordering is a fuzzy ordering which is reflexive. A fuzzy weak ordering is a fuzzy preordering in which x y @? @m"s(x,y) > 0 or @m"s(y,x) > 0. Various properties of similarity relations and fuzzy orderings are investigated and, as an illustration, an extended version of Szpilrajn's theorem is proved.

2,369 citations

### Additional excerpts

...Within the theory of Fuzzy Logic the first researche r to point the relevance of these sets was Zadeh when he discussed the concept of granu larity [24]....

[...]

••

2,099 citations