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Robustness on intuitionistic fuzzy connectives

Abstracts

The main contribution of this paper is concerned with the robustness of intuitionistic fuzzy connectives in fuzzy reasoning. Starting with an evaluation of the sensitivity in n-order functions on the class of intuitionistic fuzzy sets, we apply the results in the intuitionistic (S,N)-implication class. The paper formally states that the robustness preserves the projection functions in this class.

robustness; δ-sensitivity; perturbation; fuzzy logic; intuitionistic fuzzy logic; intuitionistic fuzzy connectives


A ánalise da robustez de conectivos fuzzy intuicionista consiste na principal contribuição deste trabalho. A partir da avaliação da sensibilidade de funções n-arias na classe dos conjuntos fuzzy intuicionistas, como proposto por Atanassov, os principais resultados são aplicados na correspondente extensão intuicionista das (S,N)-implicações fuzzy. O trabalho mostra que a robustez preserva as funções de projeções nesta classe de implicações fuzzy.

robustez; δ-sensibilidade; perturbação; lógica fuzzy; lógica fuzzy intuicionística; conectivos fuzzy intuicionistas


Robustness on intuitionistic fuzzy connectives* * This work is supported by FAPERGS (Ed. PqG 06/2010, 11/1520-1), and partial information of this work was published in the proceedings of XXXIV CNMAC. ** Corresponding author: Renata Hax Sander Reiser.

R.H.S. ReiserI, ** * This work is supported by FAPERGS (Ed. PqG 06/2010, 11/1520-1), and partial information of this work was published in the proceedings of XXXIV CNMAC. ** Corresponding author: Renata Hax Sander Reiser. ; B. BedregalII

ICentro de Desenvolvimento Tecnológico, CDTEC, UFPEL - Computação/Campus Porto, 354, 96001-970 Pelotas, RS, Brazil. E-mail: reiser@inf.ufpel.edu.br

IIDepartamento de Informática e Matemática Aplicada, DIMAP, UFRN, Campus Universitário, s/n, 59072-970 Natal, RN, Brazil. E-mail: bedregal@dimap.ufrn.br

ABSTRACT

The main contribution of this paper is concerned with the robustness of intuitionistic fuzzy connectives in fuzzy reasoning. Starting with an evaluation of the sensitivity in n-order functions on the class of intuitionistic fuzzy sets, we apply the results in the intuitionistic (S,N)-implication class. The paper formally states that the robustness preserves the projection functions in this class.

Keywords: robustness, δ-sensitivity, perturbation, fuzzy logic, intuitionistic fuzzy logic, intuitionistic fuzzy connectives.

RESUMO

A ánalise da robustez de conectivos fuzzy intuicionista consiste na principal contribuição deste trabalho. A partir da avaliação da sensibilidade de funções n-arias na classe dos conjuntos fuzzy intuicionistas, como proposto por Atanassov, os principais resultados são aplicados na correspondente extensão intuicionista das (S,N)-implicações fuzzy. O trabalho mostra que a robustez preserva as funções de projeções nesta classe de implicações fuzzy.

Palavras-chave: robustez, δ-sensibilidade, perturbação, lógica fuzzy, lógica fuzzy intuicionística, conectivos fuzzy intuicionistas.

1 INTRODUCTION

Robustness or sensitivity can be conceived as a fundamental property of a logical system stating that the conclusions are not essentially changed if the assumed conditions varied within reasonable parameters. It is a relevant research area with important contributions [30, 32, 23, 33, 18].

This paper considers the δ-sensitivity (or pointwise senstivity) study as presented in [19] but related to intuitionistic fuzzy connectives (IFCs), providing logical foundations to support robust fuzzy applications based on the Atanassov's Intuitionistic Fuzzy Logic [1,4] (IFL). Additionally, the δ-sensitivity of inference systems based on IFL can be analogously studied in the concepts of multi-valued fuzzy logic - the Interval-valued Atanassov's Fuzzy Logic [12] and integrated to challenging approach concerned with the Interval-valued Intuitionistic Fuzzy Logic [3].

Thus, in systems based on fuzzy rules, each linguistic term of an input linguistic variable is associated with a given fuzzy set [31]. Since the definition of these fuzzy sets is highly subjective, such fuzzy system should be stable in the sense that smooth changes performed on the input fuzzy sets should result only a slight change on their outputs. So, an immediate question follow as: how does one can ensure stability of such systems by applying Fuzzy Logic (FL) and the corresponding intuitionistic extension?

As a contribution to elucidate this question, this paper considers the robustness analysis defined on δ-sensitivity and related to Atanassov's intuitionistic fuzzy connectives. Such analysis can improve the stability study of systems based on Atanassov's intuitionistic fuzzy rules.

1.1 Relevance of the robustness analysis based on δ -sensitivity

In this paper we are analysing the δ-sensitivity of the steps of the fuzzy-rule inference engine, dealing with logical connectives as algebraic n-order function, meaning that desirable logical and intuitionistic properties are described by algebraic properties.

The main idea related to the δ-sensitivity study of an IFC was to provide an logical approach as foundation to applied situations where the designer does not have complete knowledge about of linguistic variables modelling the relationship of membership and non-membership functions involved in an application.

Such proposed logical approach is performed by a Δf(x,δ) operator related to n-order function f : UnU, taking into account an input x and a δ-parameter. Thus, it provides an interpretation of a function disturbance of fuzzy connective (FC), which is modelled by a δ-parameter and a function f taking an input x, respectively. Indeed, a study on the influence and sensitivity of such δ-parameter in a knowledge base leads to an improvement in the fuzzy system (faced on, e.g., variances related to noise or temporal devices).

Thus, as FCs (mainly negations, t-norms and t-conorms, implications and coimplications) are important elements in the fuzzy reasoning, the corresponding investigation of the δ-sensitivity related to the pointwise analysis on the arguments of such operators, in terms of [19] and [25], will be carried out in this work.

Firstly, this paper investigates how to measure the robustness of IFCs using the sensitivity of FCs and it also derives the best perturbation parameters of intuitionistic fuzzy reasoning. Based on these results, an extension of such approach to N-dual constructions of IFCs is performed.

1.2 Applications of the robustness analysis in Fuzzy Logic

In the research area of fuzzy control, one of the most important problems is the analysis of stability and robustness of fuzzy controllers [17].

Significant works have been developed in this research area of fuzzy control, whose main research problem is related to the analysis of stability and robustness of fuzzy controllers, e.g. [10,20] and [32]. In [30], the concepts of maximum and average robustness of fuzzy sets werealready proposed.

Sensitivity analysis has become a major tool in the assessment of the reliability of engineering structures. Given an input-output system, the question is which input variables have the most decisive influence on the output on such systems. In [23], methods of modelling correlations and interactivity in such systems are investigated. In [18] the fundamental property of robustness of interval-valued fuzzy inference is studied. Moreover, robust interval matrices over (max, min)-algebra (fuzzy matrices) are studied and equivalent conditions for interval fuzzy matrices to be robust are presented in [22].

The notion of δ-equalities of fuzzy sets is used in [10] to study robustness of fuzzy reasoning based on fuzzy implication operators, generalized modus ponens and generalized modus tollens. Other relations among the robustness of fuzzy reasoning, fuzzy conjunctions and classes of implication operators were presented in [17].

More recently, the robustness of fuzzy reasoning from the perspective of perturbation of membership functions is considered in [21] also including a method for judging the most robust elements of different classes of fuzzy connectives. Additionally, a new method for sensitivity analysis of fuzzy transportation problems is proposed in [8].

1.3 Main contribution and paper outline

This paper considers the notion of δ-sensitivity of fuzzy connectives in the Atanassov's intuitionistic fuzzy approach [2], which is characterized by the non-complementary relationship between the membership and non-membership functions.

Since δ-sensitivity on interpretation of IFCs is closed related to truth and non-truth in conditional fuzzy rules, this work is focused not only in the representable Atanassov's intuitionistic fuzzy t-norms and implications but also in their corresponding dual fuzzy connectives which are representable intuitionistic fuzzy t-conorms and coimplications.

As the main result, the δ-sensitivity of the (S,N)-intuitionistic fuzzy implication class is introduced, based on the study of δ-sensitivity of both classes, the intuitionistic fuzzy negations and t-conorms. Moreover, the paper is extending the work in [26] to the dual Atanassov's intuitionistic approach.

The preliminaries describes the basic concepts of FCs and IFCs. The δ-sensitivity of FCs and general results of robustness of IFCs are stated in Sections 3 and 4, respectively. Final remarks are reported in the conclusion.

2 PRELIMINARIES

We start by recalling some basic concepts of FCs and IFCs we are going to use in our subsequent developments.

2.1 Fuzzy connectives

Firstly, notions concerning t-(co)norms, (co)implications and dual functions are reported based on [15] and [16].

2.1.1 Fuzzy negations

Let U = [0,1] be the unit interval of real numbers. Recall that a function N : UU is a fuzzy negation if it satisfies the properties:

N1:N(0) = 1 and N(1) = 0; N2: If x>y then N(x) < N(y), ∀x,yU.

A fuzzy negation satisfying the involutive property:

N3:N(N(x)) = x, ∀xU,

is called a strong fuzzy negation (SFN), e.g. the standard negation NS(x) = 1 - x. When x = (x1,x2, ...,xn) ∈ Un and N is a fuzzy negation, the following notation is considered:

Let N be a negation. The N-dual function of f : UnU is given by:

Notice that, when N is involutive, (fN)N = f, that is the N-dual of fN is the function f. In addition, a function f for which f = fN is called self-dual function.

2.1.2 Triangular norms and conorms

A function T : U2U is a triangular-norm (t-norm) if and only if it satisfies, for all xU, the following properties.

T1: T(x,1) = x;

T2: T(x,y) = T(y,x);

T3: T(x,T(y,z)) = T(T(x,y),z);

T4: if x<x', T(x,y) <T(x',y).

The notion of a triangular conorm (t-conorm) S : U2 U can be defined in the same manner with the exception that the identity T1 should be replaced by S1: S(0,x) = x, for all xU.

Let N be a fuzzy negation on U. The mappings TN,SN : U2U denoting the N-dual functions of a t-norm T and a t-conorm S, respectively, are defined as:

2.1.3 Fuzzy implications and coimplications

An implicator operator I : U2U extends the classical implication function:

I0: I(1,1) = I(0,1) = I(0,0) = 1, I(1,0) = 0.

Definition 1. [16] When x,y,z U2, a fuzzy implication (J) I : U2 U is an implicator verifying the properties from I1 to I4 described in the following:

I1: I(x,y)> I(z,y) if x < z (first place antitonicity);

I2: I(x,y)< I(x,z) if y < z (second place isotonicity);

I3: I(0,y) = 1 (dominance of falsity);

I4: I(x,1) = 1 (boundary condition);

Analogously, the notion of a coimplicator J : U2 U can be defined as an extension of the classical coimplication function. Thus, such operators satisfy the corresponding boundary conditions:

J0: J(0,0) = J(1,0) = J(1,1) = 0, J(0,1) = 1.

It is immediate that a fuzzy coimplication is an coimplicator analogously defined as a fuzzy implication, replacing I3 and I4 in Definition 1 by

J3: J(x,0) = 0 and

J4: J(1,y) = 0, respectively.

There exist many classes of fuzzy (co)implication functions (see, e.g., [15] and [9]). In thispaper we consider the class of (S,N)-implications defined in [16] as follow:

such that S is a t-conorm and N is a fuzzy negation. If N is a SFN, then IS,N is called a strong implication or an S-implication.

Additionally, when SN is the N-dual function of the t-conorm S, the corresponding N-dual functions are (S,N)-coimplications given by

The dual construction (T,N)-coimplication can be analogously defined.

2.2 Intuitionistic Fuzzy Connectives

These preliminaries consider the IFC concepts in IFL, by applying the strategy to deal with logical connectives as algebraic mappings, meaning that desirable logical and intuitionistic properties are described in terms of algebraic properties of connectives (negation, conjunction, disjunction, implication and coimplication).

According to [1], an Atanassov's fuzzy intuitionistic fuzzy set (IFS) AI in a non-empty, universe χ, is expressed as

AI = {(x, µA(x),νA(x)) : x ∈ χ, µA(x) + νA(x)) < 1}.

Thus, an intuitionistic fuzzy truth value of an element x in an IFS AI is related to the ordered pair (µA(x),νA(x)). Moreover, an IFS AI generalizes a fuzzy set A = {(x, µA(x)) | x ∈ χ}, since νA(x), which means that the non-membership degree of an element x is less than or equal to the complement of its membership degree µA(x), and therefore νA(x) is not necessarily equal to 1 - µA(x).

Additionally, a function πA : χ → U, called an Atanassov's intuitionistic fuzzy index (IFIx) of an element x in an IFS A, is given as

Let = {(x1,x2) ∈ U2 : x1< NS(x2)} be the set of all intuitionistic fuzzy values and : U be the projection functions on , which are given by () = (x1,x2) = x1 and () = (x1,x2) = x2, respectively.

Thus, for all = (1, ...,n) ∈ n, such that i = (xi1,xi2) and xi1< NS(xi2) when 1 < i < n, consider : nUn as the projections given by:

Consider also the order relation

x1< y1 and x2> y2 such that = (0,1) and = (1,0) , for all , [2].

2.2.1 Intuitionistic fuzzy negations

An Atanassov's intuitionistic fuzzy negation (IFN shortly) NI : satisfies, for all ,, the following properties:

NI1: NI() = NI(0,1) = and NI() = NI(1,0) = ;

NI2: If > then NI() < NI().

In addition, NI is a strong Atanassov's intuitionistic fuzzy negation (SIFN) if it also verifies the involutive property:

NI3: NI(NI()) = , ∀.

Consider NI as IFN in and : n. For all = (1, ..., n) ∈ n, the NI-dual intuitionistic function of , denoted by NI : n, is given by:

In addition, when

I is a SIFN, is a self-dual intuitionistic function.

By [5, Theorem 1] [11, 12], a function NI : is a strong intuitionistic fuzzy negation (SIFN) if and only if there exists a (SFN) N : UU expressed as:

Additionally, if N = NS, Eq. (2.8) can be reduced to

2.2.2 Intuitionistic fuzzy t-(co)norms

A function (SI) TI : 2 is an Atanassov's intuitionistic fuzzy triangular (co)norm (t-(co)norm shortly), if it is a commutative, associative and increasing function with neutral element () .

Consider now the t-representability concept proposed in [12, Def. 5], see also some results of [5, Def. 3]. An intuitionistic t-conorm SI : 2 and t-norm TI : 2 is t-representable when both conditions are held:

(i) there exist t-norms T', T : U2U and t-conorms S', S : U2U such that, for all x,yU, the respective equations

are verified; and

(ii) for all = (x1,x2), = (y1,y2) ∈ , each one of such intuitionistic fuzzy connectivesis given by the corresponding expressions

2.2.3 Intuitionistic fuzzy implications

A binary function II : 2 satisfying the conditions:

II0: II(,) = II(,) = II(,) = and II(,) = ;

is called an Atanassov's intuitionistic fuzzy implicator.

According with [9, Definition 3], an Atanassov's intuitionistic fuzzy implication II : 2 is an Atanassov's intuitionistic fuzzy implicator such that, the analogous conditions from II1 to II4 reported in Definition 1 are verified together with the additional property:

II5: If = (x1,x2) such that = (y1,y2) ∈ , x1 + x2 = 1 and y1 + y2 = 1, it holds that = 0, with π : χ → U is the IFIx given by Eq. (2.6).

Thus, recovering Definition 1 of a fuzzy implication in the sense of J. Fodor and M. Roubens' work [16], an Atanassov's intuitionistic fuzzy implication also reproduces fuzzy (co)implications if, for all = (x1,x2), = (y1,y2) ∈ we have x1 = NS(x2) and y1 = NS(y2). According to [2] and [12], another way of defining an operator II is to consider boundary conditions in II0 and properties II1 and II2.

Based on [5, Theorem 4] and [11], a function II : 2 is a representable Atanassov's intuitionistic (S,N)-implication based on a strong negation NI : if and only if there exist (S,N)-implications Ia,Ib : U2U, such that for all = (x1,x2), = (y1,y2) ∈ U, II is expressed as:

Dually, in the same manner, an Atanassov's intuitionistic fuzzy coimplication JI can be defined. Moreover, a function JI : 2 is a representable intuitionistic (T,N)-coimplication based on a strong negation NI iff there exist (T,N)-coimplications Ja,Jb : U2U, such that for all = (x1,x2), = (y1,y2) ∈ U, JI is expressed as:

When N = NS and Ja = IaN and Jb = IbN, it holds that JI = IINI is a self NI-dual intuitionistic fuzzy operator.

3 POINTWISE SENSITIVITY OF FUZZY CONNECTIVES

The selection of values using e.g. Likert's scale is an important way to elicit degrees of uncertainty related to based-rule fuzzy systems, frequently expressed by composition performed on fuzzy connectives. Thus, it is desirable that the result of the fuzzy logical operation does not change much if slight changes (or small deviations) are performed in the inputs. This sensitive study leads to the least sensitive or the most robust fuzzy inference rule.

Based on [19] and [25], the study of a δ-sensitivity of n-order function f at point x on the domain U is considered in the following, in the context of robustness of fuzzy logic, mainly related to the class of (S,N)-implications.

Definition 2. [19, Def. 1] Let f : Un U be an n-order function, δ ∈ U and x = (x1,x2, ...xn), y = (y1,y2, ...yn) ∈ Un. The δ-sensitivity of f at point x, denoted by Δf(x,δ), is defined by

wherever ∨(x,y) = max{ |xi - yi| : i = 1,...,n}.

Now, we investigate the δ-sensitivity in FCs, in terms of Definition 2. For that, the binary minimal and maximal fuzzy aggregations were considered: ∧, ∨: U2U such that min(a,b) = ab and max(a,b) = ab, respectively.

The δ-sensitivity of binary functions at point xU2 are based on the monotonicity property analysis performed in their both arguments. In this paper, such analysis considers the t-(co)norms and fuzzy (co)implications. Thus, three more intuitive results related to the δ-sensitivity of these fuzzy connectives, previously presented in [19], are reported in the sequence, which are also followed by a brief discussion including some exemplification.

Proposition 1. [19, Theorem 2] Let f : U U be a reverse order function, i.e., x < y f(x)> f(y), for all x,y U. The δ-sensitivity of f at point x is given by

for δ ∈ U. In particular, Eq. (3.2) holds for a fuzzy negation function.

Henceforth, in order to provide an easier notation, when f : U2U and x = (x,y) ∈ U2, consider the following notations:

Proposition 2. [19, Theorem 1] Consider f : U2 U, δ ∈ U and x = (x,y) ∈ U2.

(i) If f is a monotone function, i.e, x < x', y < y' f(x,y)< f(x',y') for all x,y U, then it follows that

(ii) If f verifies both properties, 1-place antitonicity and 2-place isotonicity, then:

Based on [19], Proposition 3 formalizes a consequence of Proposition 2:

Proposition 3. [19, Corollary 1] Let T, S and IS,N be a t-norm, t-conorm and an (S,N)-implication. When x U2 and δ ∈ U, the next statements are true:

(i) the δ-sensitivity of a t-norm T and a t-conorm S at point x, respectively, are both defined by Eq. (3.3);

(ii) the δ-sensitivity of an (S,N)-implication IS,N at point x, is defined by Eq. (3.4).

Remark 1. Let δ ∈ U. Based on Eq. (3.3), we have the next δ-sensitivity analysis:

(i) when x = (0,1), the following is true:

(ii) when x = (1,0), by the commutativity of a t-(co)norm, we have the same results:

(iii) when x = (0,0) then the related expressions of a t-(co)norm are given as:

Additionally, based on Eq. (3.4) and taking x = (0,1) and x = (1,0), we obtain the corresponding equalities:

3.1 Pointwise sensitivity of N -dual fuzzy connectives

In the preceding section and according with the duality principle stated in Eq. (2.2), we described the definitions as foundations to study the pointwise sensitivity of N-dual FCs as follows.

Proposition 4.[25, Proposition 6] Let f : U2U be a second-order function and N be the standard fuzzy negation. For all x = (x,y) ∈ Un, the following equalities hold:

Taken at a strong fuzzy negation N, Proposition 5 states that the sensitivity of a n-order function f at a point x is equal to the sensitivity of its dual function fN taking the complement of x.

Proposition 5. [25, Theorem 1] Consider f : U2 U, δ ∈ U and x = (x,y) ∈ U2. Let Δf(x,δ) be the sensitivity of f at point x. If N is the standard fuzzy negation (N = NS in Eq. (2.1)) and fN is the N-dual function of f then the sensitivity of fN at point x is given by

Proposition 6. [25, Proposition 7] Let N be the standard fuzzy negation, fN be N-dual function related to f : U2 U, δ ∈ U and x = (x,y) ∈ U2. The sensitivity of fN at point x is given by the following cases:

(i) if f is increasing w.r.t. its variables then we have that:

(ii) if f is decreasing w.r.t. its first variable and increasing with its second variable then we have:

Proposition 7. [25, Proposition 8] Let (T)N, (S)N, (IS,N)N be N-dual functions related to a t-norm T, a t-conorm S and an implication IS,N, respectively. If x U2 and δ ∈ U, the statements as follow hold:

(i) Δ(T)N(x,δ) and Δ(S)N(x,δ) are both defined by Eq. (3.11);

(ii) ΔI(S,N)N(x,δ) is defined by Eq. (3.12).

Remark 2. Consider δ ∈ U and the pair (I(S,N),I(S,N)N) of mutual N-dual functions. By Eqs. (3.8) and (3.9) stated in Remark 1, it follows the expressions:

which are examples of Eq. (3.12) when x = (1,0) and x = (0,1), respectively.

4 ROBUSTNESS OF INTUITIONISTIC FUZZY CONNECTIVES

The sensitivity of fuzzy connectives contributes to measure the robustness of fuzzy reasoning directly linked to the selection of implication operators. When a fuzzy connective is modelled by a continuous function fI, one can consider the modulus of continuity of fI, by using a modulus of continuity of fI to any n-order intuitionistic fuzzy connective.

In order to provide a formal definition of robustness which can be applied to n-order Atanassov's intuitionistic fuzzy operators (as averages and medians aggregation functions) we introduce the definition of the δ-sensitivity of f : n at point = (1,2, ...,n) ∈ n.

Thus, δ-sensitivity of an intuitionistic operator f at point n is defined in terms of its left-projection and right-projection , which are related to the δ-sensitive of the membership and non-membership degrees of an element x ∈ χ associated with the IFS f(n).

Definition 3. Let fI : nbe an n-order function, δ = (δ1, δ2) ∈ U2and = (1,2, ...,n) ∈ n. The δ-sensitivity of fI at point , denoted by ΔfI(,δ), is defined by

wherever ∨ (x, y) = max{|xi − yi| : i = 1, . . . , n}.

The next proposition states that pointwise sensitivity is preserved by the projection functions applied to intuitionistic fuzzy negation that is t-representable in the same sense of [5,12] and [11].

Proposition 8. Let NI : be a representable intuitionistic negation as defined by Eq. (2.8). When δ = (δ12) ∈ U2 and 2, the δ-sensitivity of NI at point

, is defined by

Proof. Straightfoward from Definition 3 and Proposition 1. □

In the following, the study of a δ-sensitivity of an Atanassov's intuitionistic t-(co)norm and an intuitionistic fuzzy implication at point = (x1, x2) = ((x11,x12),(x11,x12)) on the domain 2 extends the work introduced in [19] related to the class of such binary intuitionistic fuzzy connectives which are representable by their corresponding fuzzy connectives [5,11] and [12].

Proposition 9. Let (SI) TI :

2 be a representable Atanassov's intuitionistic fuzzy t-(co)norm as defined by (Eq. (2.11b)) Eq. (2.11a). When δ = (δ12) ∈ U2 and 2, the δ sensitivity of TI at point
, denoted by ΔTI(,δ), is given by

Analogously, the δ sensitivity of SI at point is given by

Proof. For all , 2 given as = (1,2), such that 1 = (x11,x12), x11< NS(x12) and 2 = (x21,x22), x21< NS(x22); = (1, 2), such that 1 = (y11,y12), y11< NS(y12) and 2 = (y21,y22), y21< NS(y22). It holds that:

Therefore, and . Analogously, it can be proved that for δ-sensitivity of SI at point which means,

Remark 3. According with Eqs. (4.4) and (4.3) in Proposition 10, we obtain the expressions of the δ-sensitivity of an intuitionistic fuzzy t-(co)norm as follows:

(i) when = ((0,0),(0,0)) and = (δ12), it holds that

(ii) when = (,) or = (,) and = (δ12), .

Now, we study the robustness of an (S,N)-implication II at point 2.

Proposition 10. Let (JI) II :

2 be a t-representable Atanassov's intuitionistic ((T,N)-implication) (S,N)-implication as defined by (Eq. (2.13)) Eq. (2.12). When = (δ12) ∈ U2 and 2, the δ-sensitivity of II at point is defined by

Analogously, the δ-sensitivity of JI at point is defined by

Proof. Let II be a representable (S,N)-implication obtained by the standard fuzzy negation NS and a fuzzy (S,N)-implications Ia,Ib, as defined by Eq. (2.12), then:

Therefore, for all ,2, it follows that

In analogous manner, Eq. (4.6) can be proved.

Remark 4. Based on results in Remark 2, we are able to analyse the δ-sensitivity of an Atanassov's intuitionistic fuzzy ((T,N)-coimplication) (S,N)-implication (JI) II as follows:

(i) when = (,) and = (δ12) then

(ii) when = (,) and = (δ12) then

4.1 Robustness of N-dual intuitionistic fuzzy connectives

Now, consider δ = (δ1, δ2) ∈ U2, = (1,2), = (1,2) ∈ n in the following.

Proposition 11. Let fI : n be an n-order function and ΔfI(,δ) be the sensitivity of fI at point . When NI = NIS(as stated in Eq. (2.9)) and fINI the NI-dual function of fI, the δ-sensitivity of at point is given by

Proof. Straightforward from Definition 3 and duality principle. □

Proposition 12. Let (SI) TI : n be a representable Atanassov's intuitionistic fuzzy t-(co)norm given as Eqs. (2.11a) and (2.11b), and S(x,δ)) ΔT(x,δ) be the δ-sensitivity of a t-(co)norm (S) T at point x. When N = NS and (SN) TN is the N-dual function of (S) T, the δ-sensitivity of at point is given by

Analogously, the δ-sensitivity of at point is given by

Proof. Straightforward from Proposition 11.

Remark 5. Based on results in Propositions 11 and 12 , we are able to analyse the δ-sensitivity of an intuitionistic fuzzy t-(co)norm (SI) TI as follows:

(i) when = (,), by Eqs. (3.5) and (3.6) we have that:

(ii) when = ((0,0),(0,0)) we have that:

Proposition 13. Let (JI) II : nbe a representable Atanassov's intuitionistic fuzzy (co)implication and J(x,δ)) ΔI(x,δ) be the δ-sensitivity of a (co)implication (J) I at point x. According with Proposition 11, when N is the standard fuzzy negation (N = NS in Eq. (2.1)) and (JN) IN is the N-dual function of (J) I, the δ-sensitivity of at point is given by

Analogously, the δ-sensitivity of at point is given by

Proof. Straightforward from Proposition 12. □

Remark 6. Based on results in Propositions 11 and 12, we are able to analyse the δ-sensitivity of an Atanassov's intuitionistic fuzzy t-(co)norm (SI) TI as follows:

(i) when = (,), according with Remark 4(i) it follows the expressions:

(ii) when = (,) then by Remark 4(ii) it follows other dual expressions:

5 CONCLUSION

The main contribution of this paper is concerned with the study of robustness on Atanassov's intuitionistic fuzzy operators mainly used in fuzzy reasoning based on IFL. Taking the class of strong fuzzy negation (standard negation), the paper formally states that the sensitivity of an n-order Atanassov's intuitionistic fuzzy connective at a point xUn preserves its projections related to the sensitivity of its fuzzy approach at the same point. The work of estimating its sensitivity to small changes is related to reducing sensitivity in the corresponding fuzzy connectives.

Our current investigation clearly aims to contemplate two approaches: (i) the sensitivity of fuzzy inference dependent on intuitionistic fuzzy rules based on intuitionistic fuzzy connectives; and (ii) the extension of the robustness studies to other main classes of (co)implications: R-(co)implications and QL-(co)implications.

Received on September 27, 2012

Accepted on May 15, 2014

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  • *
    This work is supported by FAPERGS (Ed. PqG 06/2010, 11/1520-1), and partial information of this work was published in the proceedings of XXXIV CNMAC.
    **
    Corresponding author: Renata Hax Sander Reiser.
  • Publication Dates

    • Publication in this collection
      10 Sept 2014
    • Date of issue
      Aug 2014

    History

    • Received
      27 Sept 2012
    • Accepted
      15 May 2014
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