systemic approach to fuzzy logic formalization for approximate reasoning

Information Sciences 181 (2011) 1045–1059

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Information Sciences

journal homepage: www.elsevier .com/locate / ins

Systemic approach to fuzzy logic formalization for approximate reasoning

Rafik Aliev a,⇑, Alex Tserkovny b

a Department of Control Systems, Intelligent Systems Research Laboratory, Azerbaijan State Oil Academy, Baku, Azerbaijanb Dassault Systems, Boston, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 7 August 2009Received in revised form 16 November 2010Accepted 19 November 2010

Keywords:Fuzzy logicImplicationAntecedentConsequentModus-ponensFuzzy conditional inference ruleStabilityContinuity

0020-0255/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.ins.2010.11.021

⇑ Corresponding author. Tel.: +994 125984509.E-mail addresses: [email protected] (R. Aliev), a

L.A. Zadeh, E.H. Mamdani, M. Mizumoto, et al., R.A. Aliev and A. Tserkovny have proposedmethods for fuzzy reasoning in which antecedents and consequents involve fuzzy condi-tional propositions of the form ‘‘If x is A then y is B’’, with A and B being fuzzy concepts(fuzzy sets). A formulation of fuzzy antecedent/consequent chains is one of the most impor-tant topics within a wide spectrum of problems in fuzzy sets in general and approximatereasoning, in particular. From the analysis of relevant research it becomes clear that for thispurpose, a so-called fuzzy conditional inference rules comes as a viable alternative. In thisstudy, we present a systemic approach toward fuzzy logic formalization for approximatereasoning. For this reason, we put together some comparative analysis of fuzzy reasoningmethods in which antecedents contain a conditional proposition with fuzzy concepts andwhich are based on implication operators present in various types of fuzzy logic. We alsoshow a process of a formation of the fuzzy logic regarded as an algebraic system closedunder all its operations. We examine statistical characteristics of the proposed fuzzy logic.As the matter of practical interest, we construct a set of fuzzy conditional inference rules onthe basis of the proposed fuzzy logic. Continuity and stability features of the formalizedrules are investigated.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

In our daily life we often make inferences where antecedents and consequents are represented by fuzzy sets. Such infer-ences cannot be realized adequately by the methods, which are based either on two-valued logic or many-valued logic. Inorder to facilitate such an inference, Zadeh [76–81] suggested an inference rule called a ‘‘compositional rule of inference’’.Using this inference rule, Zadeh, Mamdani [48], Mizumoto et al. [26,54,55], Aliev and Tserkovny [1–3] suggested severalmethods for fuzzy reasoning in which the antecedent contain a conditional proposition involving fuzzy concepts:

Ant 1 : If x is P then y is Q

Ant 2 : x is P0

--------------------------Cons : y is Q 0:

ð1Þ

Those methods are based on implication operators present in various fuzzy logics. This matter has been under a thoroughdiscussion for the last couple decades. Some comparative analysis of such methods was presented in [6–11,26,29,34,36–38,50,51,54,55,67,74,75,85]. A number of authors proposed to use a certain suite of fuzzy implications to form fuzzy conditional

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[email protected] (A. Tserkovny).

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1046 R. Aliev, A. Tserkovny / Information Sciences 181 (2011) 1045–1059

inference rules [2–4,25–27,40,42,43,48,54,55]. The implication operators present in the theory of fuzzy sets were investigatedin [2–5,12–23,28,30–33,39,41,44,46,47,49,51–53,56–59,61–66,68,70–73,75,82–84]. On the other hand, statistical features offuzzy implication operators were studied in [60,69]. In turn, the properties of stability and continuity of fuzzy conditional infer-ence rules were investigated in [24,27,35,40].

However, a systemic approach toward fuzzy logic formalization for an approximate reasoning task has never been pre-sented. The intention of this study is to fill this gap. We will begin with a formation of a fuzzy logic regarded as an algebraicsystem closed under all its operations. In the sequel An investigation of statistical characteristics of the proposed fuzzy logicwill be presented. Special attention will be paid to building a set of fuzzy conditional inference rules on the basis of the fuzzylogic proposed in this study. Next, continuity and stability features of the formalized rules will be investigated.

In what follows, we introduce some required notation. x and y are the names of objects, while P, P0, Q and Q0 are fuzzyconcepts represented by fuzzy sets defined in the universes of discourse U, U, V and V, respectively. This form of inferencemay be viewed as a generalized modus ponens, which reduces to modus ponens when P0 = P and Q0 = Q. Let P and Q be fuzzysets in U and V, respectively, while the corresponding fuzzy sets are represented as P � UjlP: U ? [0,1], Q � VjlQ: V ? [0,1],where

P ¼Z

UlPðuÞ=u; Q ¼

ZVlQ ðvÞ=v : ð2Þ

Let �, [, \, : and � denoted a Cartesian product, union, intersection, complement and bounded-sum for fuzzy sets, respec-tively. Then the following fuzzy relations in U � V can be derived from the fuzzy conditional proposition ‘‘If x is P then y is Q’’in Ant 1 of (1). The fuzzy relations Rm and Ra were proposed by Zadeh [45,47], Rc by Mamdani [48], Rs, Rg are by Mizumoto[26,54,55], while, RL2, RL3 are by Aliev and Tserkovny [1–3].

Rm ¼ ðP � QÞ [ ð:P � VÞ ¼Z

U�VðlPðuÞ ^ lQ ðvÞÞ _ ð1� lPðuÞÞ=ðu;vÞ; ð3Þ

Ra ¼ ð:P � VÞ � ðU � QÞ ¼Z

U�V1 ^ ð1� lPðuÞ þ lQ ðvÞÞ=ðu;vÞ; ð4Þ

Rc ¼ ðP � QÞ ¼Z

U�VðlPðuÞ ^ lQ ðvÞÞ=ðu;vÞ; ð5Þ

Rs ¼ ðP � V)s

U � QÞ ¼Z

U�VðlPðuÞ!s lQ ðvÞÞ=ðu; vÞ; ð6Þ

where

lPðuÞ!s lQ ðvÞ ¼1; lPðuÞ 6 lQ ðvÞ;0; lPðuÞ > lQ ðvÞ:

(

Rg ¼ ðP � V)g

U � QÞ ¼Z

U�VðlPðuÞ!g lQ ðvÞÞ=ðu;vÞ; ð7Þ

and

lPðuÞ!g lQ ðvÞ ¼1; lPðuÞ 6 lQ ðvÞ;lQ ðvÞ; lPðuÞ > lQ ðvÞ:

(

Rb ¼ ð:P � V [ U � QÞ ¼Z

U�Vð1� lPðuÞÞ _ lQ ðvÞ=ðu;vÞ; ð8Þ

RD ¼ ðP � V)D

U � QÞ ¼Z

U�V½lPðuÞ!D lQ ðvÞ�=ðu; vÞ; ð9Þ

where

lPðuÞ!D lQ ðvÞ ¼1; lPðuÞ 6 lQ ðvÞ;lQ ðvÞlP ðuÞ

; lPðuÞ > lQ ðvÞ:

(

R� ¼ ðP � V)�

U � QÞ ¼Z

U�V½lPðuÞ!� lQ ðvÞ�=ðu; vÞ; ð10Þ

where

lPðuÞ!� lQ ðvÞ ¼ ðlPðuÞ ^ lQ ðvÞÞ _ ð1� lPðuÞ ^ 1� lQ ðvÞÞ _ ðlQ ðvÞ ^ 1� lPðuÞÞ:

R} ¼ ðP � V)}

U � QÞ ¼Z

U�V½lPðuÞ!} lQ ðvÞ�=ðu; vÞ; ð11Þ

R. Aliev, A. Tserkovny / Information Sciences 181 (2011) 1045–1059 1047

where

lPðuÞ!} lQ ðvÞ ¼1; lPðuÞ < 1 or lQ ðvÞ ¼ 1;0; lPðuÞ ¼ 1 or lQ ðvÞ < 1:

(

RL1 ¼ ðP � V)L1

U � QÞ ¼Z

U�V½lPðuÞ!L1

lQ ðvÞ�=ðu; vÞ; ð12Þ

where

lPðuÞ!L1lQ ðvÞ ¼

1� lPðuÞ; lPðuÞ < lQ ðvÞ;1; lPðuÞ ¼ lQ ðvÞ;lQ ðvÞ; lPðuÞ > lQ ðvÞ:

8><>:

RL2 ¼ ðP � V)L2

U � QÞ ¼Z

U�V½lPðuÞ!L2


where


1; lPðuÞ 6 lQ ðvÞ;ð1� lPðuÞÞ ^ lQ ðvÞ; lPðuÞ > lQ ðvÞ:

(

RL3 ¼ ðP � V)L3

U � QÞ ¼Z

U�V½lPðuÞ!L3


where


1; lPðuÞ 6 lQ ðvÞ;lQ ðvÞ

lP ðuÞþð1�lQ ðvÞÞ; lPðuÞ > lQ ðvÞ:

(

A necessary observation to be made in the context of this discussion is that with the only few exceptions for S-logic (6)and G-logic (7), and L1–L3 (12)–(14), all other known fuzzy logics (3)–(5) and (8)–(11) do not satisfy either the classical‘‘modus-ponens’’ principle, or other criteria which appeal to the human perception of mechanisms of a decision making pro-cess being formulated in [54]. The proposed fuzzy logic comes with an implication operator, which satisfies the classicalprinciple of ‘‘modus-ponens’’ and meets some additional criteria being in line with human intuition.

The material of the study is organized as follows. In Section 2, we offer some required prerequisites. Section 3 is dedicatedto a formalization of the proposed fuzzy logic. Next, Section 4 provides a certain introduction to a power sets based featuresof the fuzzy logic. The statistical analysis of proposed fuzzy logic is reported in Section 5. Section 6 covers the formalizationof fuzzy conditional inference rules where the rules exploit the proposed fuzzy logic. Section 7 is devoted to investigations ofstability and continuity of proposed fuzzy conditional inference rules.

2. Preliminary considerations

Consider a continuous function F(p,q) = p � q which defines a distance between p and q where p, q assume values in theunit interval. Notice that Fðp; qÞ 2 ½�1;1�, where F(p,q)min = �1 and F(p,q)max = 1. The normalized version of F(p,q) is definedas follows:

Fðp; qÞnorm ¼ Fðp; qÞ � Fðp; qÞmin

Fðp; qÞmax � Fðp; qÞmin ¼Fðp; qÞ þ 1

2¼ p� qþ 1

2: ð15Þ

It is clear thatF(p,q)norm 2 [0,1]. This function quantifies a concept of ‘‘closeness’’ between two values (potentially the onesfor the truth values of antecedent and consequent), defined within unit interval, which therefore could play significant role inthe formulation of the implication operator in a fuzzy logic.

Definition 1. An implication is a continuous function I from [0,1] � [0,1] into [0,1] such that for "p, p0, q, q0, r 2 [0,1] thefollowing properties are satisfied

(I1) If p 6 p0 then I(p,q) P I(p0,q) (antitone in first argument),(I2) If q 6 q0 then I(p,q) 6 I(p,q0) (monotone in second argument),(I3) I(0,q) = 1 (falsity),(I4) I(1,q) 6 q (neutrality),(I5) I(p, I(q,r)) = I(q, I(p,r)) (exchange),(I6) I(p,q) = I(n(q),n(p)) (contra positive symmetry), where n(�) – is a negation, which could be defined as n(q) = T(:Q) =

1 � T(Q).


Definition 2 [7]. Given a fuzzy implication operator ? and a fuzzy subset Q of universe U, the fuzzy power set PQ of Q isgiven by the membership function lPQ, where

lPQ P ¼\x2U

ðlPx! lQ xÞ: ð16Þ

The degree to which P is subset of Q is defined as

pðP # QÞ ¼ lPQ P:

Definition 3 [7]. From Definition 2, the degree to which the fuzzy sets P and Q is the same, or their degree of similarity, is

pðP QÞ ¼ pðP # QÞ ^ pðP QÞ: ð17Þ

The following proposition is immediate.

Proposition 1 [7].

pðP QÞ ¼\x2U

ðlPx$ lQ xÞ: ð18Þ

Definition 4 [7]. The degree of disjointness of P and Q, or degree to which P and Q are disjointed, in the first and secondsense, is defined as follows

ð1Þ pðPdisj1QÞ ¼ pðP # Q CÞ ^ pðQ BCÞ; ð19Þð2Þ pðPdisj2QÞ ¼ pðP \ Q ;Þ: ð20Þ

Definition 5 [7]. A degree to which a set is a subset of its complement is defined in the form p(P # PC).

Definition 6 [7]. A degree to which a set is disjointed from its complement, in terms of the two interpretations providedabove is defined as p(Pdisj1PC) and p(Pdisj2PC)).

3. The fuzzy logic

Let us define the implication operation

Iðp; qÞ ¼ 1� Fðp; qÞnorm; p > q;

1; p 6 q;

(ð21Þ

where F(p,q)norm is expressed by (15). Before showing that operation I(p,q) satisfies axioms (I1)–(I6), let us show some basicoperations encountered in proposed fuzzy logic.

Let us designate the truth values of the antecedent P and consequent Q as T(P) = p and T(Q) = q, respectively. The relevantset of proposed fuzzy logic operators is shown in Table 1. To obtain the truth values of these expressions, we use well knownlogical properties such as p ? q = :p _ q; p ^ q = :(:p _ :q) and alike.

In other words, we propose a new many-valued system, characterized by the set of union ([) and intersection (\) opera-tions with relevant complement, defined as T(:P) = 1 � T(P). In addition, the operators ; and " are expressed as negations ofthe [ and \, respectively. It is well known that the implication operation in fuzzy logic supports the foundations of decision-making exploited in numerous schemes of approximate reasoning. Therefore let us prove that the proposed implication oper-ation in (21) satisfies axioms (I1)–(I6). For this matter, let us emphasize that we are working with a many-valued system,which values for our purposes are the elements of the real interval R ¼ ½0;1�. For our discussion the set of of truth valuesV11 = {0,0.1,0.2, . . . ,0.9,1} is sufficient. In further investigations, we use this particular set V11. Table 2 shows the implicationoperation developed in the proposed fuzzy logic.

Theorem 1. Let a continuous function I(p,q) be defined by (21) taking values present in Table 2, i.e.,

Iðp; qÞ ¼ 1� Fðp; qÞnorm; p > q

1; p 6 q

(¼

1�pþq2 ; p > q;

1; p 6 q:

(ð22Þ

where F(p,q)norm is a defined by (1).

Then axioms (I1)–(I6) are satisfied and, therefore (22) is an implication operation.

Table 1

Name Designation Value

Tautology PI 1Controversy PO 0Negation :P 1 � PDisjunction P _ Q pþq

2 ; pþ q – 1;1; pþ q ¼ 1

�Conjunction P ^ Q pþq

2 ; pþ q – 1;0; pþ q ¼ 1

�Implication P ? Q 1�pþq

2 ; p – q;1; p ¼ q

�Equivalence P M Q minððp� qÞ; ðq� pÞÞ; p – q;

1; p ¼ q

�Pierce arrow P ; Q 1� pþq

2 ; pþ q – 1;0; pþ q ¼ 1

�Shaffer stroke P " Q 1� pþq

2 ; pþ q – 1;1; pþ q ¼ 1

�

Table 2

p ? q 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1

0 1 .55 .6 .65 .7 .75 .8 .85 .9 .95 1.1 .45 1 .55 .6 .65 .7 .75 .8 .85 .9 .95.2 .4 .45 1 .55 .6 .65 .7 .75 .8 .85 .9.3 .35 .4 .45 1 .55 .6 .65 .7 .75 .8 .85.4 .3 .35 .4 .45 1 .55 .6 .65 .7 .75 .8.5 .25 .3 .35 .4 .45 1 .55 .6 .65 .7 .75.6 .2 .25 .3 .35 .4 .45 1 .55 .6 .65 .7.7 .15 .2 .25 .3 .35 .4 .45 1 .55 .6 .65.8 .1 .15 .2 .25 .3 .35 .4 .45 1 .55 .6.9 .05 .1 .15 .2 .25 .3 .35 .4 .45 1 .551 0 .05 .1 .15 .2 .25 .3 .35 .4 .45 1


Proof

(I1) "p,p0 2 [0,1] jp0 P p > q) I(p,q) � I(p0,q) = 1 � p + q � 1 + p0 � q = p0 � p P 0) I(p,q) P I(p0,q),whereas q P p0 P p) Iðp; qÞ � Iðp0; qÞ 0) Iðp; qÞ Iðp0; qÞ.

(I2) "q,q0 2 [0,1]jq 6 q0 < p) I(p,q) � I(p,q0) = 1 � p + q � 1 + p � q0 = q � q0 6 0) I(p,q) 6 I(p,q0),whereas p 6 q 6 q0 ) Iðp; qÞ � Iðp; q0Þ 0) Iðp; qÞ Iðp; q0Þ.

(I3) I(0,q) 1, q P 0.

(I4) Ið1; qÞ ¼q2 ; q – 1;1; q ¼ 1:

�) Ið1; qÞ 6 q;

(I5) Notice that since p ? q = k(1 � p + q), where k = 0.5jp > q or 1jp 6 q, then "p,q,r 2 [0,1]jp > q > r) p ? (q ? r) =k(p ? (q ? r)) = k(p ? (1 � q + r)) = k(1 � p + 1 � q + r) = k(2 � p � q + r),whereas kðq! ðp! rÞÞ ¼ kðq! ð1� pþ rÞÞ ¼ kð1� qþ 1� pþ rÞ ¼ kð2� p� qþ rÞ.

(I6) IðnðqÞ;nðpÞÞ ¼ ð1� qÞ ! ð1� pÞ ¼qþ1�p

2 ; 1� q > 1� p1; 1� q 6 1� p

�¼

1�pþq2 ; p > q;

1; p 6 q:

�h

In addition, the proposed fuzzy logic is characterized by the following properties:Commutativity of both conjunction (\) and disjunction ([) operations, i.e.: p ^ q = q ^ p and

p _ q ¼ q _ p:

Assotiativity of these operations:

p ^ ðq ^ rÞ ¼ ðp ^ qÞ ^ r andp _ ðq _ rÞ ¼ ðp _ qÞ _ r:

Partial idempotency:

p ^ p ¼p; p > 0:5;0; p 6 0:5;

�


p _ p ¼p; p < 0:5;1; p P 0:5:

�

Partial distributivity:

ðp _ qÞ ^ r ¼ ðp ^ rÞ _ ðq ^ rÞ; if pþ qþ 2r 6 1;ðp _ qÞ ^ r – ðp ^ rÞ _ ðq ^ rÞ; if pþ qþ 2r > 1;

ð23Þ

p _ ðq ^ rÞ ¼ ðp _ qÞ ^ ðp _ rÞ; if 2pþ qþ r 6 1;p _ ðq ^ rÞ– ðp _ qÞ ^ ðp _ rÞ; if 2pþ qþ r > 1:

ð24Þ

To prove (23), we note that

pþ qþ 2r 6 1jðp _ qÞ ^ r ¼ pþ q2

� �^ r ¼

pþq2 þ r

2¼ pþ qþ 2r

4: ð25Þ

On the other hand, we have

ðp ^ rÞ _ ðq ^ rÞ ¼ pþ r2

� �_ qþ r

2

� �¼ pþ r

2þ qþ r

2

� �.2 ¼ pþ qþ 2r

4: ð26Þ

Therefore expression (25) is the same as (26).To prove (24), we carry out the following calculations,

2pþ qþ r 6 1jp _ ðq ^ rÞ ¼ p _ qþ r2

� �¼ pþ qþ r

2

� �.2 ¼ ð2pþ qþ rÞ

4; ð27Þ

ðp _ qÞ ^ ðp _ rÞ ¼ pþ q2

� �^ pþ r

2

� �¼ ð2pþ qþ rÞ

4: ð28Þ

Therefore expression (27) is the same as (28).DeMorgan theorems,

:ðp ^ qÞ ¼ :p _ :q;

:ðp _ qÞ ¼ :p ^ :q;

hold as well. To prove them, let us notice that

:ðp ^ qÞ ¼ : pþq2

� �1

(¼ 1� pþq

2

� �1

(¼

2�p�q2

1

(: ð29Þ

On the other hand

:p _ :q ¼:pþ:q

2

1

�¼

2�p�q2

1

(: ð30Þ

Therefore expression (29) is the same as (30).By analogy

:ðp _ qÞ ¼ : pþq2

� �0

(¼ 1� pþq

2

� �0

(¼

2�p�q2

0

(: ð31Þ

On the other hand

:p ^ :q ¼:pþ:q

2

0

�¼

2�p�q2

0

(: ð32Þ

Therefore expression (31) is the same as (32).It should be mentioned that the proposed fuzzy logic could be characterized by yet some other three features.p ^ 0 0, p 6 1. Whereas p _ 1 1, p P 0 and ::p = p. As a conclusion, we should admit that all above features confirm

that resulting system can be applied to V11 for every finite and infinite n up to that (V11,:,^,_,?) is then closed under all itsoperations.

4. Fuzzy power sets and fuzzy logic

Considering (16)–(18) and taking into account (22), we can formulate the following propositions.

Proposition 2 (Degree of possibility of set-inclusion).


pðP # QÞ ¼1�lP xþlQ x

2 ; lPx > lQ x;

1; lPx 6 lQ x:

(ð33Þ

Proposition 3 (Degree of possibility of set-equality).

pðP QÞ ¼

1�lPxþlQ x2 ; lPx > lQ x;

1; lPx ¼ lQ x;

1�lQ xþlPx2 ; lPx < lQ x:

8>>><>>>:

ð34Þ

From (34) it is clear that p(P Q) = 1 in case when lPx = lQQ and p(P Q) 6 0.5 otherwise. As it was mentioned in [9],there seem to be two plausible ways to define the degree to which sets P and Q may be said to be disjointed. One is thedegree to which each is a subset of the other’s complement. The second is the degree to which their intersection isempty.

Proposition 4 (Degree of disjointness of P and Q, or degree to which P and Q are disjointed). For the case (1) from (19)

pðP # Q CÞ ¼ p! ð1� qÞ ¼1�pþ1�q

2 ; p > 1� q

1; p 6 1� q

(¼

2�ðpþqÞ2 ; pþ q > 1;

1; pþ q 6 1;

(ð35Þ

pðQ BCÞ ¼ q! ð1� pÞ ¼1�qþ1�p

2 ; q > 1� p

1; q 6 1� p

(¼

2�ðpþqÞ2 ; pþ q > 1;

1; pþ q 6 1:

(ð36Þ

Therefore from (35) and (36), the definition (1) can arise in the form

pðPdisj1QÞ ¼2�ðpþqÞ

2 ; pþ q > 1;1; pþ q 6 1:

(ð37Þ

For case (2) from (20) and given that

p ^ q ¼pþq

2 ; pþ q > 1;0; pþ q 6 1:

�

We have

pðPdisj2QÞ ¼ pðp ^ q 0Þ ¼1; pþ q 6 1;0; pþ q > 1:

�ð38Þ

Proposition 5. Based on Definition 5 the degree to which a set is a subset of its complement is formulated as follows:

pðP # PCÞ ¼ p! ð1� pÞ ¼1�pþ1�p

2 ; p > 1� p1; p 6 1� p

(¼

1� p; p > 0:5;1; p 6 0:5:

�

Proposition 6. The degree to which a set is disjointed from its complement is based on the Definition 6. From expressions (37) and(38) we make an observation: p(Pdisj1PC) 1 whereas p(Pdisj2PC) 0.

5. Statistical properties of the fuzzy logic

In this section, we discuss some properties of the proposed fuzzy implication operator (21), assuming that the two prop-ositions (antecedent/consequent) in a given compound proposition are independent of each other and the truth values of thepropositions are uniformly distributed [20] in the unit interval In other words, we assume that the propositions P and Q areindependent from each other and the truth values v(P) and v(Q) are uniformly distributed across the interval [0,1]. Letp = v(P) and q = v(Q). Then the value of the implication I = v(p ? q) could be represented as the function I = I(p,q).

Because p and q are assumed to be uniformly and independently distributed across [0,1], the expected value of the impli-cation is

EðIÞ ¼ZZ

R

Iðp; qÞdpdq; ð39Þ


Its variance is equal to

VarðIÞ ¼ E½ðI � EðIÞÞ2� ¼ZZ

R

ðIðp; qÞ � EðIÞÞ2dpdq ¼ E½I2� � E½I�2; ð40Þ

where R ¼ fðp; qÞ : 0 6 p 6 1;0 6 q 6 1g. From (39) and given (21) as well as the fact that Iðp; qÞ ¼ I1ðp; qÞ; p > q;I2ðp; qÞ; p 6 q

�. We

have the following

EðI1Þ ¼ZZ

R

I1ðp; qÞdpdq ¼Z 1

0

Z 1

0

1� pþ q2

dpdq ¼ 12

Z 1

0

Z 1

0ð1� pþ qÞdp

� �dq

¼ 12

Z 1

0p� p2

2þ q

� �jp¼1p¼0

� �dq

¼ 1

212þ q2

2

��q¼1

q¼0

" #¼ 1

2: ð41Þ

Whereas E(I2) = 1. Therefore E(I) = (E(I1) + E(I2))/2 = 0.75.From (40) we have

I21ðp; qÞ ¼

14ð1� pþ qÞ2 ¼ 1

4ð1� 2pþ 2qþ p2 � 2pqþ q2Þ;

EðI21Þ ¼

ZZR

I21ðp; qÞdpdq ¼ 1

4

Z 1

0

Z 1

0ð1� 2pþ 2qþ p2 � 2pqþ q2Þdp

� �

dq ¼ 14

Z 1

0p� 2

p2

2þ p3

3� 2

p2

2qþ 2qþ q2

��p¼1

p¼0dq ¼ 1

4

Z 1

0

13þ qþ q2

� �dq ¼ 1

4q3þ q2

2þ q3

3

��q¼1

q¼0¼ 7

24:

Here EðI22Þ ¼ 1. Therefore EðI2Þ ¼ ðEðI2

1Þ þ EðI22ÞÞ=2 ¼ 31

48.From (40) and (41) we have VarðIÞ ¼ 1

12 ¼ 0:0833.Both values of E(I) and Var(I) demonstrate that the proposed fuzzy implication operator could be considered as one of the

fuzziest from the list of the exiting implications [32]. In addition, it satisfies the set of important Criteria I–IV, which is notthe case for the most implication operators mentioned above.

6. The fuzzy logic and fuzzy conditional inference

As it was mentioned in [26] ‘‘. . . in the semantics of natural language there exist a vast array of concepts and humans veryoften make inferences antecedents and consequences of which contain fuzzy concepts . . .’’ A formalization of methods forsuch inferences is one of the most important issues in fuzzy sets theory, in particular Artificial Intelligence, in general.For this purpose, let U and V (from now on) be two universes of discourses and corresponding fuzzy sets P and Q are the sameas in (2).

Given (2), a binary relationship for the fuzzy conditional proposition of the type: ‘‘If x is P then y is Q’’ for proposed fuzzylogic is defined as

RðA1ðxÞ;A2ðyÞÞ ¼ P � V ! U � Q ¼Z

U�VlPðuÞ=ðu;vÞ !

ZU�V

lQ ðvÞ=ðu;vÞ ¼Z

U�VðlPðuÞ ! lQ ðvÞÞ=ðu;vÞ: ð43Þ

Given (21), expression (43) reads as

lPðuÞ ! lQ ðvÞ ¼1�lPðuÞþlQ ðvÞ

2 ; lPðuÞ > lQ ðvÞ;1; lPðuÞ 6 lQ ðvÞ:

(ð44Þ

It is well known that given a unary relationship R(A1(x)) = P one can obtain the consequence R(A2(y)) by applying a compo-sitional rule of inference (CRI) to R(A1(x)) and R(A1(x),A2(y)) of type (43):

RðA2ðyÞÞ ¼ P � RðA1ðxÞ;A2ðyÞÞ ¼Z

UlPðuÞ=u �

ZU�V

lpðuÞ ! lQ ðvÞ=ðu;vÞ ¼Z

V

[u2U

½lPðuÞ ^ ðlPðuÞ ! lQ ðvÞÞ�=v : ð45Þ

In order to have Criterion I satisfied, that is R(A2(y)) = Q from (45), the equality
[u2U
½lPðuÞ ^ ðlPðuÞ ! lQ ðvÞÞ� ¼ lQ ðvÞ ð46Þ

has to be satisfied for any arbitrary v in V. To satisfy (46), it becomes necessary that the inequality

lPðuÞ ^ ðlPðuÞ ! lQ ðvÞÞ 6 lQ ðvÞ ð47Þ


holds for arbitrary u 2 U andv 2 V. Let us define a new method of fuzzy conditional inference of the following type:

Ant 1 : If x is P then y is Q

Ant 2 : x is P0

-------------------------Cons : y is Q 0:

ð48Þ

where P, P # U and Q, Q # V. Fuzzy conditional inference in the form given by (48) should satisfy Criteria I–IV presented inthe Appendix. It is clear that the inference (48) is defined by the expression (45), whenR(A2(y)) = Q0.

Theorem 2. If fuzzy sets P # U and Q # V are defined by (43) and (44) , respectively and R(A1(x),A2(y)) is expressed as

RðA1ðxÞ;A2ðyÞÞ ¼ P � V!L4

U � Q ¼Z

U�VlPðuÞ=ðu;vÞ!L4

ZU�V


U�VðlPðuÞ!L4

lQ ðvÞÞ=ðu; vÞ;

where


1�lP ðuÞþlQ ðvÞ2 ; lPðuÞ > lQ ðvÞ;

1; lPðuÞ 6 lQ ðvÞ;

(ð49Þ

then Criteria I, II, III and IV-1 [26] are satisfied.

Proof. For Criteria I–III let R(A1(x)) = Pa(a > 0) then

RðA2ðyÞÞ ¼ Pa � RðA1ðxÞ;A2ðyÞÞ ¼Z

Ula

PðuÞ=u �Z

U�VlpðuÞ!L4


V

[u2U

laPðuÞ ^ ðlPðuÞ!L4

lQ ðvÞÞ ,

v :

ð50Þ

9U1;U2 � UjU1 [ U2 ¼ U; jU1 \ U2 ¼ ; ) 8u 2 U1jlPðuÞ > lQ ðvÞ; 8u 2 U2jlPðuÞ 6 lQ ðvÞ: ð51Þ

From (50) and given subsets from (51), we have

RðA2ðyÞÞ ¼Z

V

[u2U1

laPðuÞ ^

1� lPðuÞ þ lQ ðvÞ2

� � �v

" #_Z

V

[u2U2

laPðuÞ ^ 1

�=v

" #: ð52Þ

Let us introduce the following function

f ðu; vÞ ¼1� lPðuÞ þ lQ ðvÞ

2: ð53Þ

Then the following relationship is satisfied:

8u 2 U1 laPðuÞ ^ f ðu; vÞ ¼

laPðuÞ; la

PðuÞ 6 f ðu; vÞ;f ðu; vÞ; la

PðuÞ > f ðu;vÞ;

�� ð54Þ

8u 2 U2 laPðuÞ ^ 1 ¼ la

PðuÞ;�� ð55Þ

From (54) and (55) we have

ð52Þ ¼Z

V

[u2U2

laPðuÞ=v

" #¼Z

Vla

Q ðvÞ=v ¼ Qa:

For Criteria IV-2 [26] let R(A1(x)) = :P = 1 � P then

RðA2ðyÞÞ ¼ :P �RðA1ðxÞ;A2ðyÞÞ ¼Z

Uð1�lPðuÞÞ=u �

ZU�V

lpðuÞ!L4lQ ðvÞ=ðu;vÞ ¼

ZV

[u2U

½ð1�lPðuÞÞ^ ðlPðuÞ!L4lQ ðvÞÞ�=v :

ð56Þ

From (56) and given subsets (51) we have

RðA2ðyÞÞ ¼Z

V

[u2U1

ð1� lPðuÞÞ ^1� lPðuÞ þ lQ ðvÞ

2

� � �v

" #_Z

V

[u2U2

ð1� lPðuÞÞ ^ 1 �

=v" #

¼Z

V

[u2U1

½ð1� lPðuÞÞ ^ f ðu;vÞÞ�=v" #

_Z

V

[u2U2

½ð1� lPðuÞÞ ^ 1�=v" #

: ð57Þ


Since ð1� lPðuÞÞ ^ f ðu;vÞ �

6 ð1� lPðuÞÞ ^ 1� �

, therefore

ð5:17Þ ¼Z

V

[u2U2

½ð1� lPðuÞÞ ^ 1�=v" #

¼Z

V1=v

¼ unknown: �

Theorem 3. If fuzzy sets P # U and Q # V are defined by (43) and (44), respectively, and R(A1(x),A2(y)) is defined as

RðA1ðxÞ;A2ðyÞÞ ¼ ðP � V!L4

U � QÞ \ ð:P � V!L4

U � :QÞ ¼Z

U�VðlPðuÞ!L4

lQ ðvÞÞ ^ ðð1� lPðuÞÞ!L4ð1� lQ ðvÞÞÞ=ðu;vÞ:

ð58Þ

where

ðlPðuÞ!L4lQ ðvÞÞ ^ ðð1� lPðuÞÞ!L4

ð1� lQ ðvÞÞÞ ¼

1�lP ðuÞþlQ ðvÞ2 ; lPðuÞ > lQ ðvÞ;

1; lPðuÞ ¼ lQ ðvÞ;1�lQ ðvÞþlP ðuÞ

2 ; lPðuÞ < lQ ðvÞ:

8>><>>:

Then Criteria I, II, III and IV-2 [26] are satisfied.

Proof

9U1;U2;U3 � UjU1 [ U2 [ U3 ¼ U; jU1 \ U2 \ U3 ¼ ; ) 8u 2 U1jlPðuÞ > lQ ðvÞ; 8u 2 U2jlPðuÞ ¼ lQ ðvÞ;8u 2 U3jlPðuÞ < lQ ðvÞ: ð59Þ

Let us introduce the following functions

h1ðu;vÞ ¼1� lPðuÞ þ lQ ðvÞ

2; h2ðu;vÞ ¼

1� lQ ðvÞ þ lPðuÞ2

: ð60Þ

Therefore from (58)–(60) for Criteria I–III let R(A1(x)) = Pa(a > 0) then

RðA2ðyÞÞ ¼ Pa � RðA1ðxÞ;A2ðyÞÞ ¼Z

Ula

PðuÞ=u �Z

U�VðlPðuÞ!L4

lQ ðvÞÞ ^ ðð1� lPðuÞÞ!L4ð1� lQ ðvÞÞÞ=ðu; vÞ

¼Z

V

[u2U

½laPðuÞ ^ ðlPðuÞ!L4

lQ ðvÞÞ ^ ðð1� lPðuÞÞ!L4ð1� lQ ðvÞÞÞ�=v: ð61Þ

From (60), (61) and given subsets by (59) we have

RðA2ðyÞÞ ¼Z

V

[u2U1

½laPðuÞ ^ h1ðu;vÞ�=v

" #_Z

V

[u2U2

laPðuÞ ^ 1

�=v

" #_Z

V

[u2U3

½laPðuÞ ^ h2ðu;vÞ�=v

" #: ð62Þ

Then the following is satisfied

8u 2 U1 laPðuÞ ^ h1ðu;vÞ ¼

laPðuÞ; la

PðuÞ 6 h1ðu; vÞ;h1ðu;vÞ; la

PðuÞ > h1ðu;vÞ;

�� ð63Þ

8u 2 U2 laPðuÞ ^ 1 ¼ la

PðuÞ;�� ð64Þ

8u 2 U3 laPðuÞ ^ h2ðu;vÞ ¼

laPðuÞ; la

PðuÞ 6 h2ðu;vÞ;h2ðu;vÞ; la

PðuÞ > h2ðu; vÞ:

�� ð65Þ

From (63)–(65) we have

ð61Þ ¼Z

V

[u2U2

laPðuÞ=v

" #¼Z

Vla

Q ðvÞ=v ¼ Qa: �

For Criteria IV-2 let R(A1(x)) = :P = 1 � P then

RðA2ðyÞÞ ¼ :P � RðA1ðxÞ;A2ðyÞÞ ¼Z

Uð1� lPðuÞÞ=u �

ZU�VðlPðuÞ!L4

lQ ðvÞÞ ^ ðð1� lPðuÞÞ!L4ð1� lQ ðvÞÞÞ=ðu;vÞ

¼Z

V

[u2U

½ðlPðuÞ!L4lQ ðvÞÞ ^ ðð1� lPðuÞÞ!L4

ð1� lQ ðvÞÞÞ�=v: ð66Þ


From (62), (66) and given subsets from (59) we have

RðA2ðyÞÞ ¼Z

V

[u2U1

½ð1� lPðuÞÞ ^ h1ðu; vÞÞ�=v" #

_Z

V

[u2U2

½ð1� lPðuÞÞ ^ 1�=v" #

_Z

V

[u2U3

½ð1� lPðuÞÞ ^ h2ðu;vÞÞ�=v" #

ð67Þ

As a conclusion we have

½ð1� lPðuÞÞ ^ h1ðu;vÞ� 6 ½ð1� lPðuÞÞ ^ 1�� and ½ð1� lPðuÞÞ ^ h2ðu;vÞ� 6 ½ð1� lPðuÞÞ ^ 1��;

therefore

ð67Þ ¼Z

V

[u2U2

½ð1� lPðuÞÞ ^ 1�=v" #

¼Z

Vð1� lQ ðuÞÞ=v

¼ :Q :

Theorems 2 and 3 show that fuzzy conditional inference rules, defined in (58) could adhere with human intuition to thehigher extent as the one defined by (49). The major difference between mentioned methods of inference might be explainedby the difference between Criteria IV-1 and IV-2. In particular, a satisfaction the Criterion IV-1 means that in case of logicalnegation of an original antecedent we achieve an ambiguous result of an inference, whereas for the case of the CriterionIV-2 there is a certainty in a logical inference.

7. Stability and continuity of fuzzy conditional inference

In this section, we revisit the fuzzy conditional inference rule (48). It will be shown that when the membership function ofthe observation P is continuous, then the conclusion Q depends continuously on the observation; and when the membershipfunction of the relation R is continuous then the observation Q has a continuous membership function. We start with somedefinitions. A fuzzy set A with membership function lA : R! ½0;1� ¼ I is called a fuzzy number if A is normal, continuous,and convex. The fuzzy numbers represent the continuous possibility distributions of fuzzy terms of the following type

A ¼Z

R

lAðxÞ=x:

Let A be a fuzzy number, then for any h P 0 we define xA(h), the modulus of continuity of A by

xAðhÞ ¼ maxjx1�x2 j6h

jlAðx1Þ � lAðx2Þj: ð68Þ

An a-level set of a fuzzy interval A is a non-fuzzy set denoted by [A]a and is defined by ½A�a ¼ ft 2 RjlAðtÞP ag for a 2 (0,1]and [A]a = cl(supplA) for a = 0. Here we use a metric of the following type

DðA;BÞ ¼ supa2½0;1�

dð½A�a; ½B�aÞ; ð69Þ

where d denotes the classical Hausdorff metric expressed in the family of compact subsets of R2, i.e. d([A]a, [B]a) =max{ja1(a) � b1(a)j, ja2(a) � b2(a)j}, whereas[A]a = [a1(a),a2(a)], [B]a = [b1(a),b2(a)j]. When the fuzzy sets A and B have finitesupport {x1, . . . ,xn}, then their Hamming distance is defined as

HðA;BÞ ¼Xn

i¼1

jlAðxiÞ � lBðxiÞj:

In the sequel we will use the following lemma.

Lemma 1 [15]. Let d P 0 be a real number and let A, B be fuzzy intervals. If D(A,B) 6 d, Then

supt2RjlAðtÞ � lBðtÞj 6 maxfxAðdÞ;xBðdÞg:

Consider the fuzzy conditional inference rule with different observations P and P0:

Ant 1: If x is P then y is Q
Ant 1: If x is P then y is Q Ant 2: x is P Ant 2: x is P0
Cons: y is Q
Cons: y is Q0
According to the fuzzy conditional inference rule, the membership functions of the conclusions are computed as

lQ ðvÞ ¼[u2R½lPðuÞ ^ ðlPðuÞ ! lQ ðvÞÞ�; lQ 0 ðvÞ ¼

[u2R½lP0 ðuÞ ^ ðlPðuÞ ! lQ ðvÞÞ�


or

lQ ðvÞ ¼ supu2R½lPðuÞ ^ ðlPðuÞ ! lQ ðvÞÞ�; lQ 0 ðvÞ ¼ supu2R½lP0 ðuÞ ^ ðlPðuÞ ! lQ ðvÞÞ�: ð70Þ

The following theorem shows the fact that when the observations are closed to each other in the metric D(�) of (69) type, then therecan be only a small deviation in the membership functions of the conclusions.

Theorem 4 (Stability theorem). Let d P 0 and let P, P0 be fuzzy intervals and an implication operation in the fuzzy conditionalinference rule (70) is of type (44). If D(P,P0) 6 d, then

supv2RjlQ ðvÞ � lQ 0 ðvÞj 6maxfxPðdÞ;xP0 ðdÞg:

Proof. Given an implication operation in the fuzzy conditional inference rule (70) is of type (44), for the observation P wehave

Q ¼ P � RðA1ðxÞ;A2ðyÞÞ ¼Z

UlPðuÞ=u �

ZU�V


ZV

[u2U

½lPðuÞ ^ ðlPðuÞ!L4lQ ðvÞÞ�=v : ð71Þ

It was shown above that for U;V � R; 9U1;U2 � UjU1 [ U2 ¼ U; jU1 \ U2 ¼ ; ),

8u 2 U1jlPðuÞ > lQ ðvÞ;8u 2 U2jlPðuÞ 6 lQ ðvÞ;

8u 2 U1jlPðuÞ ^ f ðu;vÞ ¼lPðuÞ;lPðuÞ 6 f ðu;vÞ;f ðu;vÞ;lPðuÞ > f ðu; vÞ;

�

8u 2 U2jlPðuÞ ^ 1 ¼ lPðuÞ;

where f(u,v) is from (53). Applying the observation P0 to (71) we obtain the following

Q 0 ¼ P0 � RðA1ðxÞ;A2ðyÞÞ ¼Z

UlP0 ðuÞ=u �

ZU�V


ZV

[u2U

½lP0 ðuÞ ^ ðlPðuÞ!L4lQ ðvÞÞ�=v: ð72Þ

We also have

8u 2 U1 lP0 ðuÞ ^ f ðu; vÞ ¼lP0 ðuÞ;lP0 ðuÞ 6 f ðu;vÞ;f ðu; vÞ;lP0 ðuÞ > f ðu; vÞ;

��8u 2 U2jlP0 ðuÞ ^ 1 ¼ lP0 ðuÞ:

From (71) and (72), we see that the difference of the values of conclusions for both P and P0 observations for arbitrary fixedv 2 RjlQ ðvÞ � lQ 0 ðvÞj. is defined as follows

8u 2 U1 jlQ ðvÞ � l0Q ðvÞj ¼jlPðuÞ � l0PðuÞj;0;

��8u 2 U2jjlQ ðvÞ � l0Q ðvÞj ¼ jlPðuÞ � l0PðuÞj;

and therefore from Lemma 1 we have

supv2RjlQ ðvÞ � lQ 0 ðvÞj ¼ sup

u2RjlPðuÞ � lP0 ðuÞj 6 maxfxPðdÞ;xP0 ðdÞg �

Theorem 5 (Continuity theorem). Let binary relationship Rðu; vÞ ¼ lpðuÞ!L4lQ ðvÞ be continuous. Then Q is continuous and

xQ(d) 6xR(d) for each d P 0.

Proof. Let d P 0 be a real number and let v1;v2 2 R such that jv1 � v2j 6 d. From (68) we have xQ ðdÞ ¼maxjv1�v2 j6djlQ ðv1Þ � lQ ðv2Þj. Then

jlQ ðv1Þ � lQ ðv2Þj ¼ j supu2R½lPðuÞ ^ ðlPðuÞ!L4

lQ ðv1ÞÞ� � supu2R½lPðuÞ ^ ðlPðuÞ!L4

lQ ðv2ÞÞ�j

6 supu2R½lPðuÞ ^ jðlPðuÞ!L4

lQ ðv1ÞÞ � ðlPðuÞ!L4lQ ðv2ÞÞj� 6 sup

u2R½lPðuÞ ^xRðjv1 � v2jÞ�

6 supu2R½lPðuÞ ^xRðdÞ� ¼ xRðdÞ: �

8. Concluding remarks


In this study, we proposed the fuzzy logic as an algebraic system closed under all its operations in which truth values of animplication operator need to be based on truth values of both the antecedent and the consequent. This implication operatorhas to be considered to be one of the fuzziest implication from a list of implication operators known so far. It was shown thatthe fuzzy logic presented here forms a basis for fuzzy conditional inference rules, which satisfy the set of known importantcriteria and seem to be suitable for capturing human intuition. The important features of stability and continuity of the pro-posed fuzzy conditional inference rules were also investigated.

Appendix A

Criterion I
Ant 1: If x is P then y is QAnt 2: x is P————————-Cons: y is Q.
Criterion II-1
Ant 1: If x is P then y is QAnt 2: x is very P—————————Cons: y is very Q.
Criterion II-2
Ant 1: If x is P then y is QAnt 2: x is very P————————–Cons: y is Q.
Criterion III
Ant 1: If x is P then y is QAnt 2: x is more or less P—————————–Cons: y is more or less Q.
Criterion IV-1
Ant 1: If x is P then y is QAnt 2: x is not P——————————Cons: y is unknown
Criterion IV-2
Ant 1: If x is P then y is QAnt 2: x is not P————————-Cons: y is not Q.
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