(T; N) and Residual Fuzzy Co-Implication in Dual Heyting Algebra with Applications

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1 Preprints ( O PEER-REVIEWED Posted: 3 August 2016 Article (; ) Residual Fuzzy Co-Implication in Dual Heyting Algebra with Applications Iqbal H. ebril Department of Mathematics cience Faculty aibah University Medina audi Arabia; iqbal501@hotmail.com; el.: Abstract: Recently many authors have been interested to introduce fuzzy implications over t-norms t-conorms. In this paper we introduce ( ) residuum fuzzy implication for Dubois t-norm Hamacher's t-norm. Also new concepts so-called ( ) residual fuzzy co-implication in dual Heyting Algebra are investigated. ome examples as well as application are discussed as well. Keywords: Fuzzy implications; implication; residuum t-norm; ( ) co-implication PAC: 0101 co-implication; residual 1 Introduction In fuzzy logic the basic theory of connective AD ( ) OR ( ) O ( ) are often modeled as (strong negations t-norm t-conorms). An important notion in fuzzy set theory is that of t-norm ( ) t-conorms ( ) strong negations ( ) C that are used to define a generalized intersection union negation of fuzzy sets (see [5] [6]). Implication co-implication functions play an important notion in fuzzy logic approximate reasoning fuzzy control intuitionistic fuzzy logic approximate reasoning of expert system (see ([1] [2] [3] [4] [7] [8] [9]). he notion of t-norm t-conorm turned out to be basic tools for probabilistic metric spaces (see [10] an [11]) but also in several other parts have found diverse applications in the theory of fuzzy sets fuzzy decision making in models of certain many-valued logics or in multivariate statistical analysis. (see [12] [13] [10]). 2 Preliminaries he logic connectives like negation is interpreted by a strong negation conjunction by a triangular norm disjunction by triangular conorm. [14]

2 Preprints ( O PEER-REVIEWED Posted: 3 August of riangular orm riangular conorm he conjunction in fuzzy logic it is often modeled as follows: Definition 2.1. [10]: A mapping from [ 01] 2 into [ 01 ] is a triangular norm (in short t-norm) iff are commutative nondecreasing in both arguments associative which satisfies ( p1) = p p [ 01]. Also disjunction in fuzzy logic is often modeled as follows: Definition 2.2. [10] A mapping from [ 01] 2 into [ 01 ] is a triangular conorm (in short t-conorm) iff are commutative nondecreasing in both arguments associative which satisfies ( p0) = p p [ 01]. Proposition 2.1. [10] A mapping is a triangular conorm iff there exists a triangular norm such that ( p q) = 1 ( 1 p1 q) pq t-conorm of. 01. In this case is called the dual he stard examples of t-norm dual t-conorms are stated in the following: Dual t-conorm () t--norm ( ) M pq = min( pq ) (Minimum t-norm) ( p q) = pq (Probabilistic Product t-norm) p if q = 1 W ( p q) = q if p = 1 0 if pq [01). (Drastic or weak t-norm) min( pq ) if p+ q 1 ( p q) = 0 if p + q < 1. (ilpotent t-norm) L( p q) = max( p + q 10) (Lukasiewicz t-norm) = p q M max p q (Maximum t-conorm) p q = p+ q pq (Probabilistic sum t-conorm) p if q =1 W ( p q) = q if p = 1 1 otherwise. (Drastic or largest t-conorm) ( pq) ( pq) p+ q< 1 max if = 0 if p + q 1. (ilpotent t-conorm) = ( + ) L pq min p q1 (Bounded um t-conorm)

Preprints (www.preprints.org) O PEER-REVIEWED Posted: 3 August 2016 3 of 19 0 if p = q = 0 H ( p q) = pq otherwise.

3 Preprints ( O PEER-REVIEWED Posted: 3 August of 19 0 if p = q = 0 H ( p q) = pq otherwise. p + q pq (Hamacher t-norm) D α pq ( p q) = α (01) max( pq α) (Dubois-Prade t-norm) 0 if p = q = 0 H ( p q) = p + q 2pq otherwise. 1 pq (Hamacher t-conorm) ( pq) ( 1 p)( 1 q) ( p q α ) D = 1 α max 1 1 α ( 01 ). (Dubois Prade t-conorm) For other family of t-norm (not needed here) we refer the reader to [12] for instance. If 1 2 there is at least one pair 2 ( pq ) [01] such that 1 ( p q) < 2 ( p q) then we briefly write. 1 < 2 With this the above t-norms satisfy the next known chain of inequalities W < L < < H < M. wo t-norm 1 2 are called comparable if 1 2 or 2 1 holds. he above chain of inequalities shows that W L H M are comparable. It is not hard to see that for example are not comparable while W M comparable with W < < M. 2.3 egation Function he truth table of the classical negation is given in following table. p 0 1 p 1 0 Definition 2.3. [15] A mapping from [ 01] into [ 01] is a negation function iff:

4 Preprints ( O PEER-REVIEWED Posted: 3 August of 19 = =0; 2. p q if p q. p q 01. (Monototonicity) A negation function is strict iff: 1. ( p ) is continuous; 2. ( p) < ( q) if p > q. p q 01. A strict negation function is strong or volutive iff: ( ) = 1. p p p 01. A negation function is weak iff is not strong. ( C p p) Example 2.1. [15] he strong negation = 1 strict negation but not strong 2 ( k ( p) = 1 p ) weaker negation D ( p) 1 1if p =0 = 0if p >0. strongest negation 1ifp< 1 D ( p) =. 2 0if p =1. Definition 2.4. [13] Let be a t-norm be a t-conorm. A mapping from [ 01] into [ 01 ] defined by = { = } p sup r 01 \ p r 0 for every p 01 = { = } p inf r 01 \ p r 1 for every p 01 are called the natural negation of respectively. 3 ( Ν ) Co-Implication his section will be devoted to introduce the concept of ( ) co implication. he relation between classical logic fuzzy logic as well as some examples are also discussed. Definition 3.1. [15] A mapping I from [ 01] 2 into [ 01 ] is fuzzy implication if pqr [01] the following conditions are satisfied:

5 Preprints ( O PEER-REVIEWED Posted: 3 August of 19 I1: I 11 = I 01 = I 00 = 1 I 10 = 0. I 2: I p q I r q if p r. I3: I p q I p r if q r. he set of all fuzzy implications is denoted by FI. In classical logic the main two ways to defining an implication 憭 in Boolean lattice ( L ) are p q p q p q max { r / p r q}. [18] he ( ) implication residual implication is generalization of these material implications to fuzzy logic. Definition 3.2. [18] A mapping I from [ 01] 2 into [ 01 ] is called an ( ) there exist a fuzzy negation a t-conorm such that I p q = p q p q 01. implication if Definition 3.3. [18] Let a left-continuous t-norm. hen the residual implication or R-implication derived form is given by (R) i.e. ( r p) ( ) =sup{ [ 01 ] / ( ) } pq I pq r r p q q r I ( p q) pqr Remark 3.1. [18] It easy to check that for every left-continuous t-norm the only operation I ( ) pq satisfies (R) is called { } I p q = max r 01 / r p q where the right side exists pq 01. Definition 3.4. [19] A mapping from [ 01] 2 into [ 01 ] is a fuzzy co implication if pqr [01] the following conditions are satisfied: 1: (11) = (1 0) = (0 0) = 0 (01) = 1.

6 Preprints ( O PEER-REVIEWED Posted: 3 August of 19 2: ( p q) ( r q) if p r. 3: ( p q) ( p r) if q r. he set of all fuzzy co implication is denoted by Co FI. From the definition 3.4. (1 q) = ( p 0) = 0 ( p p ) = 0 pq [01]. Lemma 3.1. If a mapping from [ 01] 2 into [ 01] satisfies ( 1) ( 2) then the mapping :[01] [01] defined by is a fuzzy negation. ( p) = ( p1) p [01] he following properties are generalization of fuzzy implication fuzzy co implication from classical logic. Definition 3.5. [18] A fuzzy implications I fuzzy co-implications is said to satisfy the following most important properties pqr [01]. ( 1 ) = ; I q q ( ( )) = ( ( )); I pi qr I qi pr ΝΡ (0 q) = q; ΕΡ ( p ( q r)) = ( q ( p r)); (Co-P) (Co-EP) I ( p p ) = 1; ( ΙΡ ) ( p p ) = 0; ( ) = 1 ; I pq p q ΟΡ ( p q) = 0 p q. (Co-IP) (Co-OP) Co-implication are extensions of the Boolean co-implication (p that p is not necessary for q ). (see [20]) q meaning Proposition 3.1. he operator 憭 ( material co-implication ) is generated by Boolean negation 憭 conjunction 憭 : q p q p. he ( ) co-implication is generalization of this material co-implication to fuzzy logic. In the following table we can see the truth table for the classical co-implication p q p q q p q p

Preprints (www.preprints.org) O PEER-REVIEWED Posted: 3 August 2016

A mapping from [ 01] 2 into [ 01] is called an ( ) co-implication if there exists a t-norm a fuzzy negation such that ( p q) = ( q ( p)) pq [01].

7 Preprints ( O PEER-REVIEWED Posted: 3 August of Definition 3.6. A mapping from [ 01] 2 into [ 01] is called an ( ) co-implication if there exists a t-norm a fuzzy negation such that ( p q) = ( q ( p)) pq [01]. A relation between fuzzy negations ( ) implication is given in the next proposition. Proposition 3.2 Let be an ( ) implication then =. Proof. For any p [01] ( p) = ( p1) = (1 ( p)) = ( p). Example 3.1. In the following examples we assume that C is a strong negation. t-norm C C I C M ( pq ) ( p q) ( q p) = min 1 M I C M M C C ( p q) Π C ( p q) = q pq Π C C I Π

8 Preprints ( O PEER-REVIEWED Posted: 3 August of 19 q if q = 1 W ( p q ) W C ( p q) = 1 p if q = 1 0 otherwise. I W C W C min q1 p if p < q ( p q ) C ( p q) = 0 if p q. I C C L( p q ) L C ( p q) = max ( q p0) L I C L C H ( p q ) H C ( 1 p) q ( p q) = H I 1 p + C H qp C

16 9 of 19 Dα ( p q) α (01) Dα C Example 3.2.

5 α D C I Dα C 1) A fuzzy negation p = p then the basic ( 2) implications 1 2 2 I ( )

9 Preprints ( O PEER-REVIEWED Posted: 3 August of 19 Dα ( p q) α (01) Dα C Example 3.2. For t-norm t-conorm q( 1 p) ( q p ) ( p q) = max α D C I Dα C 1) A fuzzy negation p = p then the basic ( 2) implications I ( ) co-implications M 2 2 M are: 2 2 = ( ) I M ( pq ) = min ( q1 p ) I ( pq ) max 1 p q M 2 2 I M 2 I M 2 2) A fuzzy negation ( p) 3 1 if p = 0 = 0otherwise. hen the basic ( ) implications ( ) 3 Ι co-implications M 3 3 M 3 are: I M 3 1 if p 0 ( p q) = q if p > 0. M 3 q if p = 0 ( p q) = 0 if p > 0. I M 3 M 3

Preprints (www.preprints.org) O PEER-REVIEWED Posted: 3 August 2016 10 of 19 As noted Ι is the least ( ) M 3 implications 3 is the least ( ) M 3 3 co-implications.

10 Preprints ( O PEER-REVIEWED Posted: 3 August of 19 As noted Ι is the least ( ) M 3 implications 3 is the least ( ) M 3 3 co-implications. 3) A fuzzy negation 4 ( p ) = 1 if p < 1 0if p =1. then the basic ( ) implications I co-implications. M M I M 4 1 if p < 1 ( p q) = q if p = 0. M 4 q if p < 1 ( p q) = 0 if p = 1. I M 4 M 4 As noted I is the greatest ( ) M 4 implications 4 greatest ( ) M 4 4 co-implications. 4. Residual Fuzzy Co-Implication in Dual Heyting Heyting algebra logic is the system on Heyting algebras Brouweriaun algebras. Heyting algebra L 01 is lattice with the bottom 0 the top 1 the binary operation called implication such that p qr L p q is the relative pseudocomplement of a with respect to c. hat is to say p r q p q p qr L. In other words the set of all b L such that p r q contains the greatest element denoted by p q. Precisely { } p q = sup r L\ p r q.

11 Preprints ( O PEER-REVIEWED Posted: 3 August 2016 he dual of Heyting algebra is called Brouwerian algebra ( 01) 11 of 19 L is a lattice with 0 1 the binary operation called co-implication algebra. atisfying p qr L. in dual Heyting p r q p q. he set of all r p q. Precisely L such that p r q contains the smallest element denoted by { } p q = r L p r q inf \. Definition 4.1. Let is the t-conorm of right continuous. hen the residual co-implication ( R -implication) derived from is { } p q =inf r 01 \ r p q p q 01. ( R ) R -implication come from residuted lattices based on residuation property ( R P ) that can be written as ( r p) he operation ( ) q if only if r ( p q). ( R P ) x y is called residual co-implication of the t-conorm. We now list the residual co-implication associated to the stard left-continuous t-norms previously introduced. Applying the above concepts to the stard t-norms we obtain the following interesting results. (1) Residuum of the Maximum t-conorm ( ) M p q is ( p q ) ( p q) M (2) Residuum of the Probabilistic sum t-conorm ( p q) M 0 if p q = M y otherwise. I M is

p q ) ( p q) max( 0 q p) L L = L L (4) Residuum of the

12 Preprints ( O PEER-REVIEWED Posted: 3 August of 19 ( p q) ( ) 0 if p q p q = q p otherwise. 1 p (3) Residuum of the Bounded um t-conorm ( ) L p q is I ( p q ) ( p q) max( 0 q p) L L = L L (4) Residuum of the ilpotent t-conorm ( ) p q is ( p q ) ( p q) 0 if p q = min ( 1 pq ) otherwise (5) Residuum of the Hamacher t-conorm ( ) H p q is

13 Preprints ( O PEER-REVIEWED Posted: 3 August of 19 (6) Residuum of the Dubois-Prade t-conorm ( ) D p q is ( p q ) ( ) D0.5 0 if p q D p q = αq α 0.5 max q + 1 if p < q 1 p (7) Residuum of the Hamacher s parametric t-conorm is ( p q) α D 0.5 is D0.5 ( p q) α 5 ( ) 0 if p q p q = αq p + (1 α) q α if p < q 1 ( 2 α) p + ( 1 α) p α Co-Implication Residual Co-implication Properties α In this section we introduce some properties for ( ) co-implication. co-implication residual Proposition 5.1. For a left continuous t-norm then is left-continuous.

14 Preprints ( O PEER-REVIEWED Posted: 3 August of 19 Proof: Let be left-continuous. Assume that there exist pq [01] such that ( p) < q ( p ε ) ε (0 x ]. his contradicts the left-continuity of hence must be left-continuous. Proposition 5.2. For a left continuous t-norm then the supremum in the definition of is the maximum i.e. = = p ( p) max{ t 01 \ p q 0} where the left side exists for all p [ 01 ]. 01 Proof: From the previous proposition since ( p) ( p) for all p ( p ( p )) = 0 that means by definition that the supremum is the maximum. 01 one has Proposition 5.3. For a left continuous t-norm then pq [ 01] equivalence holds: ( p) q ( p q) = 0 the following Proof: uppose that q for some p [ 01] we consider two cases: () i > q t > q: ( p t ) = 0 ( p q) = 0. (By monotonicity of ) ( ii ) = q q { t 01 \ ( p t ) = 0} ( p q) = 0 or q {t 01 \ ( p t) = 0}. herefore there exists an increasing sequence ( t i) i such that ti < q ( p t i ) = 0 for all i limt i i = q. By the left continuity of we get which is a contradiction. ( p q) = ( plim t ) = ( p t )limt = 0 i i i i i On the other side assume that is a left continuous t-norm for some pq [ 01 ]. ( p q) = 0 q { t 01 \ ( p t) = 0}

15 Preprints ( O PEER-REVIEWED Posted: 3 August of 19 q max{ t 01 \ ( p t) = 0} q ( p). he proof is complete. Proposition below states how a ( ) implications gives rise to a fuzzy ( ) co-implication vice-versa. Proposition 5.4. A mapping from [ 01] 2 into [ 01 ] is a ( ) co- implication with strong negation iff ( p q) = ( I ( q p)) for some I fuzzy (strong) negation. Conversely I from [ 01]2 into [ 01 ] is a ( ) implication iff I ( pq ) = ( ( q p)) for some fuzzy (strong) negation. heorem 5.1. For t-norm then Co FI. Proof: We have to show that 1 2 3in definition of fuzzy co-implication are satisfied for all pqr [01]. 1 : 2 : 11 = 1 0 = 0 0 = 0 01 = 1. p r ( p) ( r) ( q p ) ( q r ) ( p q) ( r q) 3 :. q r ( q p ) ( q p ) ( p q) ( p r). heorem 5.2. All ( ) co-implications are fuzzy implications satisfy (Co-P) (Co-EP). Proof: If an ( ) co-implications then (0 q) = ( q1) = q.

16 Preprints ( O PEER-REVIEWED Posted: 3 August of 19 Also ( p ( q r)) = ( ( q r) ( p)) = ( ( rq ) ( p)) = ( ( p) ( r ( q))) = ( ( ( p) r) q ) = ( ( r ( p)) q ) = ( ( p r) ( q)) = ( q ( p r)). heorem 5.3. If I satisfies ( IP ) with strong negation then Proof: ( p p) = ( p ( x)) = ( ( p) p) = ( ( p ( p))) satisfies (Co-IP). = ( ( ( p) p)) = ( I ( p p)) = (1) = 0. heorem 5.4. If I satisfies ( OP ) with strong negation then Proof: We would like to prove that Let ( p q) = 0 p q. p q ( p) ( q) I ( ( p) ( q)) = 1 by (OP) satisfies (Co-OP). ( I ( ( p) ( q))) = (1) ( ( p ( q))) = 0 ( ( ( ( p) q))) = 0 q ( p) = 0 ( p q) = 0. heorem 5.6. For t-norm a fuzzy negation then ( p p) = 0 p 01 iff ( p ( p)) = 0 for all p 01. Proof: If p p p ( ) = 0 01 then Conversely if p p p ( p ( p)) = ( p p) = 0 p 01. ( ) = 0 01 then ( p x) = ( p ( p)) = 0 p 01.

17 Preprints ( O PEER-REVIEWED Posted: 3 August of 19 heorem 5.7. For a left continuous t-norm a continuous fuzzy negation then iff ( p p) = 0 p 01 ( p) ( p) p 01. Proof: Let is a left continuous t-norm for a continuous fuzzy negation then hen ( p p) = ( p ( p)) = 0. By monotonicity of if then ( p ( p )) = 0. ( x) = max{ t 01 : ( p t) = 0} p 01. ( p) ( p) Conversely let p p p ( ) = 0 01 then ( ) 0 ( p) { t 01 : ( p t) = 0} p p = p [ 01] if then ( p) max{ t 01 : ( p t) = 0} = ( p). heorem 5.8. For a right continuous t-conorm then Co FI. Proof: We have to show that 1 2 3in definition of fuzzy co-implication are satisfied for all pqr [01]. : 1 { r ( r ) } 2 11 = inf [01]\ 1 1 = 0 { } { } { } ( p q) ( r q). 10 = inf r [01]\ r1 0 = 0 00 = inf r [01] \ r0 0 = 0 01 = inf r [01] \ r0 1 = 1. { } { } inf { t [01] \ t p q} inf { t [01] \ ( t r) q} : p r t [01]\ t p q t [01]\ t r q

18 Preprints ( O PEER-REVIEWED Posted: 3 August of 19 3 { } ( p q) ( p r). { } { t t p q} { t ( t p) r} : q r t [01]\ t p q t [01]\ t p r inf [01] \ inf [01] \ heorem 5.9. A co-implications satisfy (Co-P) (Co-IP). Proof: For any t-conorm pq [ 01] we get Also { } { } (0 q) = inf r [01] \ r0 q = inf r [01] \ r q = q. { } ( p p) = inf r [01]\ r p p = 0. heorem If is a right continuous then satisfy (Co-EP) Co-OP). Proof: For any right continuous t-conorm for all pqr [ 01] by using condition we have { } { } ( ) ( p ( q r)) = inf t [01]\ t p ( q r) = inf t [01]\ t p q r R { t ( t ( p q) ) r} t ( t ( q p) ) r { t ( tq p) r} { t tq pr} { } = inf [01] \ = inf [01] \ = inf [01] \ ( ) = inf [01] \ ( ) ( ) = ( q ( p r)). ow we would like to prove that ( p q) = 0 p q. If p q then ( p0 ) = p q so ( p q ) = 0. Conversely if ( p q ) = 0 then because of R condition we get p0 q i.e. p q. Conclusion here is four usual models of fuzzy implications that is () residual QL-operation ( I p q p p q ) p q ( ) = ( ( ) 01 D-operations ( I( p q) = ( ( ( p) ( q)) q) pq [ 01] implication. In this paper we introduced () residual co-implication. ow an interesting natural questions arises that to find Co-QL-operation Co-D-operations. Competing interests he authors declare that they have no competing interests.

19 Preprints ( O PEER-REVIEWED Posted: 3 August of 19 References [1] E.E. Kerre C. Huang D. Ruan. Fuzzy et Approximate Reasoning. Wu Han University Press Wu chang [2] M. Mas M. Monserrat. orrens E. rillas. A survey on fuzzy implication functions. IEEE ransactions on Fuzzy ystems 15(6): [3]. Gottwald. Areatise on Many-Valued Logic. Resarch tudies Press Baldok [4] L. soukalas R. Uhring L. Zadeh. Fuzzy eural Approaches in Engineering. Adaptive Learning ystems for ignal Processing communications control. Wiley-Interscience ew York [5] E.P. Klement R. Mesiar E. Pap riangular orms Kluwer Academic Publisher Dordrecht [6] Weber. A General Concept of Fuzzy Connectives egations Implications Based on -norms -conorms Fuzzy ets ystems 11 pp [7] B. De Baets Coimplicators the forgotten connectives atra Mountains Mathematical Publications 12: atra Mt. Math. Publ [8] Oh K. Kel A.: Coimplication its applications to fuzzy expert systems. Information ciences [9] F. Wolter On Logics with Coimplication ournal of Philosophical Logic 27 (4) [10] B. chweizer A. klar Probabilistic Metric paces orth Holl Amsterdam nd edition: Dover Publications Mineola Y [11] I. ebril Mohd.. Md. oorani A. aari An example of a probabilistic metric space not induced from a rom normed space. Bull. Malays Math. ci. oc. 2 (26) [12] M.M. Gupta. Qi heory of -norms fuzzy inference methods Fuzzy ets ystems [13] E.P. Klement R. Mesiar riangular orms atra Mountains Math. Publ. 13 (1997) [14] Fodor. Left-continuous t-norms in fuzzy logic: an overview. ournal of applied sciences at Budapest ech Hungary 1(2) [15].C. Fodor M. Roubens Fuzzy Preference Modelling Multicriteria Decision upport Kluwer Dordrecht [16] K.C. Maes B. De Baets. A contour view on uninorm properties.kybernetika [17] M. Baczynski B. ayaram Fuzzy Implications tudies in Fuzziness oft Computing Vol. 231 pringer Heidelberg [18] M. Baczynski B. ayaram ()- R-implications: a state-of-the-art survey Fuzzy ets yst [19] P. Li. Fang A survey on fuzzy relational equations part I: classification solvability Fuzzy Optimization Decision Making [20] Youg u Zhuden Wang Constructing implications coimplication on a complete lattice Fuzzy ets yst by the authors; licensee Preprints Basel witzerl. his article is an open access article distributed under the terms conditions of the Creative Commons by Attribution (CC-BY) license (

CONSERVATIVE AND DISSIPATIVE FOR T-NORM AND T-CONORM AND RESIDUAL FUZZY CO-IMPLICATION

Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 8 Issue 4(2016), Pages 78-90. CONSERVATIVE AND DISSIPATIVE FOR T-NORM AND T-CONORM AND RESIDUAL FUZZY