between implicator and coimplicator integrals,

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1 Implicator coimplicator integrals Gert de Cooman Universiteit Gent Elektrische Energietechniek Technologiepark 9, 9052 Zwijnaarde, Belgium Bernard De Baets Universiteit Gent Toegepaste Wiskunde en Informatica Krijgslaan 281 S9, 9000 Gent, Belgium bstract We introduce study implicator coimplicator integrals, investigate their possible application in defining the possibility necessity of fuzzy sets. First, the definition properties of implicators coimplicators on bounded posets are discussed. Then, in analogy with the theory of seminormed semiconormed fuzzy integrals, implicator coimplicator integrals are defined. Next, we study the properties of these dual types of integrals. We uncover an interesting relationship between implicator coimplicator integrals, seminormed semiconormed fuzzy integrals, which could also be called conjunctor disjunctor integrals. Finally, we show that coimplicator implicator integrals can be used to extend the domain of possibility measures necessity measures from sets to fuzzy sets. 1 INTRODUCTION Seminormed semiconormed fuzzy integrals were introduced in 1986 by Suárez García Gil Álvarez [10] as generalizations of Sugeno s fuzzy integral [11]. They were generalized by De Cooman Kerre [7], who also showed that they are, in a sense, the most general types of integrals satisfying a number of desired properties, that they are very well suited for combination with possibility respectively necessity measures. s their name suggests, these integrals are respectively associated with triangular seminorms triangular semiconorms. In this paper, we introduce two other types of fuzzy integrals, associated with border implicators border coimplicators, show that they too are well suited for combination with possibility respectively necessity measures. 2 EXTENDED LOGICL OPERTORS In this section, we discuss the four basic extended logical operators on a bounded poset P,, with top 1 P bottom 0 P. More extensive results concerning conjunctors disjunctors can be found in [3, 6], concerning implicators in [2, 3], concerning coimplicators in [1]. decreasing unary operator N on P, is called a negator iff N 0 P = 1 P N 1 P = 0 P. strong negator N on P, is a permutation of P that satisfies α β N α N β, α β in P. Note that an involutive, decreasing transformation is a strong negator. n increasing binary operator O on P, is called a conjunctor iff O0 P, 1 P = O1 P, 0 P = 0 P O1 P, 1 P = 1 P, a disjunctor iff O0 P, 1 P = O1 P, 0 P = 1 P O0 P, 0 P = 0 P. ny conjunctor C satisfies C0 P, α = Cα, 0 P = 0 P, while any disjunctor D satisfies D1 P, α = Dα, 1 P = 1 P. This means that by definition the values of a conjunctor are fixed on two of the four borders ; similarly for a disjunctor, but on the opposite borders. Fixing the values on the remaining borders leads to t-seminorms t-semiconorms. Definition 1 n increasing binary operator O on P, is called a t-seminorm iff for any α in P, O1 P, α = Oα, 1 P = α; a t-semiconorm iff for any α in P, O0 P, α = Oα, 0 P = α. Clearly, a t-seminorm is a conjunctor a t- semiconorm a disjunctor. hybrid monotonous binary operator O on P, with decreasing first increasing second partial mappings is called an implicator iff O0 P, 0 P = O1 P, 1 P = 1 P O1 P, 0 P = 0 P, a coimplicator iff O0 P, 0 P = O1 P, 1 P = 0 P O0 P, 1 P = 1 P.

2 Implicators are extensions of the Boolean implication p q meaning that p is sufficient for q, while coimplicators are extensions of the Boolean coimplication p q meaning that p is not necessary for q. ny implicator I satisfies I0 P, α = Iα, 1 P = 1 P, while any coimplicator J satisfies J 1 P, α = J α, 0 P = 0 P. gain, we can say that the values of an implicator or coimplicator are fixed on two of the four borders. The behaviour on one of the remaining borders is of particular interest. The partial mapping I, 0 P of an implicator I clearly is a negator, is denoted by N I, i.e. N I α = Iα, 0 P, called the negator induced by I. Similarly, the partial mapping J, 1 P of a coimplicator J is a negator, is denoted by N J, i.e. N J α = J α, 1 P, called the negator induced by J. Fixing the values on the remaining border leads to border implicators border coimplicators. Definition 2 n implicator I on P, that satisfies I1 P, β = β, β P, is called a border implicator. coimplicator J on P, that satisfies J 0 P, β = β, β P, is called a border coimplicator. Consider a strong negator N a binary operator O on P,. The dual operator of O w.r.t. N is the binary operator D N [O] on P, defined by, for any α β in P : D N [O]α, β = N 1 ON α, N β. With O N we also associate two binary operators Λ N [O] N [O] on P, defined by, for any α β in P : Λ N [O]α, β = Oβ, N α N [O]α, β = ON β, α. These notions lead to interesting relationships between our four basic types of extended logical operators. Proposition 3 i Consider a t-seminorm T, then D N [T ] is a t- semiconorm Λ N [T ] is a border coimplicator. ii Consider a t-semiconorm S, then D N [S] is a t- seminorm Λ N [S] is a border implicator. iii Consider a border implicator I, then D N [I] is a border coimplicator. Moreover, N [I] is a t- semiconorm iff N I = N 1. iv Consider a border coimplicator J, then D N [J ] is a border implicator. Moreover, N [J ] is a t- seminorm iff N J = N 1. mapping f between complete lattices L, M, is called a complete meet-morphism iff for any nonempty subset of L, finf = inf f; a complete join-morphism iff for any nonempty subset of L, fsup = sup f; a complete dual meet-morphism iff for any nonempty subset of L, finf = sup f; a complete dual joinmorphism iff for any nonempty subset of L, fsup = inf f. Note that a strong negator on a complete lattice always is a complete dual meetmorphism a complete dual join-morphism. 3 IMPLICTOR ND COIMPLICTOR INTEGRLS In the rest of this paper, will denote an arbitrary universe of discourse, V a field of subsets of, with {, } V. By L, we denote a complete lattice, with top 1 L bottom 0 L. The meet of L, will be denoted by, its join by. Moreover, T is a t-seminorm, S a t-semiconorm, I a border implicator J a border coimplicator on L,. By v we denote a L, -confidence measure on, V, i.e., a V L-mapping that is increasing: B v vb, for any B in V. If N is a strong negator, then the V L-mapping v N, defined by v N = N 1 vco, V, is a L, -confidence measure on, V, called the dual confidence measure of v w.r.t. N. mapping from to L is also called a L, -fuzzy set in. The set of all these mappings is denoted by L. The partial order on L allows us to define a partial order on L as follows: for any h 1 h 2 in L, h 1 h 2 ω h 1 ω h 2 ω. For any element µ of L, we denote the constant {µ}-mapping by µ. For any subset of, its characteristic L-mapping is denoted by χ defined by χ ω = 1 L, ω χ ω = 0 L, ω co. Let h be a L-mapping let λ be an element of L. We define the cut set Sλ h = {ω hω λ} the dual cut set Dλ h = {ω hω λ} of h at level λ. h is called V-cut-measurable iff λ LSλ h V dually V-cut-measurable iff λ LDλ h V. Note that for any, the following statements are equivalent: χ is V-cut-measurable; χ is dually V-cut-measurable; V. We call a L-mapping s V-simple iff it has a finite range s = {s 1,..., s n } is V-measurable, i.e., D k = s 1 {s k } V, k = 1,..., n. It is easily verified that for any ω in : sω = inf n I n χ Dk ω, s k = sup J χ codk ω, s k. k=1 k=1

3 Inspired by these expressions, we associate two functionals γi v ; δv J ; with the L, -confidence measure v in the following way. If V s is a V-simple L-mapping, then, with obvious notations, γ v I; s = δ v J ; s = n inf k=1 Iv D k, s k sup n k=1 J v cod k, s k. In the stard way see also [7], we use these functionals to construct two new functionals which are defined for arbitrary L-mappings, not just for V-simple mappings. Definition 4 Let h be a L-mapping let be an element of V. Then I hdv = inf{γi; v s h s} is called the I-integral or implicator integral of h on ; J hdv = sup{δj v ; s s h} is called the J -integral or coimplicator integral of h on. The following theorems give explicit formulas for the calculation of implicator coimplicator integrals. Theorem 5 Let h be a L-mapping let be an element of V. Then I hdv = inf I v B, sup hω 1 J hdv = sup J v cob, inf hω. 2 Proof. We give the proof of 2. The proof of 1 is completely analogous. Consider an arbitrary B in V, let λ B = inf hω consider the V-simple mapping s B assuming the value λ B on B 0 L elsewhere. Clearly s B h, whence J hdv δv J ; s B = J v cob, inf hω, therefore J hdv sup J v cob, inf hω. Conversely, consider a V-simple mapping s with s h, then clearly, with obvious notations, s k inf ω Dk hω, therefore also J v cod k, s k J v cod k, inf ω Dk hω, k = 1,..., n. Therefore δj v ; s supn k=1 J v cod k, inf ω Dk hω sup J v cob, inf hω, whence J hdv sup J v cob, inf hω. Theorem 6 Let h be a L-mapping let be an element of V. If h is dually V-cut-measurable, then I hdv = inf Iv λ L Dh λ, λ. 3 If h is V-cut-measurable, then J hdv = sup J v cosλ, h λ. 4 λ L Proof. We give the proof of 3. The proof of 4 is analogous. ssume that h is dually V- cut-measurable. Consider an arbitrary B in V let λ B = sup hω. Clearly B Dλ h B, whence Iv Dλ h B, λ B Iv B, sup hω. Therefore, inf Iv λ L Dh λ, λ inf I v B, sup hω. Conversely, for λ in L, sup ω D h hω λ, whence λ I v Dλ h, sup ω D hω Iv D h h λ λ, λ. Therefore, inf Iv λ L Dh λ, λ inf I v B, sup hω. We continue with a number of interesting properties of implicator coimplicator integrals. Proposition 7 Let be an element of V. Then I χ dv = N I vco J χ dv = N J vco. Proof. We give the proof of the first equality. The other equality is proven analogously. Since χ is dually V-cut-measurable, we find that, using 3, I χ dv = inf λ L IvDχ λ, λ = Iv, 1 L inf Ivco, λ λ<1 L = Ivco, 0 L = N I vco.

4 Proposition 8 Let be an element of V let µ be an element of L. Then I µdv = N I v Iv, µ J µdv = N J v J v, µ. Proof. We give the proof of the second equality. The proof of the first equality is similar. Consider any λ in L. Since, for λ µ, S µ λ = for λ µ, Sµ λ =, µ is V-cut-measurable, therefore, using 4, J µdv = sup J v cos µ λ, λ λ L = sup J v, λ sup J v, λ λ µ { λ µ J v, 1L ; µ = 1 = L J v, µ J v, 1 L ; µ < 1 L = N J v J v, µ. s a corollary of this, we have for any in V that I 0 Ldv = N I v I 1 Ldv = N I v. lso, J 0 Ldv = N J v J 1 Ldv = N J v. Moreover, if µ is any element of L, we find that if v = 0 L, then I µdv = Iv, µ J µdv = N J v µ. If v = 1 L, then I µdv = N Iv µ J µdv = J v, µ. Finally, if both v = 0 L v = 1 L, then I µdv = J µdv = µ. The following proposition is an immediate consequence of Theorem 5. It shows that implicator coimplicator integrals are increasing in their integr but decreasing in their domain of integration, which is a somewhat undesirable property. Indeed, in [7] it was shown that the only integrals which have a selection of desired properties, among which isotonicity in the integration domain, are the seminormed semiconormed fuzzy integrals. It is therefore not surprising that these new integrals have a number of less desirable properties. We shall nevertheless see further on that, like seminormed semiconormed fuzzy integrals, they have a part to play in possibility theory. Proposition 9 Let, 1 2 be elements of V. Let h, h 1 h 2 be L-mappings. ssume that 1 2 h 1 h 2. Then I h 1dv I h 2dv J h 1dv J h 2dv. lso, I 1 hdv I 2 hdv J 1 hdv J 2 hdv. To end this section, we give a number of propositions relating implicator coimplicator integrals both to each other to seminormed semiconormed fuzzy integrals. For a detailed account of the latter, we refer to [7]. In the context of this paper, we merely point out that for the L, -fuzzy T -integral of a L-mapping h on a set V: T hdv = sup T inf hω, v B, for the L, -fuzzy S-integral of h on : S hdv = inf Ssup hω, v cob. Proposition 10 Let N be a strong negator on L, let h be a L-mapping. Then D N [I] hdv N = N 1 I N hdv D N [J ] hdv N = N 1 J N hdv. Proof. We prove the first equality. The proof of the second is similar. Since D N [I] is a border coimplicator on L,, Theorem 5 tells us that D N [I] hdv N = sup D N [I] v N cob, inf hω = sup N IN 1 v N cob, N inf hω = N 1 inf I vb, supn hω, which completes the proof. The proof of the next proposition is similar therefore omitted. Proposition 11 Let N be a strong negator on L, let h be a L-mapping. Then Λ N [T ] hdv N = T hdv Λ N [S] hdv N = S hdv. Moreover, if N I = N 1, then N [I] hdv N = I hdv, if N J = N 1, then N [J ] hdv N = J hdv,

5 4 POSSIBILITY ND NECESSITY OF FUZZY SETS Let us now consider an ample field R [5, 12] of subsets of, i.e., a class of subsets of that is closed under arbitrary unions under complementation. The atom [ω] R of R containing ω is the element of R defined by [ω] R = { R ω }. Note that for any, R = ω [ω] R. For any L-mapping h, the following statements are readily shown to be equivalent: h is R-cut-measurable; h is dually R-cut-measurable; h is constant on the atoms of R. Whenever h is constant on the atoms of R, we say that h is R-measurable. mapping Π from R to L is called a L, -possibility measure or simply possibility measure on, R iff for any family j j J of elements of R, Π j = sup Π j. Note that Π = 0 L. Π is called normal iff Π = 1 L. L-mapping π is called a distribution of Π iff it is constant on the atoms of R R-measurable if for any in R, Π = sup ω πω. Such a distribution is unique given by πω = Π[ω] R, ω. mapping N from R to L is called a L, -necessity measure or simply necessity measure on, R iff for any family j j J of elements of R, N j = inf N j. Note that N = 1 L. N is called normal iff N = 0 L. L-mapping ν is called a distribution of N iff it is constant on the atoms of R if for any in R, N = inf ω co νω. gain, such a distribution is unique. It is given by νω = Nco[ω] R, ω. For more information about possibility necessity measures, we refer to [4, 8, 9, 13] In this section, Π is a L, -possibility measure on, R, with distribution π; N is a L, -necessity measure on, R with distribution ν. Furthermore, for the border implicator I, we assume that its first partial mappings are complete dual join-morphisms that its second partial mappings are complete meet-morphisms. For the border coimplicator J, it will dually be assumed that its first partial mappings are complete dual meet-morphisms that its second partial mappings are complete join-morphisms. Note that this implies that N I is a complete dual joinmorphism N J a complete dual meet-morphism. The following results tell us what happens if we associate an implicator integral with Π, a coimplicator integral with N. Theorem 12 Let be an element of V let h be a L-mapping. Then I hdπ = inf I πω, sup hx ω x [ω] R J hdn = N J N sup J ω νω, inf hx. x [ω] R Proof. We give the proof of the second equality. The proof of the first equality is analogous, but somewhat less complicated. Since the first partial mappings of J are complete dual meet-morphisms, we find, taking into account Theorem 5, J hdn = sup J N cob, inf hx B R x B = sup J N inf νω, inf hx B R x B = N J N sup J inf νω, inf hx B R x B Since moreover it is easily verified that for µ L, J χ cob ω νω, µ = J νω, χ B ω µ, since the second partial mappings of J are complete join-morphisms, we find that sup J inf νω, inf hx B R x B = sup sup J χ cob ω νω, inf hx x B ω B R = sup J ω = sup J ω νω, sup χ B ω inf hx B R x B νω, inf, x [ω] R hx The following propositions give interesting expressions for implicator coimplicator integrals associated with possibility respectively necessity measures, when the integr is R-measurable. They immediately follow from the theorem above. Proposition 13 Let be an element of V let h be an R-measurable L-mapping. Then I hdπ = inf Iπω, hω ω J hdn = N J N sup J νω, hω. ω Proposition 14 Let h be an R-measurable L- mapping. Then I hdπ = inf Iπω, hω ω

6 J hdn = sup J νω, hω. ω Moreover, for any in R, J hdn = N J N J hdn. s a corollary of this proposition, we have for any R- simple L-mapping s that I sdπ = γπ I ; s J sdn = N J N δ N J ; s. Proposition 15 Let be an element of R let h j j J be a family of R-measurable L- mappings. Then inf h j sup h j are L- measurable. Moreover, I inf h jdπ = inf I h j dπ J sup h j dn = sup J h j dn. Proposition 16 Let j j J be a family of elements of R let h be a R-measurable L- mapping. Then I hdπ = inf I h j dπ j j J j hdn = sup J hdn. j In [7] it was shown that seminormed semiconormed fuzzy integrals can be used to extend the domain of possibility respectively necessity measures from measurable sets to measurable fuzzy sets. Propositions 7 15 tell us that something similar can be done with implicator coimplicator integrals. For any R-measurable L mapping h, define Π I h = I hdπ N J h = J hdn. Then clearly Π I is infimum preserving N J supremum preserving. Moreover, for any in R, Π I χ = N I Πco N J χ = N J Nco. n implicator integral therefore allows us to turn a possibility measure Π into an extended necessity measure Π I, dually, a coimplicator integral allows us to turn a necessity measure N into a possibility measure N J. If N I N J are strong negators, then Π I is the extension to fuzzy sets of the dual necessity measure Π N 1 of Π w.r.t. N 1 I I dually, N J is the extension to fuzzy sets of the dual possibility measure N N 1 J of N w.r.t. N 1 J. cknowledgements Gert de Cooman is a Postdoctoral Fellow of the Belgian National Fund for Scientific Research NFWO. He would like to thank the NFWO for partially funding the research reported on in this paper. References [1] B. De Baets. Coimplicators, the forgotten connectives. Tatra Mountains Math. Publ. submitted. [2] B. De Baets. Model implicators their characterization. In N. Steele, editor, Proceedings of the First ICSC International Symposium on Fuzzy Logic Zürich, Switzerl, May 26 27, 1995, pages ICSC cademic Press, [3] B. De Baets. Oplossen van vaagrelationele vergelijkingen: een ordetheoretische benadering [Solving fuzzy relational equations: an ordertheoretic approach]. PhD thesis, Universiteit Gent, in Dutch. [4] G. de Cooman. Possibility theory I III. International Journal of General Systems. in press. [5] G. de Cooman E. E. Kerre. mple fields. Simon Stevin, 67: , [6] G. de Cooman E. E. Kerre. Order norms on bounded partially ordered sets. The Journal of Fuzzy Mathematics, 2: , [7] G. de Cooman E. E. Kerre. Possibility necessity integrals. Fuzzy Sets Systems, 77: , [8] G. de Cooman, D. Ruan, E. E. Kerre, editors. Foundations pplications of Possibility Theory Proceedings of FPT 95, Ghent, Belgium, December 1995, Singapore, World Scientific. [9] D. Dubois H. Prade. Théorie des possibilités. Masson, Paris, [10] F. Suárez García P. Gil Álvarez. Two families of fuzzy integrals. Fuzzy Sets Systems, 18:67 81, [11] M. Sugeno. The Theory of Fuzzy Integrals Its pplications. PhD thesis, Tokyo Institute of Technology, Tokyo, [12] P.-Z. Wang. Fuzzy contactability fuzzy variables. Fuzzy Sets Systems, 8:81 92, [13] L.. Zadeh. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Systems, 1:3 28, 1978.