Kybernetika. Masahiro Inuiguchi Calculations of graded ill-known sets. Terms of use: Persistent URL:

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1 Kybernetka Masahro Inuguch Calculatons of graded ll-known sets Kybernetka, Vol. 50 (04), No., 6 33 Persstent URL: Terms of use: Insttute of Informaton Theory and Automaton AS CR, 04 Insttute of Mathematcs of the Czech Academy of Scences provdes access to dgtzed documents strctly for personal use. Each copy of any part of ths document must contan these Terms of use. Ths document has been dgtzed, optmzed for electronc delvery and stamped wth dgtal sgnature wthn the project DML-CZ: The Czech Dgtal Mathematcs Lbrary

2 K Y B E R N E T I K A V O L U M E 5 0 ( 0 4 ), N U M B E R, P A G E S CALCULATIONS OF GRADED ILL-KNOWN SETS Masahro Inuguch To represent a set whose members are known partally, the graded ll-known set s proposed. In ths paper, we nvestgate calculatons of functon values of graded ll-known sets. Because a graded ll-known set s characterzed by a possblty dstrbuton n the power set, the calculatons of functon values of graded ll-known sets are based on the extenson prncple but generally complex. To reduce the complexty, lower and upper approxmatons of a gven graded ll-known set are used at the expense of precson. We gve a necessary and suffcent condton that lower and upper approxmatons of functon values of graded ll-known sets are obtaned as functon values of lower and upper approxmatons of graded ll-known sets. Keywords: ll-known set, lower approxmaton, upper approxmaton Classfcaton: 03E7, 6E5, 68T37. INTRODUCTION Varous models have been proposed to represent uncertanty: probablty theory [8], fuzzy sets [0], belef functons [7], possblty theory [], random sets [5], rough sets [6], and so on. Most of those models treat the uncertanty of a sngle-valued varable whle others treat the uncertanty of a set-valued varable. Person s heght, weght and age, the stock prce at expraton and the cost of a cab rde between certan places, etc., are consdered sngle-valued varables because the true values of those are unque. On the other hand, someone s favorte food, person s belongngs, the day when a person stays n Osaka, canddate for the research topc, and so on are consdered set-valued varables because the true values are not always unque. The former s called a dsjunctve varable whle the latter s called a conjunctve varable (see [3, 9]). Whle a dsjunctve varable takes a value of the unverse, a conjunctve varable takes a subset of the unverse. Then the set of possble realzatons of conjunctve varables becomes a collecton of subsets and the number of possble realzatons s exponentally many. Therefore, unlke that of a dsjunctve varable, the treatment of the uncertanty of a conjunctve varable becomes complex. Moreover, a conjunctve varable mght be consdered less encountered n the real world than a dsjunctve varable. Because of those possble reasons, conjunctve varables have been studed much less than dsjunctve varables. Nevertheless, several models such as belef functons [], ll-known sets [] DOI: /kyb

3 Graded ll-known sets 7 and graded ll-known sets [4] to represent uncertanty of conjunctve varables have been proposed. A varaton range of a sngle-valued (dsjunctve) varable can be seen as a conjunctve varable (see [4]). For example, to predct the varaton range of a stock prce, we should treat t as a conjunctve varable. Moreover, the predcted values of stock prce by many experts can also be treated by a conjunctve varable. In addton, from ts orgnal defnton, a conjunctve varable s useful to represent a set-valued varable, e. g., a satsfactory range, feasble range, members of a group, and so on. From these ponts of vew, conjunctve varables can be encountered as often as dsjunctve varables. In ths paper, we concentrate on the graded ll-known set as a model of conjunctve varable and nvestgate the calculatons of graded ll-known sets. An ll-known set s a subset whose members are not known exactly. They can be represented by a famly of subsets that can be true. A graded ll-known set s an ll-known set represented by a famly of subsets wth possble degrees. In other word, a graded ll-known set can be represented by a possblty dstrbuton on the power set. The treatments of graded llknown sets are prmtvely very complex because ther manpulatons are defned n the power set n prncple. Namely, the number of elements n the power set s exponental, and thus the processng of graded ll-known sets usually requres an exponental order of computatons. Lower and upper approxmatons of graded ll-known sets are proposed for ts smplfed model. Generally speakng, lower approxmaton s composed of sure members whle upper approxmaton s composed of possble members. In some real world problems, we may know only lower and upper approxmatons and, n ths case, t s shown that possblty and necessty measures of graded ll-known sets are calculated by ts lower and upper approxmatons [, 4]. Because lower and upper approxmatons are defned n the unverse, the treatments of those approxmatons are much computatonally less than the treatments of graded ll-known sets. Therefore the results about possblty and necessty measures of graded ll-known sets defned by ther approxmatons are very computatonally advantageous. In ths paper, we show a smlar result about functon calculatons of graded ll-known sets. We ntroduce the extenson prncple to graded ll-known sets to calculate functon values of graded ll-known sets. The calculatons of functon values wth graded ll-known sets are performed on the power set. Therefore, as descrbed earler, t would requre a lot of computatonal efforts. In some real world applcatons, t can be suffcent to know the lower and upper approxmatons of the functon value of graded ll-known sets. From ths pont of vew, we consder the lower and upper approxmatons of functon value of graded ll-known sets. We nvestgate the necessary and suffcent condton for the lower and upper approxmatons of functon value of graded ll-known sets to be calculated by the lower and upper approxmatons of gven graded ll-known sets. Moreover, we gve some smpler suffcent condtons useful for the applcatons of graded ll-known sets to varous felds. Ths paper s organzed as follows. In Secton, we brefly revew graded ll-known sets. The man results on calculatons of graded ll-known sets are gven n Secton 3. In Secton 4, a smple example s gven. Concludng remarks are gven n Secton 5.

4 8 M. INUIGUCHI. GRADED ILL-KNOWN SETS Let X be a unverse. Let A be a crsp set whose members are not known exactly. For example, consder student partcpants of a conference held 0 years ago n a laboratory under Prof. X. Prof. X knows that there were sx students at that tme, say a, b, c, d, e and f. However, hs memory s not certan. He s sure that three students attended the conference and f was absent at the conference. Moreover, he remembers that a and b attended the conference. From ths memory, we know that the set of the student partcpants n Prof. X s laboratory was {a, b, c}, {a, b, d} or {a, b, e}. Such a crsp set wth mprecse members s called an ll-known set. To represent an ll-known set, collectng possble realzatons of A, we obtan the followng famly: A = {A, A,..., A n }, () where A s a crsp set such that A = A s consstent wth the partal knowledge about A. Gven A, we obtan a set of elements whch certanly belong to A, say A and a set of elements whch possbly belong to A, say A + are defned as A = A = =,...,n A, A + = A = =,...,n A. () We call A and A + the lower approxmaton of A and the upper approxmaton of A, respectvely. In the prevous example about the student partcpants of a conference n Prof. X s laboratory, we may defne X = {a, b, c, d, e, f}, A = {a, b, c}, A = {a, b, d} and A 3 = {a, b, e}. Then we have A = {a, b} and A + = {a, b, c, d, e}. A concdes wth the sure partcpants n Prof. X s memory and X A + = {f} concdes wth the sure nonpartcpants n Prof. X s memory. In the real world, we sometmes may know sure members and sure non-members of A only. In other words, we know the lower approxmaton A as a set of sure members and the upper approxmaton A + as a complementary set of sure non-members. Gven A and A + (or equvalently, the complement of A + ), we obtan a famly  of possble realzatons as  = {A A A A + }. (3) We note that A and A + are recovered by applyng () to the famly  nduced from A and A + by (3). On the other hand, a gven famly A of () cannot be always recovered by applyng (3) to A and A + defned by (). For example, n the example of the student partcpants of a conference n Prof. X s laboratory, A s not recovered. If all A s of () are not regarded as equally possble, we may assgn a possblty degree π A (A) to each A X so that A X, π(a) =. (4) A possblty dstrbuton π A : X [0, ] can be seen as a membershp functon of a fuzzy set A n X. Thus, we may dentfy A wth A. The ll-known set havng such a possblty dstrbuton s called a graded ll-known set.

5 Graded ll-known sets 9 For example, assume Prof. X feels {a, b, c} s most concevable and {a, b, d} s more concevable than {a, b, e} n the settng of the prevous example. Hs feelng may be expressed by a possblty dstrbuton π A ({a, b, c}) =, π A ({a, b, d}) = 0.6, π A ({a, b, e}) = 0.3 and π A (A) = 0 for any other subset A X = {a, b, c, d, e, f}. In ths case, the lower approxmaton A and the upper approxmaton A + are defned as fuzzy sets wth the followng membershp functons (see Dubos and Prade [], Inuguch [4]): µ A (x) = nf ( π A (A)), µ A +(x) = sup π A (A). (5) A X x A We have the followng property: A X x A x X, µ A (x) > 0 mples µ A +(x) =. (6) In the example of possblty dstrbuton for student partcpants of a conference n Prof. X s laboratory, we obtan µ A (a) = µ A (b) =, µ A (c) = 0.4 and µ A (x) = 0 for x {d, e, f}. On the other hand, we obtan µ A +(a) = µ A +(b) = µ A +(c) =, µ A +(d) = 0.6, µ A +(e) = 0.3 and µ A +(f) = 0. Because the specfcaton of possblty dstrbuton π A may need a lot of nformaton, as s n the usual ll-known sets, we may know only the lower approxmaton A and the upper approxmaton A + as fuzzy sets satsfyng (6). The consstent possblty dstrbuton π A for any A and A + s not unque. However, the followng possblty dstrbuton πa (A ) s the maxmal possblty dstrbuton among the consstent possblty dstrbutons ( ) πa(a) = mn nf ( µ A(x)), nf µ A +(x), (7) x A x A where we defne nf =. We dentfy the maxmal possblty dstrbuton πa (A) wth the gven fuzzy sets A and A + unless the other nformaton s avalable. For example, when π A of X = {a, b, c, d, e, f} are gven by µ A (a) = µ A (b) =, µ A (c) = 0.4 and µ A (x) = 0 for x {d, e, f}, and µ A +(a) = µ A +(b) = µ A +(c) =, µ A +(d) = 0.6, µ A +(e) = 0.3 and µ A +(f) = 0, we obtan πa ({a, b}) = 0.6, π A ({a, b, c}) =, πa ({a, b, d}) = 0.6, π A ({a, b, e}) = 0.3, π A ({a, b, c, d}) = 0.6, π A ({a, b, c, e}) = 0.3, πa ({a, b, d, e}) = 0.3, π A ({a, b, c, d, e}) = 0.3 and π A (A) = 0 for any other A X. When A and A + are obtaned from a possblty dstrbuton π A, πa obtaned from A and A + through (7) s not always same as the orgnal π A. We only have π A (A) πa (A), A X. Indeed, ths fact can be observed n the examples above. Namely, the possblty dstrbuton π A defned for the student partcpants of a conference n Prof. X s laboratory has lower and upper approxmatons A and A + whch are used for the calculaton of πa n the example above. We observe π A(A) πa (A), A X. 3. EXTENSION PRINCIPLE FOR GRADED ILL-KNOWN SETS In ths paper, we consder graded ll-known sets n real lne R and nvestgate the calculatons of graded ll-known sets n R. Graded ll-known sets n real lne R are called

6 0 M. INUIGUCHI graded ll-known sets of quanttes. The set of graded ll-known sets of quanttes s denoted by IQ. Because graded ll-known sets are characterzed by possblty dstrbutons on the power set whch can be seen as a membershp functon of a fuzzy set n the power set, the functon values of graded ll-known sets of quanttes can be defned by the extenson prncple [] n fuzzy set theory. When a functon ψ : ( R ) m R s gven, we extend ths functon to a functon from IQ m to IQ n the followng defnton. Defnton 3.. Let A, =,,..., m be graded ll-known sets of quanttes. Gven a functon ψ : ( R ) m R, the mage ψ(a, A,..., A m ) s defned by a graded ll-known set of quanttes assocated wth the followng possblty dstrbuton: π ψ(a,a,...,a m)(y ) sup mn (π A (Q ), π A (Q ),..., π Am (Q m )), f ψ (Y ), Q =,Q,...,Q m R Y =ψ(q,...,q m) 0, f ψ (Y ) =, (8) where π A s a possblty dstrbuton assocated wth graded ll-known set of quanttes A and ψ s the nverse mage of ψ. Note that, functon f : R m R can be extended to a functon f : ( R ) m R by f(a, A,..., A m ) = {f(x, x,..., x m ) x A, =,,..., m}. The extended functon f : ( R ) m R can be further extended to a functon f : IQ m IQ by Defnton 3.. The calculaton of ψ(a, A,..., A m ) s very complex because we should consder all elementary sets of power set R. Ths mples that at least an exponental order of calculatons are requested. In ths paper, we nvestgate the necessary and suffcent condton for the lower and upper approxmatons of ψ(a, A,..., A m ) to be calculated n smaller order of complexty when ψ s the extenson of f : R m R. The lower and upper approxmatons provdes the approxmated values and, n some specal cases, the exact values (see [, 4]), Therefore t s very useful to know those approxmatons. We obtan the followng theorem about the upper approxmaton. Theorem 3.. The upper approxmaton f + (A, A,..., A m ) of f(a, A,..., A m ) can be calculated by upper approxmatons of A, =,,..., m. More concretely, we obtan µ f + (A,A,...,A m)(y) = sup π f(a,a,...,a m)(y ) y Y = sup x,x,...,x m R y=f(x,x,...,x m) mn(µ A + (x ), µ A + (x ),..., µ A + m (x m ))) = µ f(a +,A+,...,A+ m) (y), (9)

7 Graded ll-known sets where µ f + (A,A,...,A m) s the membershp functon of f + (A, A,..., A m ) and µ A + s the membershp functon of the upper approxmaton A + of A. Smlarly, µ f(a +,A+,...,A+ m) s the membershp functon of the mage f(a +, A+,..., A+ m). P r o o f. It can be proved straghtforwardly from the defntons. For the lower approxmaton, we only have an nequalty as shown n the followng theorem. Theorem 3.3. The membershp functon of lower approxmaton f (A, A,..., A m ) of f(a, A,..., A m ) s not smaller than that of f(a, A,..., A m),. e., µ f (A,A,...,A m)(y) = nf y Y ( π f(a,a,...,a m)(y )) sup x,x,...,x m R y=f(x,x,...,x m) mn(µ A (x ), µ A (x ),..., µ A m (x m )) = µ f(a,a,...,a m) (y), (0) where µ A s the membershp functon of lower approxmaton A of A. µ f(a,a,...,a m) s the membershp functon of the mage f(a, A,..., A m) of fuzzy sets A, A,..., A m. P r o o f. For the sake of smplcty, we prove when m =. In cases where m, t can be proved n the same way. From the defnton, we have µ f (A,A )(y) = nf y Y sup A,A Y =f(a,a ) mn (π A (A ), π A (A )) = nf max (( π A (A )), n (π A (A ))). A,A y f(a,a ) Assume µ f(a )(y) α. By defnton of f(a,a, A ), there exst x and x such that y = f(x, x ), ( A x, n(π A (A )) α) and ( A x, n(π A (A )) α). Under ths assumpton, we prove µ f (A,A )(y) α. For all A and A such that y f(a, A ), from the assumpton, we have x A or x A. Moreover, from the assumpton, x A mples n(π A (A )) α ( =, ). Hence, applyng those to the equaton above, we obtan µ f (A,A )(y) α. The equalty of (0) does not hold generally but n specal cases. In the followng secton, we nvestgate the necessary and suffcent condton for the equalty of (0).

8 M. INUIGUCHI 4. THE MAIN RESULT AND ITS IMPLICATIONS We obtan the followng theorem. Theorem 4.. We have f (A, A,..., A m ) = f(a, A,..., A m),. e., µ f (A,A,...,A m)(y) = nf y Y ( π f(a,a,...,a m)(y )) = sup x,x,...,x m R y=f(x,x,...,x m) f and only f mn(µ A (x ), µ A (x ),..., µ A m (x m )) = µ f(a,a,...,a m) (y), () α [0, ), {f(q, Q..., Q m ) Q (A ) α, (A ) α,..., Q m (A m ) α } ( = f (A ) α, (A ) α,..., ) (A m ) α, () where (A ) α = {Q π A (Q) > α}. P r o o f. For the sake of smplcty, we prove when m =. In cases where m, t can be proved n the same way. From Theorem, we consder the necessary and suffcent condton of Ths s equvalent to µ f (A,A )(y) µ f(a,a )(y). α (0, ], µ f (A,A )(y) α mples µ f(a,a )(y) α. ( ) Then we consder the equvalent condton of (a) µ f (A,A )(y) α and that of (b) µ f(a,a )(y) α. Frst let us nvestgate the equvalent condton of (a). By defnton, we have µ f (A,A )(y) α nf Y y ( π f(a,a )(Y )) α y Y mples π f(a,a )(Y ) α π f(a,a )(Y ) > α mples y Y sup mn(π A (Q ), π A (Q )) > α mples y Y Q,Q :Y =f(q,q ) y {f(q, Q ) Q (A ) α, Q (A ) α }.

9 Graded ll-known sets 3 Now let us nvestgate the equvalent condton of (b). By defnton, we obtan ( ) µ f(a,a )(y) α sup mn µ A (x ), µ A (x ) α x,x :y=f(x,x ) ε > 0, x, x, y = f(x, x ), µ A (x ) > α ε, µ A (x ) > α ε ε > 0, x, x, y = f(x, x ), nf Q x ( π A (Q )) > α ε, =, ε > 0, x, x, y = f(x, x ), ( Q x, π A (Q ) < α + ε), ( Q x, π A (Q ) < α + ε) ε > 0, x, x, y = f(x, x ), x {Q π A (Q ) α + ε}, =, x, x, y = f(x, x ), x {Q π A (Q ) > α}, =, ( y f (A ) α, ) (A ) α. From those equvalent condtons of (a) and (b), the necessary and suffcent condton of ( ) s obtaned as α [0, ), {f(q, Q ) Q (A ) α, Q (A ) α } = f ( (A ) α, (A ) α ). The necessary and suffcent condton for f (A, A,..., A m ) = f(a, A..., A m) obtaned n Theorem 4. s not easly confrmed. Then we wll gve a suffcent condtons whch are easly confrmed. To ths end, we defne a class IQ nt IQ of graded ll-known sets of quanttes A satsfyng the followng propertes: α [0, ), A(α) = (A) α s nonempty and convex, and there exsts a famly of convex sets {Q j } j J such that Q j (A) α, j J and A(α) = j J Q j. (3) A graded ll-known set of quanttes A satsfyng (3) can be seen as an extenson of an nterval n R. Then IQ nt s consdered the set of ll-known ntervals. Then we obtan the followng theorem. Theorem 4.. Let f : R m R be contnuous and monotone (monotoncally ncreasng or monotoncally decreasng wth respect to each argument). Let A IQ nt, =,,..., m. Then we have (),. e., f (A, A,..., A m ) = f(a, A..., A m). P r o o f. By the same reason as Theorem 4., we prove when m =. Wthout loss of generalty, we assume f s monotoncally ncreasng wth respect to all arguments. From A (α) = (A ) α Q for Q (A ) α, f(a (α), A (α)) {f(q, Q ) Q (A ) α, Q (A ) α }. Then we prove y f(a (α), A (α)) mples y {f(q, Q ) Q (A ) α, Q (A ) α }. ( )

10 4 M. INUIGUCHI Because f s contnuous and A (α), =, are nonempty and convex, f(a (α), A (α)) becomes an nterval (a convex set n the real lne). Then we prove ( ) dvdng nto two cases: (a) y nf f(a (α), A (α)) and y f(a (α), A (α)) and (b) y sup f(a (α), A (α)) and y f(a (α), A (α)). Because A IQ nt, there exsts a famly Q of convex sets {Q j } j J such that Q j (A ) α and A (α) = j J Q j for =,. From the convexty of Q j, j J, =,, there exst subfamles Q = {Q j } j J Q and Q = {Q j } j J Q such that sup j J nf Q j = nf A (α) and nf j J sup Q j = sup A (α). From the monotoncty, we obtan r A (α), r A (α), y < f(r, r ) mples k J, k J, q Q k, q Q k, y < f(q, q ), r A (α), r A (α), y > f(r, r ) mples l J, l J, q Q l, q Q l, y > f(q, q ). Therefore, n case (a) y nf f(a (α), A (α)) and y f(a (α), A (α)), we have y f(q, Q ). Ths mples that y {f(q, Q ) Q (A ) α, Q (A ) α }. Smlarly, n case (b) y sup f(a (α), A (α)) and y f(a (α), A (α)), we have y f(q, Q ). Ths mples that y {f(q, Q ) Q (A ) α, Q (A ) α }. Hence, ( ) s proved. If A (α) = (A ) α (A ) α, =,,..., m for any α [0, ), we have (). From Theorem 4., we have the followng corollary. Corollary 4.3. If A (α) (A ) α, =,,..., m for any α [0, ), then we have (),. e., f (A, A,..., A m ) = f(a, A,..., A m). P r o o f. It suffces to prove that Q Q mples f(q, Q,..., Q m) f(q, Q,..., Q m). Ths s obvous from defnton, f(q, Q,..., Q m ) = {f(x, x,..., x m ) x Q, =,,....m}. When A (α) (A ) α, =,,..., m for any α [0, ), we have () wthout any condton on f. The strong condton A (α) (A ) α, =,,..., m for any α [0, ) s satsfed by a graded ll-known set of quanttes defned by lower and upper approxmatons. Ths can be understood drectly from the followng proposton. Proposton 4.4. Let A be a graded ll-known set defned by lower and upper approxmatons A and A +. Then we have { } (A) α = A [A ] α A (A + ) α, (4) where [A ] β s a weak β-level set of A,. e., [A ] β = {x µ A (x) β}, β (0, ] whle (A + ) γ s a strong γ-level set of A +,. e., (A + ) γ = {x µ A +(x) > γ}, γ [0, ).

11 Graded ll-known sets 5 P r o o f. From (7), we obtan the followng equvalences: A (A) α nf ( µ A(x)) > α and nf µ A +(x) > α x A x A (x A mples µ A +(x) > α) and (µ A (x) α mples x A) [A ] α A (A + ) α. From Proposton 4.4, we know that A(α) = [A ] α f A s defned by lower and upper approxmatons A and A +. Because A + s not related to A(α), we may have a weaker suffcent condton for A (α) (A ) α. Namely, we obtan the followng theorem. Theorem 4.5. If the possblty dstrbuton π A of a graded ll-known set of quanttes A satsfes π A (A) = nf ( µ x A A (x)), A such that nf ( µ x A A (x)) nf µ x A A + (x), =,,..., m, (5) we have (),. e., f (A, A,..., A m ) = f(a, A,..., A m), where A and A + are lower and upper approxmatons of A, respectvely, and µ A and µ A + are ther membershp functons. P r o o f. From Corollary 4.3, t suffces to prove A (α) (A ) α under condton (5). Because ( ) π A (A ) πa (A) = mn nf ( µ x A A (x)), nf µ x A A + (x) nf ( µ x A A (x)), we obtan π A (A ) > α nf ( µ x A A (x)) > α [A ] α A [A ] α A (α). Now we prove [A ] α = A (α) = (A ) α usng (5). From (6), for ε (0, α), we have nf ( µ A (x)) nf µ x [A ]αε A + (x). x [A ]αε From (5), we obtan π A ([A ] αε) = nf ( µ A (x)) α + ε > α. x [A ]αε Namely, we have [A ] αε (A ) α for any ε (0, α). From the property of weak level set, we have [A ] αε [A ] α and ε (0,α) [A ] αε = [A ] α. Hence, we obtan [A ] α = A (α)

12 6 M. INUIGUCHI 5. CASES OF F ( A, A+,..., A M, A+ M ) = F (A,..., A M ), F (A+,..., A+ M ) In ths secton, we nvestgate cases where f( A, A+, A, A+,..., A m, A + m ) = f(a, A,..., A m), f(a +, A+..., A+ m). (6) Contrary to our expectaton, (6) does not always hold. Counter examples are gven as follows. Example 5.. Let us consder a functon f : R R defned by x + x, f x + x 6, f (x, x ) = 0, f x + x (6, 0], x + x 4, f x + x > 0. Let A and A be ll-known sets defned by lower approxmatons A = [, 3] and A = [, 3] and upper approxmatons A+ = [, 7] and A+ = [, 8], respectvely. Then we have [4, 8] f (A, A ) = f ( A, A+, A, A+ ), but {0} [4, 8] f (A, A ) = f ( A, A+, A, A+ ). On the other hand, we obtan f (A, A ) = [4, 6] and f (A +, A+ ) = {0} [, ]. Then we have [4, 8] f (A, A ), f (A +, A+ ). Therefore, we have f ( A, A+, A, A+ ) f (A, A ), f (A +, A+ ). Even when functon s contnuous and monotone, we have a smlar result. Consder the followng example. Example 5.. Consder a functon f : R R defned by f (x, x ) = x + x. Let A and A + ( =, ) be the same as above,. e., A = [, 3], A = [, 3], A + = [, 7] and A+ = [, 8]. We have f (A, A ) = [4, 6] and f (A +, A+ ) = [, 5]. Then [4, 6] [, ] f (A, A ), f (A +, A+ ). On the contrary, [4, 6] [, ] f ( A, A+, A, A+ ). Ths s because there s no Q R and Q R such that f (Q, Q ) = [4, 6] [, ]. From the examples above, we know that we may have π f( A,A+, A,A+,..., A m,a + m ) (Y ) = π f(a,a,...,a m),f(a + (Y ), (7),A+,...,A+ m) only for Y f( R,..., R ). The followng theorem shows that (7) holds for a convex set Y R and a monotone contnuous functon.

13 Graded ll-known sets 7 Theorem 5.3. Let f : R m R be contnuous and monotone. Let A and A + be fuzzy sets showng lower and upper approxmatons of a graded ll-known set A, =,,..., m. Then (7) holds for a convex set Y R. P r o o f. We prove (7) when m =. (7) can be proved n the same way even when m >. Let f (Y ) = {Q Q R f(q, Q ) = Y } for Y R and f (y) = {(x, x ) f(x, x ) = y} for y R. For the sake of smplcty, we defne graded llknown sets A = A, A+, =, and F = f(a, A ), f(a+, A+ ). The proof s gven n two complementary cases: () f (Y ) = and () f (Y ). Frst we consder () f (Y ) =. Suppose y Y, f (y). Then y Y, (Q Q ) f (y) mples f(q, Q ) Y. It s obvous that there exsts ˆQ and ˆQ such that ( ˆQ ˆQ ) f (y). Let q L = nf ˆQ and q R = sup ˆQ, =,. Because of the monotoncty, we have f(q L, q L ) nf Y and f(q R, q R ) sup Y. Thus, we may fnd 0 λ L and 0 λ R such that f(( λ L )q L + λ L q R, ( λ L )q L + λ L q R ) = nf Y, and f(( λ R )q R + λ R q R, ( λ R )q R + λ R q R ) = nf Y, because of the contnuty and monotoncty of f. Then we fnd convex sets Q R and Q R such that nf Q = ( λ L )q L + λ L q R, nf Q = ( λ L )q L + λ L q R, sup Q = ( λ R )q R + λ R q R, sup Q = ( λ R )q R + λ R q R, where Q and Q nclude ther nfmums f Y ncludes ts nfmum, and Q and Q nclude ther supremums f Y ncludes ts supremum. For Q and Q, because of the contnuty of f, we have f( Q, Q ) = Y. Ths contradcts f (Y ) =. Therefore we know f (Y ) = mples y Y, f (y) =. By Defnton 3., we have π f(a,a )(Y ) = 0 from f (Y ) =. Moreover, nf y Y µ f(a + )(y) = 0 because µ,a+ f(a + )(y) = 0 for f (y) = from the extenson,a+ prncple n fuzzy sets. Ths mples π F (Y ) = 0. Hence, we have (7) when f (Y ) =. We now consder a case where f (Y ). From the assumpton, we have f (A, A ) = f(a, A ) and f + (A, A ) = f(a +, A+ ). Because π F s the maxmal possblty dstrbuton of graded ll-known sets havng lower and upper approxmatons f(a, A ) and f(a+, A+ ), π f(a,a )(Y ) π F (Y ). Therefore, we prove π f(a,a )(Y ) π F (Y ). ( ) Moreover, because A = A, A+, =,, we have π f(a,a )(Y ) = sup mn (π A (Q ), π A (Q )) Q,Q R Y =f(q,q ) = sup mn Q,Q R Y =f(q,q ) ( ( mn nf ( mn nf ( µ x Q A (x)), nf ( µ x Q A (x)), nf µ x Q A + µ x Q A + ) (x), )) (x).

14 8 M. INUIGUCHI expense Expert Expert Expert 3 L k [0, 3] [8, ] [, ] U k [7, 7] [8, 8] [6, 5] ncome Expert 4 Expert 5 Expert 6 L k [0, 3] [, 5] [9, 4] U k [9, 5] [8, 6] [7, 7] Tab.. Expense and ncome estmatons (, 000 $). From Defnton 3. and Proposton 4.4, we obtan π f(a,a )(Y ) > α (Q, Q ) such that Y = f(q, Q ), [A ] α Q (A + ) α and [A ] α Q (A + ) α. (#) On the other hand, from the extenson prncple n fuzzy sets, we obtan ( ) π F (Y ) = mn nf mn µ y Y A (x ), µ A (x ), nf y Y sup x,x R y=f(x,x ) sup x,x R y=f(x,x ) ( mn µ A + In the same way of the proof of Proposton 4.4, we obtan (x ), µ A + (x ) ). π F (Y ) > α ε>0 f([a ] αε, [A ] αε) Y f((a + ) α, (A + ) α). ( ) Now we prove ( ) by showng π F (Y ) > α mples π f(a,a )(Y ) > α. For lower and upper approxmatons A and A +, we have µ A (x)>0 mples µ A + (x)=. Then we obtan [A ] α [A ] αε (A + ) α for any ε > 0 and for =,. Assume π F (Y ) > α, from ( ), we obtan f([a ] α, [A ] α) ε>0 f([a ] αε, [A ] αε) Y. In the same way that we prove f (Y ) = mples y Y, f (y) =, from the contnuty and monotoncty of f, the convexty of Y and f([a ] α, [A ] α) Y f((a + ) α, (A + ) α), we fnd [A ] α Q (A + ) α, =, such that Y = f(q, Q ) (see Fgure ). Hence, from (#), we obtan π f(a,a )(Y ) > α. 6. A SIMPLE EXAMPLE In order to gve an mage of graded ll-known sets n the real world as well as to demonstrate the effcency n computaton owng to (9) and (), we consder a vrtual proft

15 Graded ll-known sets 9 Fg.. [A ]α Q (A + )α. estmaton problem. There s a small project requrng some expenses but producng ncomes n future. To estmate the expected proft of the project, we asked sx experts. Three of them are good at the estmaton of expenses whle the other three are good at the estmaton of ncomes. Although they are experts, due to the uncertan envronment, they cannot estmate them n unvocal values. Ther estmatons are twofold: hghly possble ntervals L k and somehow possble ntervals U k such that L k U k. As shown n Table, we assume that the estmatons of expenses are L k and U k, k =,, 3 whle the estmatons of ncomes are L k and U k, k = 4, 5, 6 Usng L k and U k such that L k U k, k =,,..., 6, the possblty dstrbutons π A, =, about expenses A and ncomes A are defned by π A (A) = {k L k A U k, k [3, 3]}, =,, (8) 3 where B s the cardnalty of set B. For the normalty of π A, we assume k=3,3,3 L k k=3,3,3 U k, =,. Moreover, to satsfy (3), we assume k=3,3,3 L k, =,, otherwse Â( 3 ) = (A) 3 =, =,. Here, we note that the nformaton of experts can be modeled by a basc probablty assgnment Bpa : R [0, ], =, such that Bpa (C) = 3, f k [3, 3], C = {A L k A U k }, 0, otherwse. Then π A, =, of (8) can be seen as contour functons of Bpa, =,,. e., we have π A (A) = Bpa (C), =,. (0) A C R (9)

16 30 M. INUIGUCHI For =,, let us defne S = M = A R A R k=3,3 k=3,3,3 k=3,3 L k A L k A L k A k=3,3 U k k=3,3,3 k=3,3 or U k, k=3,3 U k, () L k A k=3,3 W = {A R L 3 A U 3, L 3 A U 3 or L 3 A U 3 }. U k, () (3) Then π A, =, are obtaned by π A (A) =, f A S, 3, f A S and A M, 3, f A M and A W, 0, otherwse. (4) From () to (4), we easly confrm that A, =, satsfy (3),. e., A IQ nt, =,. Then we can apply Theorem 4.. Let us calculate the range of proft A A and ensure Theorems 3. and 4.. Frst we apply Defnton 3. to A A. For parameters shown n Table, we obtan π AA (A), f A S S, = 3, f A S S and A M M, 3, f A M M and A W W, 0, otherwse, =, f A S,, f A S and A M, 3, f A M and A W, 3 0, otherwse, (5)

17 Graded ll-known sets 3 where we defne S = {A R [6, 7] A [4, 7]}, (6) M = {A R [7, 7] A [, 7], [6, 7] A [, 8], [8, 7] A [4, 7], [7, 7] A [3, 8], [7, 5] A [4, 8], [6, 5] A [3, 9] or [6, 4] A [4, 8]}, (7) W = {A R [9, 5] A [, 9], [6, 4] A [0, 0], [, 7] [0, 8], [8, 6] A [, 9], [8, ] A [4, 9], [0, 4] A [3, 0] or [7, 3] A [, ]}. (8) Applyng (5), we obtan the followng lower and upper approxmatons (A A ) and (A A ) + : µ (x) (AA ), f x [, ],, f x [8, ) (, 4], 3 =, f x [6, 8) (4, 7], 3 0, otherwse, µ +(x) (AA ), f x [4, 7],, f x [, 4) (7, 9], 3 =, f x [, ) (9, ], 3 0, otherwse. (9) Now let us calculate A A and A+ A+. For =,, we defne S = L k, S + = U k, M = M + = W = k=3,3,3 k=3,3 k=3,3 L k U k L k k=3,3 k=3,3 U k L k and W + = k=3,3,3 k=3,3 k=3,3 L k, U k, k=3,3,3 k=3,3,3 U k. (30) For parameters gven n Table, we obtan S = [, ], S+ = [8, 5], M = [0, ], M + = [7, 7], W = [8, 3], W + = [6, 8], S = [, 3], S+ = [9, 5], M = [0, 4], M + = [8, 6], W = [9, 5] and W + = [7, 7], We note that those sets always become closed nterval because of k=3,3,3 L k, =,. Then the lower and upper approxmatons of A, =, are obtaned as, f x S ±, µ A ± (x) = 3, f x S± and x M ±, 3, f x M ± and x W ±, 0, otherwse, (double sgn n same order). (3)

18 3 M. INUIGUCHI Applyng the extenson prncple n fuzzy set theory, we obtan µ A A (x), f x [, ],, f x [8, ) (, 4], = 3, f x [6, 8) (4, 7], 3 0, otherwse, µ A + A+ (x), f x [4, 7],, f x [, 4) (7, 9], = 3, f x [, ) (9, ], 3 0, otherwse. (3) We have (A A ) = A A and (A A ) + = A + A+. Then we could confrm Theorems 3. and 4.. Even n ths smple case, the calculaton of A A s rather complex because we should consder all combnatons of mnmal and maxmal elements of α-level sets. A part of the complexty can be observed n the defntons of M and W. On the other hand, as demonstrated above, calculatons of (A A ) and (A+ A+ ) are much smpler. 7. CONCLUDING REMARKS We nvestgated the calculatons of graded ll-known sets. We showed that the lower and upper approxmatons of functon values of graded ll-known sets are obtaned rather easly n some cases whle the exact calculatons are complex. We revealed the necessary and suffcent condton that lower and upper approxmatons of functon values of graded ll-known sets are obtaned by functon values of lower and upper approxmatons of graded ll-known sets. Usng ths condton, we gave the suffcent condtons. From one of them, we know that the lower and upper approxmatons of functon values of graded ll-known sets defned by lower and upper approxmatons are always obtaned by functon values of the gven lower and upper approxmatons. Moreover, we gave counterexamples to show that functon values of graded ll-known sets defned by lower and upper approxmatons do not always equal to graded ll-known sets defned by functon values of the gven lower and upper approxmatons whle ther lower and upper approxmatons are. However the possblty dstrbutons correspondng to those functon values may take same membershp values at functon mages of sets. We showed a suffcent condton that those membershp values are equal. The results obtaned n ths paper are valuable for applcatons of graded ll-known sets to systems optmzaton, decson makng, data analyss and so on. Those applcatons would be future topcs. ACKOWLEDGEMENT Ths work was supported by Grant-n-Ad for Scentfc Research (C), No (Receved February 8, 03)

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