Soft Retrieval and Uncertain Databases

Transcription

1 th Hawaii Intenational Confeence on System Science Soft Retieval and Uncetain Databases Ronald R. Yage Iona College Rachel L. Yage Metopolitan College of New Yok Abstact We investigate tools that can enich the pocess of ueying databases by enabling the inclusion of moe human focused consideations. We show how to include soft conditions with the use of fuzzy sets. We descibe some moe sophisticated techniues fo aggegating the satisfactions of the individual conditions based on the inclusion of impotances and the use of the OWA opeato. We discuss a method fo aggegating the individual satisfactions that can model a lexicogaphic elation between the individual euiements. We look at ueying databases in which the infomation in the database can have some pobabilistic uncetainty. 1. INTRODUCTION Database stuctues play a pevasive ole in undelying many of ou most pominent websites, as a esult the intelligent etieval of infomation fom databases is an impotant task. The use of soft concepts often allows us to model human cognitive concepts. Hee we focus on poviding a famewok fo etieving infomation fom database stuctues via soft ueying [1-4]. A uey can be seen as a collection of euiements and an impeative fo combining an object's satisfaction to the individual euiements to get the object's oveall satisfaction to the uey. We descibe some fuzzy set based methods that enable the inclusion of soft human focused concepts in the constuction of a uey [5-8]. Hee we also descibe sophisticated techniues fo aggegating the satisfactions of the individual conditions to obtain an object's oveall satisfaction to a uey. The inclusion of impotances and the use of the OWA opeato ae some tools that povide this facility. Hee we also discuss a method fo aggegating the individual satisfaction's which can model a lexicogaphic elation between the individual euiements. In addition we look at databases in which the infomation can have some pobabilistic uncetainty 2. STANDARD DATABASE QUERYING A database consists of a collection of attibutes, V j, j = 1 to and a domain X j fo each attibute. Associated with a database is a collection of objects D i, the objects in the database. Each D i is an tuple, (d i1, d i2,..., d i ). Hee d ij X j is the value of attibute V j fo object D i. A uey Q is of a collection of pais. Each pai P k = (V Q(k), F Q(k) ) consists of an attibute V Q(k ) and an associated popety fo that attibute F Q(k). Hee Q(k) indicates the index of the attibute in the k th pai. Thus if Q(k) = i then the attibute in the kth pai is V i. In addition to the collection of pais a uey contains infomation egading the elationship between the pais. We denote this infomation as Agg. The ueying pocess consists of testing each object D i to see if it satisfies the uey. The pocess of testing an object consists of two steps. The fist is the detemination of the satisfaction of each of the pais by the object. We denote this as T (P k D i ) o moe simple as t ik. The second step is to calculate the oveall satisfaction of D i to the uey Q, Sat(Q/D i ), by combining the individual t ik, guided by the instuctions contained in Agg. Hee then Sat(Q/D i ) = Agg(t i1,..., t i ) = T i The calculation of t ik, T [P k D i ] is based on detemining whethe the condition F Q(k) is satisfied based on d iq(k), the value of V Q(k) fo the i th object D i. In the standad envionment F Q(k) is epesented as a cisp subset of X Q(k) indicating the desied values of V Q(k). The detemination of satisfaction, T [P k D i ], is based on whethe d iq(k) F Q(k). The system etuns the value t ik = 1 if d iq(k) F Q(k) and t ik = 0 if d iq(k) F Q(k). Using the notation F Q(k) (d iq(k) ) to indicate membeship of d iq(k) in F Q(k) we get t ik = F Q(k) (d iq(k) ). Thus in this case each of the values t ik {0, 1} whee 0 indicates false, not satisfied, and 1 indicates tue, satisfied. In this binay envionment the aggegation pocess is constucted as a well fomed logical statement consisting of conjunctions, disjunctions, negations and implications. The oveall satisfaction Sat(Q D i ) is eual to the tuth-value of this well-fomed fomula. The calculation of this tuth-value involves the use of Min, Max and Negation. Fo example the euiement that all pais in Q ae satisfied is expessed as T i = Min k = 1 to [t ik ]. The euiement that any of the conditions need be satisfied is expessed as T i = Max k = 1 to [t ik ]. We note this is a binay envionment, T i {0, 1}, and hence we have two categoies of objects, those which satisfy the uey and those that do not /14 $ IEEE DOI /HICSS

2 3. Soft database ueying An extension of the ueying pocess involves the use of soft o flexible ueies [6, 8]. This idea extends the pocess of ueying standad databases in a numbe of ways. The most fundamental is to allow the seach citeia, the F Q(k), to be expessed as fuzzy subsets of the domain X Q(k). This allows one to epesent impecise non-cisply bounded concepts. In this case the detemination of the satisfaction of a citeia P k = (V Q(k), F Q(k) ) by an object D i, t ik, again euals the membeship gade of d iq(k) in F Q(k), t ik = F Q(k) (d iq(k) ). Howeve in this case t ik [0, 1], it takes a value in the unit inteval athe than simply in the binay set {0, 1}. An impotant implication of this is that the oveall satisfaction of the element D i will also lie in the unit inteval. This will allow fo a iche and moe disciminating odeing of the elements in egad to thei satisfaction to the uey then simply satisfied o not, hee we get a degee of satisfaction. We note the evaluation of the oveall satisfaction to a uey Q whose Agg opeato is based on a well fomed logical fomula can be implemented in this famewok. In paticula conjuncting the satisfactions to two citeia pais P k1 and P k2 is implemented by Min(t ik1, t ik2 ). Disjuncting the satisfactions to two citeia pais P k1 and P k2 is implemented by Max(t ik1, t ik2 ). The negation of the satisfaction to P ki is implemented as 1 t ik1. A second benefit obtained by using flexible ueies is the allowance fo moe sophisticated fomulations fo the opeato Agg used to detemine the oveall satisfaction to the uey [6]. Let us look at some of these extensions. A fist class of extensions is a genealization of the opeations used fo implementing the "anding" and "oing" opeato. Assume P 1 and P 2 ae two conditions whose satisfaction fo D i ae t i1 and t i2. Classically the "anding" of the satisfactions to these two citeia has been implemented using the Min opeato: P 1 and P 2 Min(t i1, t i2 ). The "oing" of the satisfactions to these two citeia has been implemented using the Max opeato: P 1 o P 2 Max(t i1, t i2 ). The t-nom opeato is a genealization of the "anding" opeato [9], it povides a class of cointensive opeatos that can be used to implement the conjunction. We ecall a t-nom is a binay opeato T: [0, 1] [0, 1] [0, 1] having the popeties: symmety, associativity, monotonicity and has one as the identity. Notable examples of t-nom in addition to the Min ae T P (a, b) = a b and T L (a, b) = Max(0, a + b - 1). Since fo any t-nom T we have T(a, b) Min(a, b) then using othe t-noms inceases the penalty fo not completely satisfying the conditions. The t-conom opeato povides a genealization of the oing opeato. A t-conom is a mapping S: [0, 1] [0, 1] [0, 1] which has the same fist thee popeties of the t-nom binay, condition fou is eplaced by S(0, a) = a, zeo as Identity. In addition to the Max othe notable examples of t- co-nom ae S P (a, b) = a + b - ab and T L (a, b) = Min(a + b, 1). We note fo any t nom S we have S(a, b) Max(a, b). Anothe extension to the standad situation is the inclusion of impotances associated with the conditions [10, 11]. Hee we assume each of the conditions P k has an associated impotance weight w k [0, 1]. The methodology fo including impotances depends on the aggegation elationship between the citeia, it is diffeent fo "anding" and "oing". Assume fo object D i we have that t ik [0, 1] as its degee of satisfaction to P k. Conside fist the case in which we desie an "anding" of the P k. That is ou aggegation impeative is expessed as (P 1, w 1 ) and (P 2, w 2 )... and (P, w ). In this case the weighted aggegation esults in an oveall the satisfaction Min(h 1,..., h ) whee h k = Max((1 - w k ), t ik ) [12]. It is inteesting to obseve the standad situation is obtained when all w k = 1. In this case we get Min k [t ik ]. We can also obseve that since Max((1 - w k ), t ik ) t ik then Min k [Max((1 - w k ), t ik )] Min k [t ik ]. Thus the use of impotances is effectively to educe the euiements, we ae easing the difficulty. We also obseve that moe geneally we can eplace Min with any t-nom T and Max with any t conom S thus we can use T [S((1 - w k ), t ik )] [10]. Conside now the case whee we have an "oing" of the citeia. Thus hee ou aggegation impeative is (P 1, w 1 ) o (P 2, w 2 ) o (P 3, w 3 ) o,..., (P, w ). In this case the weighted aggegation and the satisfaction is Max(g 1, g 2,..., g ) whee g k = Min(w k, t ik ) [10, 13]. We obseve the standad situation is obtained when all w k = 1. In this case we get Max k [t ik ]. We can also obseve that since Min(w k, t ik ) t ik then Max k [Min(w k, t ik )] Max k [t ik ]. Thus the use of impotances educes the influence of individual citeia.. We also obseve that moe geneally we can eplace Min with any t-nom T and Max with any t conom S thus we can use S [T(w k, t ik )]. An inteesting obsevation can be made with espect to the weighted "anding" and "oing" obsevations. In the case of the weighted oing, inceasing the weights associated with the citeia can only esult in an incease in the oveall satisfaction while in the case of the "anding" inceasing the weights can only esult in a decease in the satisfaction. Moe fomally if we let w k and w k be two pais of weights such that w k w k and let g k = T(w k, t ik ) and let g k = T( w k, t ik ) then fo any t conom S, we have S(g 1,..., g ) S( g 1,..., g ). On the othe hand if we let h k = S((1 w k ), t ik ) and h k = S((1-935

3 w k ), t ik ) then fo any t-nom T, T(h 1,..., h ) T( h 1,..., h ). We can make the following obsevation about the weighted "oing" aggegation in the case of the using S = Max. In this case ou oveall aggegation is Max(g 1,..., g ) whee g k = T(w k, t ik ) fo a t-nom T. We futhe obseve that if w k < 1 then fo any T and t ik we have g k < 1. Fom this we can conclude that in the case of the weighted "o" aggegation using S = Max we can neve get an oveall satisfaction of one unless at least one at the citeia conditions has impotance of one. We note this is a necessay but not sufficient condition fo getting an oveall satisfaction of one. A coesponding esult can be obtained in the case of the weighted "anding" aggegation in the case of using T = Min. Recall the definition h k = S( w k, t ik ) fo any t-conom S, fom this we can obseve that if w k < 1, then w k > 0 and hence fo any S and t ik we have h k w k > 0. Since ou oveall aggegation is Min(h 1,..., h ) we can conclude that in this case of weighted "anding" aggegation we can not get an oveall satisfaction of zeo unless at least one of the citeia conditions has impotance weight of one. An inteesting fomulation fo the weighted "oing" aggegation occus when we use S(a, b) = Min(a + b, 1) and T(a, b) = a b. In this case (P 1, w 1 ) o (P 2, w 2 ) o...o (P, w ) evaluates to Min( w k t ik, 1 ). Thus hee we take a simple weighted sum and then bound it by the value one. A coespondingly inteesting fomulation fo the weighted "anding" can be had if we use T(a, b) = Max(a + b - 1, 0) and S( a, b) = a + b - ab. In this case T(g 1,..., g ) = 1 - Min[ w k t ik, 1] The OWA opeato [14] can povide a useful fomulation fo the aggegation pocess in flexible ueying of databases. We ecall the OWA opeato is aggegation opeato F: I n I so that F(a 1,..., a n ) = j w j b j whee b j is the j th lagest of the a i and w j ae collection of n weights such that w j [0, 1] and j w j = 1. We note that if is an index function so (j) is the index of the j th lagest a i, then b j = a (j) and hence F(a 1,..., a n ) = j w j a (j). It is common to efe to the collection of w j using an n vecto W whose j th component is w j, in this case we efe to W as the OWA weighting vecto. We obseve that if w 1 = 1 and w j = 0 fo j 1 then F(a 1,..., a n ) = Max i [a i ]. If w n = 1 and w j = 0 fo j n then F(a 1,..., a n ) = Min i [a]. If w j = 1/n fo all the j then F(a 1,..., a n ) = 1 n i a i. The OWA opeato is a mean opeato, it is monotonic, symmetic and bounded, It is also idempotent The OWA opeato can be vey diectly used in flexible ueying. Assume we have a uey with n component pais, P k = (V Q(k), F Q(k) ) whee F Q(k) is a fuzzy subset ove the domain of V Q(k). As in the peceding fo any object D i we can obtain t ik = F Q(k) (d iq(k) ). We can then povide an aggegation of these using the OWA opeato, F(t i1, t i2,..., t in ). Essentially this povides a kind an aveage of the individual satisfactions. We can denote this OWA W (D i ). Diffeent choices of the OWA weights will esult in diffeent types of aggegation. If we use the weights such that w 1 = 1 then we get Max k [t ik ] which is essentially an "oing" of the components. If we use a weighting vecto W such that w n = 1 we get Min k [t ik ] which is an "anding" of the conditions. If w j = 1/n then we ae taking a simple aveage of the citeia satisfactions. Using uantifies and paticulaly linguistic uantifies we can use the OWA opeato to povide a vey ich class of aggegation opeatos [15]. A uantifie o popotion can be seen as a value in the unit inteval. In [16] Zadeh genealized the concept of uantifie by intoducing the concept of linguistic uantifies. Examples of these linguistic uantifies ae: most, about half, all, few and moe then pecent. Zadeh [16] suggested that one can epesented these linguistic uantifies as fuzzy subsets of the unit inteval. This if R is a linguistic uantifie we can epesent it as a fuzzy subset R of the unit inteval so that fo any y I, R(y) is the degee to which the popotion y satisfies the concept R. An impotant class of uantifies ae egula monotonic uantifies. These uantifies have the popeties: R(0) = 0, R(1) = 1 and R(x) R(y) fo x > y. Thus fo these uantifies the satisfaction inceases as the popotion inceases. Examples of these ae "at least p", "most", "some", "all" and "at least one". In [15] Yage showed how to use the fuzzy subsets associated with these uantifies to geneate the weights of an OWA opeato. In paticula he suggested we obtain the weights as w j = R( j j 1 ) - R( n n ). It can be shown if R is egula monotonic then the weights sum to one and lie in the unit inteval. Using these ideas we can obtain a uantifie guided appoach to flexible ueying. We can expess a uey as collection of n pais, P k = (V Q(k), F Q(k) ) and an aggegation impeative in tems of a linguistic uantifie R descibing the popotion of these pais we euie to be satisfied. We then expess this uantifie R as a fuzzy subset of the unit inteval, R. Using this R we obtain a collection of OWA weights, w j = R( j j 1 ) - R( ) fo j = 1 to n. Once having these weights n n we can evaluate the satisfaction of each element D i as OWA R (D i ). In [17] we showed how to extend this to the case impotance weighted citeia. 4. Lexicogaphic fomulated ueies In the database uey stuctue pesented hee we impose a pioity odeing on the conditions composing the uey. In this appoach conditions highe in pioity odeing play a 936

4 moe impotant ole in detemining the oveall satisfaction of an object to the uey in the same way that lettes in ealie positions a play in detemining the alphabetical ode. The techniue we shall develop will be called a LEXicogaphic Aggegation (LEXA) uey and it will make use of the pioitized aggegation opeato intoduced in [18-20]. This appoach will allow us to model diffeent uey impeatives than in the peceding. Hee again we shall conside a uey Q to consist of a collection of pais P k and an aggegation impeative. Each pai as in the peceding is of the fom (V Q(k), F Q(k) ) whee V Q(k) indicates an attibute name and F Q(k) is a fuzzy subset of the domain of V Q(k) denoting the desied values fo V Q(k). Again fo each object D i = (d i1,..., d i ) we can obtain the values t ik = F Q(k) (d ik ), the satisfaction of the pai P k by D i. Hee we will conside an aggegation impeative in the spiit of a lexicogaphic odeing. This aggegation impeative euies an odeing of the P k indicating thei pioity in fomulating the oveall satisfaction to the uey. While ou lexicogaphic aggegation uey famewok will wok in the flexible envionment we shall initially conside its pefomance in the binay situation. We now descibe the basic binay LEXA uey. In this case it is assumed that the t ik ae binay eithe 1 o 0, P k is satisfied by D i o not. In the following we shall assume thee ae m elements in the database, D i fo i = 1 to m. As we subseuently see the output of a LEXA uey is an odeing of the elements in D i accoding to thei satisfaction to lexicogaphic uey. A lexicogaphic aggegation uey assumes a linea pioity among the citeia, the P k, with espect to thei impotance. Fo simplicity we shall assume that the P k have been indexed accoding to this pioity odeing: P 1 > P 2 >... > P. The following algoithm descibes the pocess fo foming the odeed list L 1) Initiate: k = 1, D = {D i i = 1 to m}, list L empty 2) Set F = 3) Test all elements D i D with espect P k and place all those fo which t ik = 0 in F. 4) Place all elements in F tied at the top available level of L. 5) Set D = D - F 6) If D = Stop 7) Set k = k + 1 8) If k > a) Place all emaining elements in D tied as top level of L b) Stop 9) Go to step 2 The esult of this is the list L which is an odeed list of the satisfactions of the elements in D to the lexicogaphic uey Q. We note the highe up the list the bette the satisfaction. Thee is anothe way we can geneate the satisfaction odeing of the elements in D unde the pioity odeing P 1 >... > P. Fo each D i, stating fom P 1 and peceding in inceasing ode find the fist index k fo which t ik = 0. Assign D i a scoe S i whee S i = if D i meets no failues S i = k - 1 othewise Using this method each D i gets a scoe S i {0, 1,..., }. If we ode the elements by thei value fo S i we get L. We note that S i is the numbe of citeia D i satisfied befoe it meets failue. This method fo evaluating the scoe of satisfaction of an element D i to a LEXA uey is moe useful then the peceding as it povides a scoe is addition to an odeing. It is inteesting to note thee is only one situation in which an element D i can attain the maximal scoe S i =, if all t ik = 1. On the othe hand thee ae many ways D i can get the minimal scoe of S i = 0. In paticula any D i which has t i1 = 0 will have S i = 0 egadless of the values of t ik fo k 1. We futhe obseve that since 0 S i we can povide a kind of nomalization by defining G i = S i. Hee we ae getting a value in the unit inteval. We shall intoduce a moe geneal implementation of the lexicogaphic aggegation type uey that will be appopiate fo the case whee the t ik ae values fom [0, 1] athe then being binay values fom {0, 1}. In this envionment fo each element D i we shall obtain a scoe G i indicating its satisfaction to ou LEXA uey. Again hee we have a uey Q consisting of condition pais, P k = (V Q(k), F Q(k) ). In addition we have a pioity odeing ove the pais guiding the lexicogaphic aggegation, P 1 > P 2 > P 3... > P. Assume fo object D i we have t ik = F Q(k) (d ik ) as the degee to which D i satisfies P k. Ou pocedue fo detemining G i is to calculate G i = w ik t ik Whee the w ik ae detemined as follows. 1) Set u i1 = 1 k 2) u ik = t i( j 1) fo k = 2 to j=2 3) w ik = 1 u ik The impotant obsevation hee is that the w ik is popotional to the poduct of the satisfactions of the highe pioity conditions. We note that we can expess u ik = u ik-1 H ik-1. We can make some obsevations. Fo k 1 < k 2 we always have w ik1 w ik2. Thus a lowe pioity condition can't have a bigge weight than one that is highe. We obseve that if t ik1 = 0 then w ik2 = 0 fo all k 2 > k 1 We can show that G i = 1 if and only t ik = 1 fo all k. 937

5 We also obseve that G i = 0 iff t i1 = 0. It is impotant to emphasize that this is independent of the satisfaction to the othe citeia. Zeo satisfaction to the highest pioity citeion means zeo oveall satisfaction, thee is no possibility fo compensation by othe citeia. Let us look at this pocess fo the binay case to see that it coectly genealizes the evaluation in the binay case we pesented ealie. Conside the list of satisfactions: t i1, t i2,..., t i which hee we assume ae eithe one o zeo. Assume t ik is the fist one of these that is zeo, hence t ij = 1 fo j < k. In this case we get that u ij = 1 fo j k and u jk = 0 fo j > k. Hence G i = w ij t ij = 1 1 u ij t ij = t ij Futhemoe since t ij = 1 fo j < k and t ij = 0 fo j = k we get G i = 1 k k 1 1 t ij = 1 = k 1 as desied. In the special case whee all t ij = 1 then all u ij = 1 and all w ij = 1 and hence G i = = 1. As discussed in [18] the appoach we suggested can be extended to the case when the odeing among the pais is a weak odeing, we allow ties among the conditions pais with espect to thei pioity. Hee we shall denote a condition pai as P (k, j) whee all pais with the same k value ae tied in the pioity odeing. The j value is just an indexing distinguishing among the tied pais. Thus hee we ae assuming the pioity odeing is such that fo k 1 < k 2 that P (k1, j) > P (k 2, i) and P (k 1, j) = P (k 1, i) fo all j and i. We let n k denote the numbe of pais with k in the fist tem, thus n 1 is the numbe of conditions with the highest pioity. We shall also let n = n k. Hee we let t i(k, j) denote the satisfaction of the object D i to the condition pai P (k, j). In this case we obtain the value G i as n k G i = w ik ( t i( j,) ) =1 Again it is a weighted sum of the satisfaction to each of the pais. We emphasize that each pai with the same k has the same weight w ik. The pocedue we use to obtain the weights is simila to the ealie one except in the fist step: 1. Calculate: H ij = Max [t i( j,) ] =1ton j (It is the satisfaction value of the least satisfied pai at the j level) 2. u i1 = 1 k u ik = H i(j 1) fo k = 2 to 3. w ik = 1 n u ik Let us see how this plays out in the pue binay case whee all t i(k, ) {0, 1}. Let b be the k value at which we meet the fist condition fo which t i(k, ) = 0. In this case we have that H ij = 1 fo j < b and H ij = 0 fo j = 0. Hee then n while u i1 = 1 we have fo k = 2 to that u ik = H i(j 1). Fom the above we get that u ik = 1 2 k b u ik = 0 k > b Thus we get w ik = 1 n fo k = 1 to b and w ik = 0 fo k > b. b 1 Fom this we get G i = w ik ( t i(k,) ) = n ( t i(k,) ) =1 =1 Fo all k < b all the elements have t i(k,) = 1 hence G i = 1 b 1 n b n ( n k + t i(b,) ) =1 We note at least one element in the second tem is zeo. We see that the numeato of G i is eual to the numbe of elements in the pioity classes highe than the fist class whee we meet failue plus all the pais satisfied in the class that we meet failue. An altenative expession of this numeato is the sum of all satisfied pais in pioity classes up to and including the class when we meet ou fist failue. We hee point to elated wok by Chomicki [22-26] on pefeences in databases and the on bipola ueies by Dubois and Pade [27, 28] and Zadozny and Kacpzyk [29, 30] 5. Queying pobabilistic databases In the peceding we have discussed seveal diffeent paadigms fo fomulating uestions to databases. In paticula we consideed a uey to be a collection of condition pais, P k = (V Q(k), F Q(k) ), and an agenda fo aggegating the satisfactions to these pais by a database element D i. We added flexibility by allowing F Q(k) to be a fuzzy subset of the domain X Q(k) of V Q(k). In this flexible case we obtained the degee of satisfaction of P k by D i to be t ik = F Q(k) (x iq(k) ) [0, 1] whee x iq(k) is the value of V Q(k) fo D i. Thus t ik is the degee to which F Q(k) is tue given the value of V Q(k) fo D i is x iq(k). We now shall conside the situation in which the value of V Q(k) fo D i, x iq(k), is andom. Moe specifically if the domain X Q(k) = {y Q(k)1, y Q(k)2,..., y Q(k)Q(k) } then the knowledge of the value of V Q(k) fo object D i, x iq(k), is best expessed as a pobability distibution P iq(k) whee Pob(x iq(k) = y Q(k)j ) = p j/iq(k). In the special case whee n k n k 938

6 P j/q(k) = 1 fo some j then we have the peceding situation in which x iq(k) is exactly y Q(k)j. Initially in the following discussion we shall assume all the Q(k) ae distinct. We shall subseuently conside the case whee a uey can involve multiple occuences of the same attibute. The appoach we shall follow in this pobabilistic situation is in the spiit of the possible wods appoach used in [31-39]. We shall associate with evey D i a collection Z of tuples. In paticula Z = X Q(1) X Q(2)... X Q(). That is each element z Z is of the fom z = (z 1, z 2,..., z ) whee z k X Q(k). Fo each object D i in the database we now associate a pobability distibution PD i ove the space Z so that the pobability of z, PD i (z) = Pob(x iq(k) = z k ). We want to make one comment hee. In the peceding we assumed all the V Q(k) in a uey whee distinct. This is not a necessay euiement. Howeve in the case whee the we have a multiple occuences of the same attibute in the uey cae must be taken in the fomulation of the possible wolds. In the peceding we fomulated Z = X Q(1) X Q(2) X Q(), hee each element z Z is of the fom z = (z 1, z 2,, z ) whee each z k X Q(k). In ode to undestand which happens when all V Q(k) ae not distinct we conside the situation whee Q(1) = Q(2), hee then V Q(1) = V Q(2). Hee the associated Z space has the additional euied that fo all z Z, z 1 = z 2. Thus any element in Z must have the same value of V Q(1) and V Q(2). In addition we must make some modifications in the calculation of PD i (z). In paticula we must note duplicate the pobabilities and avoid using them twice, thus PD i (z) must be based on the poduct of the distinct pobabilities and hence we have PD i (z) = Pob(x iq(k) = z k ) fo all distinct Q(k) Anothe comment we want to make is egading the elationship between distinct attibute values. Implicit in the peceding has been an assumption of independence between the pobability distibution of distinct attibutes. In some cases this may not be tue. Fo example some values of V 1 may not be allowable unde V 2 while some values fo V 1 may mandate a paticula value fo V 2. Such elationships euie us to modify to possible elements in Z and condition the pobability distibution in known ways. Nevetheless when including these special conditions we still end up with a subset of tuples fom Z with associated pobabilities, as a esult in the following we shall neglect any special elationships between the attibutes as they don't effect the subseuent discussion no the basic ideas of the appoach intoduced. We now ecall that ou uey consists of a collection of constaints P k = (V Q(k), F Q(k) ) whee each F Q(k) is a fuzzy subset ove the space X Q(k). We now apply these constaints to the space of possible wolds Z. In paticula we tansfom the space Z to F so that each tuple z = (z 1,..., z ) is tansfomed to new tuple F(z) = (F Q(1) (z 1 ), F Q(2) (x 2 ),, F Q()( z )) We note each element F Q(k) (x k ) [0, 1]. Thus each tem in F(z) is a tuple of values dawn fom the unit inteval, F(z) is a subset the space I. In addition fo each element D i in the database we can associate a pobability distibution ove the space F. In paticula fo each D i we associate PD i (z) with the tuple F(z). Thus now fo any element D i we have a pobability distibution on the space F of satisfactions to the uey components. We note that if two z tuples, z 1 and z 2, tansfom into the same value, F(z 1 ) = F(z 2 ), we epesent these in F with just one, F(z 1 ), and use the sum of the pobabilities z 1 and z 2 In ode to povide an intuitive undestanding of the discussion to follow we shall use the following database to illustate ou ideas Example 3. We conside a database with thee attibutes V 1, V 2 and V 3. The domain of these ae espectfully: X 1 = {a, b, c}, X 2 = {ed, blue}, X 3 = {10, 20} Let D be one object in the database and let x 1, x 2, x 3 denote the values of V 1, V 2, V 3 fo this object. In paticula fo each of these, x 1, x 2, x 3, we have a pobability distibution x 1 : Pob(a) = 0.4, Pob(b) = 0.5 and Pob(c) = 0.1 x 2 : Pob(ed) = 0.7 and Pob(blue) = 0.3 x 3 : Pob(10) = 0.6 and Pob(20) = 0.4 In this example the set of Z of possible wolds ae Z = {(a, ed, 10), (a, ed, 20), (a, blue, 10), (a, blue, 20), (b, ed, 10), (b, ed, 20), (b, blue, 10), (b, blue, 20), (c, ed, 10), (c, ed, 20), (c, blue, 10), (c, blue, 20)}. Fo the element D we get the pobability of each of these components as shown in Table 1. Befoe peceding we want to point out that fo any othe object D* in the database the set of possible wolds Z will be the same, the diffeence between D and D* will be in the pobabilities associated with the elements in Z. We now conside ou uey as consisting of thee components: P 1 = (V 1, F 1 ), P 2 = (V 2, F 2 ) and P 3 = (V 3, F 3 ) whee each F k is a fuzzy set ove the space X k defining the euied condition. Below ae the associated fuzzy subsets F 1 = { 1 a, 0.6 b, 0.2 c }, F 2 = { 0.8 ed, 1 blue }, F 3 = { 1 10, } We now can apply these constaints to ou set Z of possibilities and also use the pobabilities fo D and shown the esults in Table 2. Table 1. Pobabilities of Elments in Z Element in Z Pobability (a, ed, 10) (0.4) (0.7) (0.6) = (a, ed, 20) (0.4) (0.7) (0.4) = (a, blue, 10) (0.4) (0.3) (0.6) = (a, blue, 20) (0.4) (0.3) (0.4) = (b, ed, 10) (0.5) (0.7) (0.6) =

7 (b, ed, 20) (0.5) (0.7) (0.4) = 0.14 (b, blue, 10) (0.5) (0.3) (0.6) = 0.09 (b, blue, 20) (0.5) (0.3) (0.4) = 0.06 (c, ed, 10) (0.1) (0.7) (0.6) = (c, ed, 20) (0.1) (0.7) (0.4) = (c, blue, 10) (0.1) (0.3) (0.6) = (c, blue, 20) (0.1) (0.3) (0.4) = Table 2. Pobabiities of Tansfomed Elements Element in Z Tansfom F(z) Pobability (a, ed, 10) (1, 0.8, 1) (a, ed, 20) (1, 0.8, 0.2) (a, blue, 10) (1, 1, 1) (a, blue, 20) (1, 1, 0.2) (b, ed, 10) (0.6, 0.8, 1) 0.21 (b, ed, 20) (0.6, 0.8, 0.2) 0.14 (b, blue, 10) (0.6, 1, 1) 0.09 (b, blue, 20) (0.6, 1, 0.2) 0.06 (c, ed, 10) (0.2, 0.8, 1) (c, ed, 20) (0.2, 0.8, 0.2) (c, blue, 10) (0.2, 1, 1) (c, blue, 20) (0.2, 1, 0.2) We now conside the issue of evaluating the uey Q fo the object D in the database. Let us ecapitulate the situation. Associated with D we have a space F of tuples T j = (F 1 (z 1 ),..., F (z )}, each tuple is an element fom the space I. That is each tuple is a collection of values that the unit inteval. In addition associated with each tuple in F we have a pobability. Paenthetically we note that the space F of tuples is the same fo each element D i in the database, the only distinction between the elements in the database ae the pobabilities associated with each of the tuples. At this point we can efomulate moe simply. We have a subset F I with abitay element T j = (t j1, t j2,..., t j ). Futhemoe fo any database element D i we have a pobability distibution ove the space F whee p ij is the pobability associated with the tuple T j fo the database object D i. We n z note that p ij = 1 whee n z is the cadinality of the space Z. In addition associated with a uey is an aggegation impeative that dictates how we aggegate the satisfactions to the individual components, in paticula Agg(T j ) = Agg(t j1, t j2,..., t j ). Afte applying the Agg opeation to the tuples in the space F we end up with a collection of scala values, Agg(F), consisting of the elements Agg(T j ). Fo each D i we have a pobability distibution ove the set of Agg(T j ). In paticula fo each D i and each Agg(T j ) we have p ij as its pobability. We now illustate the peceding with ou ealie example and conside the application of diffeent aggegation impeatives. Example 4. We have the collection of 12 tuples and thei associated pobabilities. In Table 3 we conside thee aggegation impeatives the fist impeative is an "anding" of all conditions, hee Agg(T j ) = Min k (t jk ), the second impeative is an aveage of all conditions, Agg(T j ) = 1 t jk and the thid impeative is an "oing" of all conditions, hee Agg(T j ) = Max k (t jk ).. Table 3. Aggegated Value of Elements in F Tuple in F Pobability and Aveage oing T 1 = (1, 0.8, 1) T 2 = (1, 0.8, 0.2) T 3 = (1, 1, 1) T 4 = (1, 1, 0.2) T 5 = (0.6, 0.8, 1) T 6 = (0.6, 0.8, 0.2) T 7 = (0.6, 1, 1) T 8 = (0.6, 1, 0.2) T 9 = (0.2, 0.8, 1) T 10 = (0.2, 0.8, 0.2) T 11 = (0.2, 1, 1) T 12 = (0.2, 1, 0.2) In a simila way we can implement any aggegation impeative. Combining the pobabilities of tuples with the same value fo thei anding we get Table 4. Table 4. Pobabilities of anded Value Value Pob Combining the pobabilities of the tuples with the same value fo thei oing we get Table 5 Table 5. Pobabilities of oed Value Value Pob Let us now summaize ou situation fo a uey we obtained a collection T j of tuples. Associated with the uey is an aggegation opeation that convets T j into a single value Agg(T j ) indicating the oveall satisfaction of the tuple. Fo simplicity we denote these as Agg(T j ) = a j. Essentially we have a collection of degees of satisfaction, a j, of each possible wold to the uey. Finally fo each element D i in the database we have a pobability distibution ove the a j. Thus the satisfaction of the uey by the database object D i is a pobability distibution a 1 p i1 a 2 p i2... a p i. We emphasize that the set a j, which ae the satisfactions of a possible wold to the uey, is the same fo all D i, the diffeence between the D i is eflected in the pobability 940

8 distibution ove the a j, the pobabilities they assign to a possible wold. A natual uestion that aises is how to ode the D i with espect to thei satisfaction to the uey. Hee we look to [40] fo some ideas. If D and ˆD ae two database objects then fo each of these we have a pobability distibution ove the set A = {a 1,..., a }. P ˆP a 1 p 1 ˆP1 a 2 P 2 ˆP2 a j p j ˆPj a p ˆP In the following fo simplicity we shall assume the a j have been indexed so that they ae in inceasing ode, a j > a k if j > k. In the following we shall use the notation D > ˆD if D is a moe satisfying altenative than ˆD. What is clea is that if p j = 1 and ˆP k = 1 and j > k than D > ˆD. One appoach to odeing the P and ˆP is to use the cumulative distibution function, CDF, and the idea of stochastic dominance [41, 42]. j We define CDF P (a j ) = p i, this the pobability that fo i=1 object D the actual satisfaction will be less o eual a j. j Similaly we define CDF ˆp (a j ) = ˆp i. Using this we shall i=1 say D > ˆD if CDF P (a j ) CDF ˆp (a j ) CDF P (a j ) < CDF ˆp (a j ) fo all j fo at least one j We note that if CDF P (a j ) CDF ˆp (a j ) then j j p i ˆp i this implies i=1 i=1 1 p i 1 ˆp i i = j+1 i= j+1 which futhe implies that p i ˆp i. Thus we see i= j+1 i= j+1 that unde this condition object D neve has less pobability associated with the highe satisfaction then ˆD. This povides a easonable justification fo asseting that D > ˆD. While the CDF povides a way fo odeing the elements if the conditions ae met, usually the condition CDF P (a j ) CDF ˆp (a j ) is not satisfied fo all j. This effectively means that the use of stochastic dominance does not povide the kind of geneal appoach necessay to cove all cases. Anothe appoach to compaing the two database elements is a scalaization. Hee we associate with each element a distinct value and then compae these values. Since these ae scala values we ae able to ode them. One appoach to scalaization is to use the expected value of satisfaction of each the elements. Hee EV(D) = a j p j and EV( ˆD ) = a j ˆp j. We then say D > ˆD if EV(D) > EV( ˆD ). If EV(D) = EV(D) then we say they ae tied. It is inteesting to show that if CDF P (a j ) CDF ˆp (a j ) fo all j then EV(D) EV( ˆD ). We shall illustate this fo the case when = 4, the extension to the moe geneal case of will be obvious. Since we assumed a i > a j fo i > j then we can expess a 2 = a > 0 a 3 = a > 0 a 4 = a > 0 We see 4 EV(D) = a j p j = a 1 p 1 + (a )p 2 + (a )p 3 + (a )p EV(D) = a 1 p j + 2 p j + 3 p j + 4 p j j=2 j=3 j=4 As we have aleady shown if CDF p (a i ) CDF ˆp (a i ) fo all i then 4 4 p j ˆp j fo all i. Fom this we see that if j=4 j=4 CDF P (a i ) CDF ˆp (a i ) fo all i then EV(D) EV( ˆD ). A moe geneal appoach to scalaization of the pobabilities of the database elements can be had if we use some ideas fom the OWA aggegation opeato. Hee we let [0, 1] be a measue ou optimism. The moe optimistic the moe we ae anticipating the highe valued satisfactions to occu. Hee then if = 1 we ae always anticipating that a will occu. While if = 0 we ae always anticipating a 1 will occu. We now associate with a given a function f(x) = x 1 whee =. We see that if 0 then and if 1 then 0 and if = 1/2 then = 1. Using this function we obtain a set of OWA weights as follows. Letting S j = p k we get k =i w j = f(s j ) - f(s j + 1 ) fo j = 1 to Using these weights we obtain EV (D) = w j a j. We fist see that if = 1/2 then = 1 and hence w j = p k p k = p j. Hee we get the usual expected k = j k = j+1 941

9 value. We see that if 0, =, since S j < 1 fo all j > 1 we have (S j ) 0 fo j < 1 and (S i ) = 1 hence in this case w 1 = 1 w j = 0 fo j > 1 Fom this we obtain EV 0 (D) = a 1, it is the least satisfaction. If 1, = 0, w = 1, if p > 0 and hence ED 1 (D) = a. Hee we see that in using EV fo the exteme optimistic and pessimistic of all database elements D k will have the same value fo EV (D k ) howeve fo othe values 0 < < 1 as in the case of = 0.5 each of the D k will get its own uniue value fo EV (D k ). Hee then choosing will detemine the odeing of the D k. 6. Conclusion Hee we povided a famewok fo soft ueying of databases. We descibed a soft uey as a collection of euied conditions and an impeative fo combining an objects satisfaction to individual conditions to get its oveall satisfaction. We investigated tools that can enich this pocess by enabling the inclusion of moe human focused consideations. 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