Uncertain database retrieval with measure-based belief function attribute values

Information Sciences 501 (2019) 761–770

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

Uncertain database retrieval with measure-based belief function attribute values Ronald R. Yager a,∗, Naif Alajlan b, Yakoub Bazi b a b

Machine Intelligence Institute, Iona College, New Rochelle, NY 10801 ALISR Laboratory, College of Computer and Information Sciences, King Saud University, Riyadh Saudi Arabia

a r t i c l e

i n f o

Article history: Received 28 August 2018 Revised 27 March 2019 Accepted 28 March 2019 Available online 19 April 2019 Keywords: Large data Database retrieval Uncertain attribute values Theory of evidence Interval-valued

a b s t r a c t We discuss how the Dempster-Shafer belief structure provides a framework for modeling an uncertain value x˜ from some domain X. We note how it involves a two-step process: the random determination of one focal element (set) guided by a probability distribution and then the selection of x˜ from this focal element in some unspecified manner. We generalize this framework by allowing the selection of the focal element to be determined by a random experiment guided by a fuzzy measure. In either case the anticipation that x˜ lies in some subset E is interval-valued, [Bel(E), Pl(E)]. We next look at database retrieval and turn to issue of determining if a database entity with an uncertain attribute value satisfies a desired value. Here we model our uncertain attribute value as x˜ and our desired value as a subset E. In this case the degree of satisfaction of the query E by the entity is [Bel(E), Pl(E)]. In order to compare these interval-valued satisfactions we use the Golden rule representative value to turn the intervals into scalars. We describe an application involving retrieval from a uncertain database. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Given the vast amount of information being digitally captured [11], databases, particularly relational databases [2,19], provide a very useful and popular way of storing this information. A major benefit of relational databases is their highly structure nature. An important task in modern technological applications is the retrieval of objects from a database. The retrieval process involves that of finding entities in the database that satisfy a query Q. It is based on finding objects in database whose attribute values match those in the query. A simple query would be to find all people in the database who were born in New York City. This involves matching the place of birth of an object in the database with New York City, if they match, are same, the person is an entity we are looking. In many cases the information in the database has some uncertainty, for example, if a person has as their place of birth the East Coast of America. In this case the task of matching is not easy. Our major concern here is with the problem of matching a query in the case when an entity in database has an uncertain attribute value. Since uncertainty can take many different forms, i.e. be random or imprecise or otherwise, in this work we consider the situation when the uncertainty in the attribute valued is modeled using a Dempster-Shafer, d-S, like belief structures [31] whose focal elements can be fuzzy sets as well as crisp sets. In addition, to further enhance the range of the types of uncertainty that can be modeled we allow the underlying uncertainty, rather then simply being probabilistic as is the case with the classic Dempster-Shafer structure, we allow it to based on a fuzzy measure [25]. ∗

Corresponding author. E-mail address: [email protected] (R.R. Yager).

https://doi.org/10.1016/j.ins.2019.03.074 0020-0255/© 2019 Elsevier Inc. All rights reserved.

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2. Dempster-Shafer belief structures The Dempster-Shafer belief structure, based on the pioneering work of Arthur Dempster [3–5] and Glenn Shafer [13,14], provides a framework for the representation of uncertain information that manifests both random and imprecise components. Assume V is an uncertain variable that takes is value in the finite space X = {xi for i = 1 to n}. A d-S framework, m, for expressing our knowledge about the variable V can be viewed as follows. We have a set F = {F1 , …, Fq } of non-empty q subsets of X called the focal elements and associated with each Fj is a value α j ∈ [0, 1] where j=1 αj = 1. A random experiment is performed on the space F where α j is the probability that Fj is the outcome of this experiment. Here we shall denote the outcome of this experiment as W, it is a random variable that takes its value in the space F. Thus W takes as its value a subset of X. Once having determined the value for W, Fj∗ , an element is chosen in some unspecified manner from Fj∗ . This chosen element is V. Thus our uncertain variable V is chosen from a randomly selected subset in some unspecified manner. Our interest here is in the probability that V lies in some subset E of X, Probm (E). As is well known with the d-S framework the imprecision in the selection of the element V from Fj∗ results in a situation in which we can only have interval values for Probm (E). Using the notation of Shafer [13] we have Probm (E) = [Belm (E), Plm (E)]. Here Belm (E) is referred to as the belief of E given m and Plm (E) is referred to as the plausibility of E given m. We note that Dempster [3,4] originally referred to Belm (E) as Low-Probm (E), lower probability of E, and Plm (E) as Upp-Probm (E), upper probability of E. Using Zadeh’s concepts of possibility and certainty [35,37] where

Poss(A/B ) = 1 if A ∩ B = ∅ Poss(A/B ) = 0 if A ∩ B =∅ and

Cert(A/B ) = 1 if B ⊆ A Cert(A/B ) = 0 if B ⊂ A then we have that

Pl m ( E ) =

q 

Poss(E/Fj )αj

j=1 q 

Belm (E ) =

j=1

Cert(E/Fj )αj

We see that Plm (E) is the expected possibility of E given Fj and Belm (E) is the expected certainty of E given Fj . Here it is quite clear that Belm (E) ≤ Plm (E) since Cert(E/Fj ) ≤ Poss(E/Fj ) for all j. For any element xi ∈ X we see that

Probm ({xi } ) = [Belm ({xi } ), Plm ({xi } )]. Here

P l m ( {x i } ) =

q 

Poss({xi }/Fj )αj =

j=1 q 

Belm ({xi } ) =

j=1

 j,xi ∈Fj

Cert({xi }/Fj )αj =

Thus Prob({xi }) = [α [ i ] ,



j,xi ∈Fj

αj



j,Fj ={xi }

αj = α[i]

αj ] where α [i] is the probability of any focal element that just contains xi . If the elements

in X are numerical values then we can obtain an expected value of V,

EV(V ) =

n 



xi Probm ({xi } ) =

i=1

n 

xi Belm ({xi } ),

i=1

n 



x i P l m ( {x i } ) ,

i=1

it is an interval value. Even though Dempster showed in [3] that Prob(E) is interval valued, Prob(E) = [Bel(E), Pl(E)], we can consider approximating Prob(E) by a scalar value in this interval. One such scalar value is the pignistic value introduced by Smets [15–17] defined q Card(F ∩E ) as Pig(E) = j=1 Card(jF ) m(Fj ). In this case we divide the weight m(Fj ), α j , associated with the focal element Fj evenly bej

tween all the elements in Fj . We observe that for any subset E of X, Pl(E) ≥ Pig(E) ≥ Bel(E). We see that Pig(E) is the expected proportion of elements in Fj that are also in E. We note the pignistic value is a probability distribution on X with q Card(F ∩{x } ) pk = Pig({xk }) = j=1 Cardj (F )k m(Fj ) [15]. j

Let us now consider some special cases of D-S belief structures. One notable example is the Bayesian belief structure [13]. In this case the focal elements are singletons Fj = {xj } and the weights α j = pj . This corresponds to a probability distribution.  For this belief structure Pl(E) = Bel(E) = Prob(E) = x ∈E pj . j

A belief structure is called a vacuous belief structure if it has one focal element F1 = X with m(F1 ) = 1 and m(Fj ) = 0 for all j = 1. For this belief structure Pl(X) = Bel(X) = 1, Pl(∅) = Bel(∅) = 0 and for all other sets E, Pl(E) = 1 and Bel(E) = 0.

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Another notable example is a simple support function [13]. Here F1 = A and F2 = X where m(F1 ) = α and m(F2 ) = 1 - α . For any arbitrary set E we see that

Pl(E ) = Poss(E/A )α + Poss(E/X )(1 − α ) Bel(E ) = Cert(E/A )α + Cert(E/X )(1 − α ) We see that there are three relationships that can exist between A and E:

1.A ∩ E = ∅, 2.A ⊆ E and 3.A ∩ E = ∅ but A ⊂ E Here we observe in these three cases that 1. Pl(E) = 1 and Bel(E) = 0 2. Pl(E) = 1 and Bel(E) = α 3. Pl(E) = 1 and Bel(E) = 0 A belief structure is called consonant if the focal elements are nested, that is we can index the focal elements so that: Fj ⊆ Fj + 1 that is F1 ⊆ F2 ⊆ F3 … ⊆ Fq In [26] we provided an interval valued Shannon-like quantification of the entropy, S(m), associated with a D-S belief structure m. In particular

S(m ) = [SL (m ), SH (m )] where

SL ( m ) = −

q  j=1 q 

SH ( m ) = −

  

m(Fj )log2 Pl Fj

j =1



 

m(Fj )log2 Bel Fj

[22] [7 ]

It can easily be shown that in the case of a Bayesian belief structure this reduces to the classical Shannon entropy,  S(m) = SL (m) = SH (m) = − ni=1 pi log2 (pi ). Consider now the entropy of a simple support function where F1 = A and F2 = X with m(F1 ) = α and m(F2 ) = 1 - α . For this we case

Plm (F1 ) = Pl(A ) = 1 Plm (F2 ) = Pl(X ) = 1 Bel(F1 ) = Bel(A ) = α Cert(A/A ) + (1 − α )Cert(A/X ) = α Bel(F2 ) = Bel(X ) = 1 Here then

SL (m ) = − − (α log2 (1 ) + (1 − α )log2 (1 ) ) = 0 SH (m ) = − − (α log(α ) + (1 − α )log2 (1 ) ) = − − α log(α ) S(m ) = [0, −α log2 (α )]. We note that since α ≤ 1 then log(α ) < 0 and hence -α log(α ) ≥ 0. In [21] Yager introduced the idea of specificity associated with a D-S belief structure. In particular the specificity assoq ciated with a belief structure m is Sp(m) = j=1 Card1(F ) m(Fj ). We see that the specificity takes its maximum value for a j

Bayesian belief structure with Card(Fj ) = 1 for all j and hence Sp(m) = 1. It takes its minimal value for the vacuous belief structure, where F1 = X and α 1 = 1, in this case Sp(m) = n1 , one over the cardinality of X. Assume m1 and m2 are two belief structures on X with focal elements F = {F1 , …, Fq } and E = {E1 , …, Er } where m1 (Fi ) = α i and m2 (Ej ) = β j . Shafer [13] suggested a procedure, called Dempster’s rule, for obtaining a fused belief structure, m, reflecting an “anding” of these two belief structures denoted m = m1  m2 . Here m is a belief structure with focal  1  elements Ak = Fi ∩ Ej = ∅ and the associated weights are m(Ak ) = 1−T F ∩E =A m1 (Fi ) m2 (Ej ) where T = i,j, F ∩F =∅ m1 ( Fi ) i

j

k

i

j

m2 (Ej ). Thus here we normalize the non-null focal elements. We see that the operator is commutative, m = m1  m2 = m2  m1 . It also can be shown that if one of the belief structures is Bayesian than m is Bayesian. While Dempster’s rule is the most popular method for combining belief structures as pointed out by Zadeh [36] at times it leads to uncomfortable results. Consider m1 and m2 where m1 has focal elements F1 = {x1 } and F2 = {x2 } where m1 (F1 ) = 0.99 and m1 (F2 ) = 0.01 and m2 has focal element E1 = {x3 } and E2 = {x2 } with m2 (E1 ) = 0.99 and m2 (E2 ) = 0.01. Applying Dempster’s rule results in this case leads to m with focal element A1 = {x2 } and m(A1 ) = 1. Clearly this can be seen to be somewhat troubling. This and other difficulties with Dempster’s rule have lead researchers to consider alternative rules for combining belief structures [6]. One alternative to Dempster’s rule suggested by Yager [22] is the following. Here we denote m = m1 ⊥ m2 . Here m has focal elements Ak = Fi ∩ Fj = ∅ and X and the associated weights are

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m ( Ak ) =

i,j,Fi ∩Ej =Ak



m (X ) =

i,j,Fi ∩Ej =X

m1 (Fi )m2 Ej for Ak = Fi ∩ Ej = ∅ and Fi ∩ Ej = X

 

m 1 ( F i )m 2 E j +

 

 i,j,Fi ∩Ej =∅

m1 ( Fi )m2 Ej

Here the weight that would normally go to the empty set is given to the focal element X. We note that for our previous example we would get m with focal elements A1 = {x2 } and A2 = X = {x1 , x2 , x3 }. In the case using Yager’s rule we have m(A1 ) = 0.0 0 01 and m(X) = 0.9999. 3. Measure-based belief structures In [25,28,29] we introduced a more general form of the Dempster-Shafer belief structure called a measure-based belief structure, MBBS, it is described in the following. Assume V, our variable of interest, is an uncertain variable taking its value in the space X = {xi for i = 1 to n}. A measure based belief structure g has the following framework expressing knowledge about the variable V. Again let F = {F1 , …, Fq } be a collection of non-empty subsets of X called the focal elements. Associated with F is a monotonic measure μ: 2F → [0, 1] having the properties: 1. μ(∅) = 0, 2. μ(F) = 1 and 3. μ(B) ≥ μ(A) if A ⊂ B. Instead of directly obtaining information about V we observe information about a related variable W, which contains V. A random experiment is performed to determine the value of the variable W ∈ F. The outcome of this experiment is governed by the measure μ so that for any subset B ⊆ F, μ(B) is the anticipation that the value of W is in the subset B. Here the term “anticipation” can be viewed as a generalization of the term probability to the case of a measure. Once having determined, via our experiment the value of W, some Fj ∈ F, we select an element x∗ from Fj in some unspecified manner. This element, x∗ , is the value of V. Thus we see the choice of V is a two-step process: the first is the determination of a set Fj from the collection of focal elements F via an random experiment governed by μ and the second is the selection from the set Fj of the element V. For any subset E of X we shall refer to Antg (E) as the anticipation that V is an element in E under the MBBS g. We observe that given W = Fj we are not able to determine whether V ∈ E, the best we can do is determine an optimistic (upper) anticipation and pessimistic (lower) anticipation that V ∈ E given W = Fj . The optimistic (upper) anticipation, which we shall refer to as the possibility that V ∈ E given W = Fj , denoted as Poss(E/Fj ), is defined as





Poss E/Fj = 1 if E ∩ Fj = ∅   Poss E/Fj = 0 if E ∩ Fj = ∅ We shall refer to the pessimistic (lower) anticipation as the certainty that V ∈ F given W = Fj , denoted as Cert(E/Fj ), it is defined as





Cert E/Fj = 1 if Fj ⊆ E   Cert E/Fj = 0 if Fj ⊂ E Here then the anticipation that V is in E given W = Fj , Ant(E/Fj ), is in the interval [Cert(E/Fj ), Poss(E/Fj )]. Taking into account the randomness involved in the determination of the value of W requires us to calculate an expected value based on the underlying measure μ. As is well known the calculation expected value in the face of measure based uncertainty involves the use of Choquet integral [1,8,10]. Here then





Expectμ Poss E/Fj











= Choqμ Poss E/Fj , j = 1toq

Following Shafer’s notation is we shall refer to this as the plausibility of E, Plg (E). Thus









Plg (E ) = Choqμ Poss E/Fj , j = 1toq . Similarly





Expectμ Cert E/Fj











= Choqμ Cert E/Fj , j = 1toq

Again following Shafer’s notation we shall refer to this as the belief of E, Belg (E) and hence









Belg (E ) = Choqμ Cert E/Fj , j = 1toq

Using this we obtain the anticipation of E given g,

Antg (E ) ∈ [Belg (E ), Plg (E )]. Let us now look at the calculation of the corresponding Choquet integrals. First consider Plg (E) = Choqμ (Poss(E/Fj ), Fj = 1 to q). Let ρ be an index function so that ρ (i) is the index of the focal element with the ith largest value for Poss(E/Fj ). Using this we get

Pl g ( E ) =

q  i=1

  (μ(Hi ) − μ(Hi−1 ))Poss E/Fρ (i)

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where Hi = {Fρ (k) /k = 1 to i}, the subset of focal elements with the i largest values for Poss(E/Fj ). However the here the situation has very special values for Poss(E/Fj ), they are either one or zero. Thus in this case if K is the number of focal elements have Poss(E/Fi ) = 1 then

Pl g ( E ) =

K 

(μ(Hi ) − μ(Hi−1 )) = μ(HK )

i=1

Here HK is the subset of focal elements that intersect with E. Let us this denote HK = HP = {Fj , Fj ∩ E = ∅). Thus we see that Plg (E) = μ(HP ). Since Cert(E/Fj ) is also either one or zero in a similar manner we get Belg (E) = μ(HB ) where HB = {Fj /Fj ⊆ E}, it is the subset of focal elements that are contained in E. Using these results we obtain

Antg (E ) =

  B   P  μ H ,μ H

Let us look at the situation for some notable examples of measures. The classic example of a measure μ is a prob ability measure. Here associated with each Fj ∈ F we have μ({Fj }) = pj and if A is a subset of F then μ(A) = F ∈A pj . j  We see Plμ (E) = μ(HP ) = j,F ∩E=∅ pj , this is the same as the plausibility for the classic D-S belief structure. Here j  Belμ (E) = μ(HP ) = j,F ⊆E pj . This is also the same as the belief for the classic D-S belief structure. j

Consider the case of a cardinality-based measure μ on F. Here we have a collection of q + 1 parameters, ak for k = 0 to q such that a0 = 0, aq = 1 and ak+1 ≥ ak . If A is a subset of F then μ(A) = a|A| . In this case Pl(E) = μ(HP ) = a|HP | and Bel(E) = μ(HB ) = a|HB | . Here |HP | is the number of focal elements that intersect with E and |HB | is the number of focal elements contained in E. A particular interesting example of cardinality-based measure is one in which ak = 0 for k <  and ak = 1 for k ≥  . This is called a tipping function. Here then since μ(HB ) ≤ μ(HP ) we have three possible values for Antg (E):



Antg (E ) = [0, 0] = 0 if

HP < ψ

P

Antg (E ) = [0, 1] if HB

< ψ

≤ H B Antg (E ) = [1, 1] = 1 if H ≥ ψ Another notable example of a measure is a maxitive or possibility measure. Here associated with each focal element Fj is value α j ∈ [0, 1] where at least one of the α j = 1. Here for any subset of focal elements, A ⊆ F we have μ(A) = Max [αj ]. j,Fj ∈A

In the case

 

Pl(E ) = μ HP = Max [αj ]





j,Fj ∩E=∅

Bel(E ) = μ HB = Max[αj ] j,Fj ⊆E

Consider the situation when E = {xi }. Here HP is the subset of focal elements that contain xi . Since HB is subset of focal elements where Fj ⊆ E and E = {xi } then the only possibility is a focal element Fi = {xi }. We recall that Antg ({xi }) = [μ(HB ), μ(HP )]. We see that if Fi is not a focal element, Fi ∈ F, then HB = ∅ and μ(HB ) = 0 hence Antg ({xi } = [0, μ(HP )]. On the other hand if Fi is one of the focal elements Fi = Fj∗ then HB = {Fj∗ } and Antg ({xi }) = [μ({Fj∗ }), μ(HP )] Consider the case where E = {xi , xk }. Here HP is the subset of focal elements that contain one or both these, in particular HP = {Fj /Fj ∩ {xi , xk } = ∅}. Since HB = {Fj /Fj ⊆ {xi , xk )} there are three potential values for Fj , {xi }, {xk } and{xi , xk }. Assume our focal elements are consonant, here our focal element are such that Fj ⊂ Fj + 1 and Fq = X. Consider the determination of Antg (E) = [μ(HB ), μ(HP )]. Let j∗ be the Min index so that Fj∗ ∩ E = ∅. In this case HP = {Fj for j = j∗ to q}. Let j∗∗ be the maximum index so that Fj∗∗ ⊆ E. Here then HB = {Fj for j = 1 to j∗∗ }. If no focal elements are contained in F then j∗∗ = 0. Assume a = [a1 , a2 ] and b = [b1 , b2 ] are two sub-intervals of the unit interval. We say a is strongly greater then b, denoted a >S b, if a1 > b2 . We say a is weakly greater then b, denoted a >ϖ b, if a1 ≥ b1 and a2 ≥ b2 and one of the ≥ is an >. It is clear if a >S b then a >ϖ b. Assume E1 and E2 are two subsets of X such that E1 ⊂ E2 . Consider a measure based belief structure g. Let HPi be the subset of focal elements that intersect Ei and let HBi be the subset of focal elements that are contained in Ei . We easily see that HP2 ⊇ HP1 and hence μ(HP2 ) ≥ μ(HP1 ). We also easily see that HB2 ⊇ HB1 and μ(HP2 ) ≥ μ(HP1 ). From this it follows that Antg (E2 ) ≥W Antg (E1 ). Zadeh introduced the idea of fuzzy sets [32]. A fuzzy subset A of X is such that associated with each x ∈ X is a value A(x) ∈ [0, 1] called the membership grade of x in A. A fuzzy subset A is called normal if there exists at least one x ∈ X such that A(x) = 1. The concept of a normal fuzzy set generalizes the idea of non-empty from crisp sets. Here we shall be interested in normal fuzzy sets. A particularly useful application of fuzzy sets is to represent linguistic terms associated with V [33,34]. Here we shall consider the situation where the focal elements F = {F1 , …, Fq } are normal fuzzy sets of X. In particular we can understand our focal elements as linguistic terms associated with the domain X.

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In [35] Zadeh extended the concepts of possibility and certainty to fuzzy sets. Assume Fj and all E are fuzzy sets of x then

 

 

 

Cert E/Fj = Min Fj (x ) ∧ E(x ) x Poss E/Fj = Max E(x ) ∧ Fj (x ) x

Theorem. If Fj is normal then Poss(E/Fj ) ≥ Cert(E/Fj ) Proof. Let

x∗

be

such

that

Fj (x∗ ) = 1.

In

this

case

Cert(E/Fj ) = Min[Fj (x ) ∨ E(x )]. Here we have Fj (x∗ ) ∨ E(x∗ )=

Poss[E/Fj ] = Max[E(x ) ∧ Fj (x )]

E(x∗ )

x

x

≥

E(x∗ ).

Consider

and hence Cert(E/Fj ) = Min[E(x ) ∨ Fj (x )] ≤ E(x∗ ). x

From this it follows that Poss(E/Fj ) ≥ Cert(E/Fj ). Assume that we have a measure μ on the space F = {F1 , …, Fq } used to determine W then

















Plμ (E ) = Expμ Poss E/Fj , j = 1toq = Choqμ Poss E/Fj , j = 1toq         Belμ (E ) = Expμ Cert E/Fj , j = 1toq = Choqμ Bel E/Ej , j = 1toq

If ρ is an index function such that ρ (i) is the index of the focal element with the ith largest value of Poss(E/Fj ) and q Hi = {Fρ (k) , k = 1 to i} then Plμ (E) = i=1 (μ(Hi ) - μ(Hi-1 ))Poss(E/Fρ (i) ). If δ is an index function so that δ (i) is the index of q the focal element with the ith largest value of Cert(E/Fj ) and Gi = {Fδ (k) for k = 1 to i) then Belμ (E) = i=1 (μ(Gi ) - μ(Gi-1 )) Cert(E/Fδ (i) ). Furthermore as we have already shown it is the case that Poss(E/Fj ) ≥ Cert(E/Fj ) for all j then Plμ (E) ≥ Belμ (E). We see that the measure based belief structure provides a framework for modeling a wide class of uncertain information. If the focal elements are singletons we can model the classical types of uncertainty such as probability and possibility. If the focal elements are not singletons we can model imprecise uncertainties. 4. Database retrieval A relational database, or more simply a database is a very common method of storing information. A fundamental feature of a relational database is its highly structured format, which makes it easy to know the meaning of each piece of data. A database has a collection of attributes or features, Vj for j = 1 to r, and a corresponding domain Xj for each attribute. Associated with a database is a collection Di , the objects in the database, these Di are sometimes referred as entities. Each Di is an r tuple (di1 , di2 , …, dir ) where dij is the value of attribute Vj for object Di . Each dij takes its value in the domain Xj . There are two useful views of a database, one is a table as shown below

V1 D1 D2 Di Dn

V2

Vj

Vr

dij

The other view is as a set, DB = {D1 , …, Dn }, here the elements in the set are the entities in the database, the tuples Di . Each of these views has its benefits. The table point of view is useful from a cognitive and computational point of view, while the set view is beneficial from a formal mathematical point of view. An important task in modern technological applications is the retrieval of objects from a database, this is sometime referred to as search. The retrieval process involves finding entities in the database that satisfy a query Q. The result of a query Q is a subset from DB, one that contains all entities that satisfy the query Q, denoted DB(Q). A simple standard query Q consists of a pair (Vj , E), where Vj is an attribute and E is a subset of the domain Xj of Vj . Here we are interested in finding entities whose value for Vj lie in E. The querying process consists of looking at each Di to see if it satisfies the query. We denote this as Tr (Q/Di ), here Tr (Q/Di ) = 1 if Di satisfies the query and Tr (Q/Di ) = 0 if not. In this simple standard query environment the calculation of Tr (Q/Di ) is based on dij . Here E is a crisp subset of Xj indicating the desired values of Vj . The system returns the value Tr (Q/Di ) = 1 if dij ∈ E and Tr (Q/Di ) = 0 if dij ∈ E. Using the notation E(dij ) to indicate the membership of dij , in E we have Tr (Q/Di ) = E(dij ). Here the answer to the query is a subset DBQ of DB so that DBQ (Di ) = E(dij ). Here Tr (Q/Di ) = 1 indicates the query is satisfied by the object Di and Tr (Q/Di ) = 0 the objective Di does not satisfy the query. Since E is crisp all objects Di ∈ DB either satisfy or do not satisfy the query. A number of extensions of the simple standard query, SSQ, are possible. A common extension is to allow the E to be a fuzzy subset of X of the domain of Vj [7,9,20,23]. As is well documented in the literature this allows for the soft querying of the database, which enables the querying by linguistic or soft concepts [7,9,20,23]. In this case the satisfaction of the query (Vj , E) by object Di , Tr (Q/Di ) = E(dij ), is the membership grade of dij in the fuzzy set E. Here Tr (Q/Di ) rather, than being simply 1 or 0, is a value in the unit interval, Tr (Q/Di ) ∈ [0, 1]. Here the satisfaction of Q by Di rather then being a simple yes or no is a matter of degree of satisfaction. The larger E(dij ) the more Di satisfies the query. The situation allows us to order the objects in the database, the Di , with regard to their satisfaction degree. Here DB(Q) is a fuzzy subset of DB.

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5. Retrieval with uncertain databases One obstacle that can interfere with the calculation of Tr (Q/Di ) occurs when there is some uncertainty in our knowledge of the value of dij [23]. Here we have a database with uncertain information. In this work we shall consider the situation where the uncertainty associated with an entity’s attribute value is represented using a measure-based belief structure. It should be clear that this is a very general formulation and allows for the modeling of many different types of uncertainty. Consider again a query Q = (Vj , E) where E is a fuzzy subset and our knowledge of dij , the attribute value of Vj for object Di is a measure-based structure, gi . In our representation of dij we have a collection of focal elements F = {F1 , …, Fq } which are fuzzy subsets of the domain Xj of the attribute Vj . Additionally we have a measure μi over F that guides the selection of a focal element from F. We denote this selected focal element using the variable W. If W = F∗ then the value of dij is selected from F∗ . Based upon our preceding work the satisfaction of Q, (Vj , E), by object Di , in the case where the value of dij is a measure based belief structure gi is imprecise and is equal to the interval [Belgi (E), Plgi (E)]. Thus here Tr (Q/Di ) = [Belgi (E), Plgi (E)] which is a sub-interval of the unit interval. Given that each object Di in the database can have an interval value for its satisfaction to the query Q it is not easy to order these interval satisfactions. One approach here is to obtain a scalar representative value for each Tr (Q/Di ), Rep(Tr (Q/Di )), and then use these representative values for comparison of query satisfaction by the different entities in the database. Here we shall use the golden rule method [27] to obtain the representative value for Tr (Q/Di ). The golden rule representative value for an interval [Belgi (E), Plgi (E)] makes use of two characteristic features of the interval. The first feature is its mean Belg (E ) + Plg (E )

i i value, mi = , and the other is its range ri = Plgi (E) – Belgi (E). Clearly the mean is related to the value of Tr (Q/Di ) 2 so that the bigger mi the better. On other hand ri characterizes the variability associated with the value Tr (Q/Di ). Here we see that if the mean is large we prefer less variability, small ri , and if the mean is small we prefer more variability, large ri . Yager used these intuitions to provide a fuzzy rule base [24] description of a corresponding representative value:

If If If If

the the the the

mean mean mean mean

is is is is

large and the range is small then Rep(Tr (Q/Di )) = 1 large and the range is large then Rep(Tr (Q/Di )) = 0.5 small and the range is large then Rep(Tr (Q/Di )) = 0.5 small and the range is small then Rep(Tr (Q/Di )) = 0

Using the Takagi-Sugeno [12,18,30] approach to implement this fuzzy rule base describing the representative value Yager [24] showed that

Rep(Tr (Q/Di ) ) = mi + (0.5 − mi )ri . Yager referred to this as the Golden Rule representative rule. Here we see that if mi  0.5 then the uncertainty ri adds to the representative value while if mi  0.5 then the uncertainty ri subtracts from the representative value. With a little algebra we can express the Golden rule representative value in terms of Belgi (E) and Plgi (E). Since mi =

Belgi (E ) + Plgi (E ) 2

and ri = Plgi (E) – Belgi (E) and with Rep(Tr (Q/Di )) = mi + (0.5 - mi )ri = mi + 0.5 ri – mi ri we see

Rep(Tr (Q/Di ) ) =

Belgi (E ) + Plgi (E ) 1 + (Plgi (E ) − Belgi (E ) ) 2 2  Belgi (E ) + Plgi (E ) − (Plgi (E ) − Belgi (E ) )Rep(Tr (Q/Di ) ) = Plgi (E ) 2 −

 1 (Plgi (E ) )2 − (Belgi (E ) )2 2

We see that this representative value has the following properties (1) If Tr (Q/Di ) is a precise value, Plgi (E) = Belgi (E) then

Rep(Tr (Q/Di ) = Plgi (E ) = Belgi (E ) (2) It is monotonic with respect to both Plgi (E) and Belgi (E)

∂ Rep(Tr (Q/Di )) = Belgi (E ) ≥ 0 ∂ Belgi (E ) ∂ Rep(Tr (Q/Di )) = 1 − Plgi (E ) ≥ 0 ∂ Plgi (E ) (3) It satisfies right shifting condition

Assume Tr (Q/Di ) = [Belgi (E ), Plgi (E )] and Tr (Q/Dk ) = [ Belgk (E ), Plgk (E ) )], if Belgi (E ) ≥ Belgk (E ) and Plgi (E ) ≥ Plgk (E ) then Rep(Tr (Q/Di ) ) ≥ Rep(Tr (Q/Dk ) )

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In the preceding we considered simple queries, one’s which involve the satisfaction of a condition associated with one attribute. Her we shall generalize our queries to involve the satisfaction to multiple attributes, we refer to these as complex queries. A complex query Q is based on a collection of q pairs. Each pair Pk = (VQ(k) , EQ(k) ) consists of an attribute VQ(k) and associated required property for the attribute, EQ(k) , which is a fuzzy subset of the domain XQ(k) of the attribute. Here Q(k) indicates the index of the attribute in the kth pair, thus if Q(k) = j then the attribute in kth pair is Vj . A complex query also contains some information on how to fuse the satisfactions to the individual pairs to get a global query satisfaction. The query process involves looking at each object Di to see how much it satisfies the query. The first step is to determine the satisfaction of each of individual pair Pk in Q by entity Di , Tr (Pk /Di ) = EQ(k) (diQ(k) ), it is the membership grade of entity Di ’s value for attribute VQ(k) in EQ(k) . Since each EQ(k) can be a fuzzy set then Tr (Pk /Di ) ∈ [0, 1]. The overall satisfaction of the query Q by Di , Tr [Q/Di ) involves the fusion of the satisfactions of the individual pairs Tr [Pk /Di ) guided by the prescribed combination procedure. As noted in [1] many combination procedures can be modeled using an aggregation function, so here we shall assume that the fusion of these indicated pair satisfactions is guided by an aggregation function. Definition. An aggregation function Agg: [0, 1]q → [0, 1] is defined so that (1) Agg(0, …, 0) = 0 (2) Agg(1, …, 1) = 1 (3) Agg(a1 , …, aq ) ≥ Agg(b1 , …, bq ) if ak ≥ bk for all k Using an aggregation function we have

Tr (Q/Di ) = Aggk (Tr (Pk /Di ) ), k = 1toq ) where Tr (Pk /Di ) = EQ(k) (dkQ(k) ). Allowing for the possibility that we may have some uncertainty in our database we have

Tr (Pk /Di ) = [Bel(Tr (Pk /Di ) ), Pl(Tr (Pk /Di ) )] As a result of the properties of the aggregation operator we have

Aggk (Tr (Pk /Di ) ) = Aggk ([Bel(Tr (Pk /Di ), Pl(Tr (Pk /Di )] ) Aggk (Tr (Pk /Di ) ) = [Aggk (Bel(Tr (Pk /Di ) ), Agg(Pl(Tr (Pk /Di ) )] Here then Aggk (Tr (Pk /Di ), k = 1 to q) is again simply an interval value. Here we shall provide an illustrative example of our technology, Example. Assume our database has three attributes V1 , V2 , V3 and the domain of these attributes are

X1 = {1, 2, 3, 4, 5} X2 = {a, b, c, d} X3 = very small (vs ), small (s ), medium(m ), big(b ), very big(vb ) Let our query Q consists of three pairs

Q = ( (V1 ,E2 ), (V2 ,E2 ), (V3 ,E3 ) ) where

0.3 0.4 0.6 0.8 1  , , , ,

11 0.62 0.23 0 4 5 E2 = a , b , c , d

0.2 0.6 0.8 1 0.6  E1 = E3 =

vs

,

s

,

m

, b,

vb

Assume that our combination of the satisfactions by the individual pairs in a weighted average

Tr (Q/Di ) = 0.5Tr (P1 /Di ) + 0.3Tr (P2 /Di ) + 0.2Tr (P3 /Di ) Consider entity D1 where our knowledge is the following. For attribute V1 the knowledge is uncertain and is modeled by a measure based belief structure with focal elements in {F11 , F12 }, where F11 = {1, 2, 3} and F12 = {3, 4, 5} and we have a measure μ1 on this set of focal elements that is a probability measure, where μ1 ({F11 }) = 0.7, μ1 ({F12 }) = 0.3 and μ1 ({F11 , F12 }) = 1. For attribute V2 again our knowledge is uncertain and captured by a measure-based belief structure with focal elements {F21 , F22 , F23 } where F21 = {a, b}, F22 = {c, d} and F23 = {b, c, d} and we have a possibility measure μ2 on these focal elements with μ2 ({F21 }) = 0.7, μ2 ({F22 }) = 1 and μ2 ({F23 }) = 0.2. In this case μ2 ({F21 , F22 }) = 1, μ2 ({F21 , F23 }) = 0.7, μ2 ({F22 , F23 }) = 1 and μ(2 {F21 , F22 , F23 }) = 1. Finally for object D1 our knowledge of V3 is precise here V3 = medium We now calculate the required possibilities and certainties. For V1 we get

R.R. Yager, N. Alajlan and Y. Bazi / Information Sciences 501 (2019) 761–770

Poss(E1 /F11 ) = 0.6 Poss(E1 /F12 ) = 1

769

Cert(E1 /F11 ) = 0.3 Cert(E1 /F12 ) = 0.6

From these values and the associated probability measure μ1 we get

Pl(E1 ) = 0.6(0.7 ) + (1 )(0.3 ) = 0.72 Bel(E1 ) = 0.3(0.7 ) + (0.6 )(0.3 ) = 0.39 thus Tr (P1 /Di ) = [0.39, 0.72]. For V2 we get

Poss(E2 /F21 ) = 1 Cert(E2 /F21 ) = 0.6

Poss(E2 /F22 ) = 0.2 Cert(E2 /F22 ) = 0

Poss(E2 /F23 ) = 0.6 Poss(E2 /F23 ) = 0

Here we have

Pl(E2 ) = Choqμ2 (Poss(E2 /F21 ), Poss(E2 /F22 ), Poss(E2 /F23 ) ) Bel(E2 ) = Choqμ2 (Cert(E2 /F21 ), Cert(E2 /F22 ), Cert(E2 /F23 ) ) We note here that

Poss(E2 /F21 )> Poss(E2 /F23 )> Poss(E2 /F22 ) hence

Pl(E2 ) = μ2 ({F21 } ) Poss(E2 /F21 ) + (μ2 ({F21 ,F23 } ) − μ2 ({F21 } ) ) Poss(E2 /F23 )+ (1 − μ2 ({F21 ,F23 } ) ) Poss(E2 /F22 ) Pl(E2 )=(0.7 )(1 ) + (0 )(0.6 ) + (0.3 )(0.2 ) = 0.7 + 0 + 0.06 ) = 0.76 We see here that

Cert(E2 /E21 ) ≥ Cert(E2 /F22 )= Cert(E2 /E23 ) From this we have

Bel(E2 ) = μ2 ({F21 } ) Cert(E2 /F21 ) + (μ2 ({F21 ,F22 } ) − μ2 ({F21 } ) Cert(E2 /F23 )+ (1 − μ2 ({F21 ,F22 } ) ) Cert(E2 /F23 ) Bel(E2 ) = (0.6 )(0.7 ) + 0 = 0.42 Tr (P2 /D1 ) = [Bel(E2 ), Pl(E2 )] = [0.42, 0.76] Finally since V3 = medium we have

Tr (P2 /D1 ) = E2 (Medium ) = 0.8 = [0.8, 0.8] Here then

Tr (Q/D1 ) = Agg(Tr (Pk /D1 ), k = 1, 3 ) Tr (Q/D1 ) = w1 Tr (P2 /D1 ) + w2 Tr (P2 /D1 ) + w3 Tr (P2 /D2 ) Tr (Q/D1 ) = 0.5[0.39, 0.72] + 0.3[0.42, 0.76] + 0.2[0.8, 0.8] Tr (Q/D1 )=[(0.5 )(0.39 ) + (0.3 )(0.42 ) + (0.2 )(0.8 ),(0.5 )(0.72 ) + (0.3 )(0.76 ) + (0.2 )(0.8 )] Tr (Q/D1 )=[0.48, 0.72] Finally we can calculate the representative value using the Golden Rule. With Tr (Q/D1 ) = [0.48, 0.72] we see m = 0.6 and r = 0.72 – 0.48 = 0.24. Using

Rep(Tr (Q/Di ) ) = m + (0.5 − m ) r We get

Rep(Tr (Q/Di ) ) = 0.6 + (0.5 − 0.6 )(0.24 ) = 0.576. 6. Conclusion We described how the Dempster-Shafer belief structure provides a framework for modeling an uncertain value x˜ from some domain X. We noted how it involved a two-step process: the first, a random determination of one focal element guided by a probability distribution and the selection of x˜ from this focal element in some unspecified manner. We generalized this framework by allowing the determination of the focal element to be determined by a random experiment guided by a fuzzy measure. In either case the anticipation that x˜ lies in some subset E was interval valued, [Bel(E), Pl(E)]. We next looked at database retrieval and turned to issue of determining if a database entity with an uncertain attribute has a desired value. Here we modeled our uncertain attribute value as x˜ and our desired value as a subset E. In this case the degree of satisfaction of the query E by the entity is [Bel(E), Pl(E)]. In order to compare these interval satisfactions we used the Golden rule representative value to turn the intervals into scalars.

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Acknowledgment The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group No (RG-1435-055). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]

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