On Top-k Queries over Evidential Data
Fatma Ezzahra Bousnina
1
, Mouna Chebbah
1,2
, Mohamed Anis Bach Tobji
1,3
, Allel Hadjali
4
and Boutheina Ben Yaghlane
1,5
1
Universit
´
e de Tunis, Institut Sup
´
erieur de Gestion, LARODEC, Bardo, Tunisia
2
Universit
´
e de Jendouba, Facult
´
e des Sciences Juridiques, Economiques et de Gestion de Jendouba, Jendouba, Tunisia
3
Univ. Manouba, ESEN, Manouba, Tunisia
4
Universit
´
e de Poitiers, Ecole Nationale Sup
´
erieure de M
´
ecanique et de l’A
´
erotechnique, LIAS, Poitiers, France
5
Universit
´
e de Carthage, Institut des Hautes Etudes Commerciales, Carthage, Tunisia
Keywords:
Evidence Theory, Evidential Databases, Top-k Queries, Ranking Intervals, Score Function.
Abstract:
Uncertain data are obvious in a lot of domains such as sensor networks, multimedia, social media, etc. Top-k
queries provide ordered results according to a defined score. This kind of queries represents an important
tool for exploring uncertain data. Most of works cope with certain data and with probabilistic top-k queries.
However, at the best of our knowledge there is no work that exploits the Top-k semantics in the Evidence
Theory context. In this paper, we introduce a new score function suitable for Evidential Data. Since the result
of the score function is an interval, we adopt a comparison method for ranking intervals. Finally we extend
the usual semantics/interpretations of top-k queries to the evidential scenario.
1 INTRODUCTION
Processing queries over uncertain data received an in-
creasing importance with the emergence of several
applications in domains like sensor networks (Consi-
dine et al., 2004; Silberstein et al., 2006), moving ob-
jects tracking (Cheng et al., 2004a) and data cleaning
(Andritsos et al., 2006). Several types of queries deal
with uncertain data like uncertain skyline query (Ding
and Jin, 2012; Elmi et al., 2014; Elmi et al., 2015),
probabilistic top-k query (Soliman et al., 2007), un-
certain range query (Chung et al., 2009), uncertain
threshold query (Cheng et al., 2004b), etc.
In this paper, our interest goes to ranking queries,
also called top-k queries. A top-k query reports the
k objects with the highest scores based on a defined
scoring function. Imperfect top-k queries are different
from top-k queries over certain data, they focus not
only on the value of the scoring function, but also on
the degree of objects’ uncertainty.
Data uncertainty can be detected in three levels
(Tao et al., 2007):
• The table level uncertainty, represented with a de-
gree of imperfection about the presence or the ab-
sence of the table in the database.
• The tuple level uncertainty, represented with a de-
gree of imperfection about the presence or the ab-
sence of that tuple.
• The attribute level uncertainty, represented by a
degree of imperfection about individual attributes.
The tuple and attribute levels are the most studied
in the literature. Lots of theories like the probability
theory (Laplace, 1812), the fuzzy sets theory (Zadeh,
1965), the possibility theory (Zadeh, 1978) and the
belief functions theory, also called the evidence
theory (Dempster, 1967; Shafer, 1976) have been
introduced in order to handle imperfection as uncer-
tainty, imprecision and inconsistency.
At the best of our knowledge, there is no work that
deals with top-k queries for Evidential Data. Simi-
larly to the probabilistic top-k queries (Soliman et al.,
2007), Evidential Top-k queries should return the k
answers that respond to an evidential query with the
highest scores based on a scoring function that takes
into consideration the degrees of imperfection in the
database.
Through this paper, we present a new type of un-
certain top-k queries; the Evidential Top-k Queries
that we apply over an evidential relation. We recall
106
Bousnina, F., Chebbah, M., Tobji, M., Hadjali, A. and Yaghlane, B.
On Top-k Queries over Evidential Data.
DOI: 10.5220/0006317701060113
In Proceedings of the 19th International Conference on Enterprise Information Systems (ICEIS 2017) - Volume 1, pages 106-113
ISBN: 978-989-758-247-9
Copyright © 2017 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

that an evidential database is a database that stores
perfect and imperfect data modeled with the theory
of evidence (Dempster, 1967; Dempster, 1968). Each
object in the evidential database is quantified with an
interval of confidence called the Confidence Level and
denoted CL (Bell et al., 1996; Bousnina et al., 2015;
Lee, 1992a; Lee, 1992b).
For this purpose, we introduce a new scoring
function for evidential data that returns an interval
bounded by a belief and a plausibility. To rank the ev-
idential scores, we rely on the method of (Wang et al.,
2005). We present also a new imperfect top-k seman-
tics specific to the evidential scenario.
Table 1 presents an example of an evidential table
that stores some users’ appreciations about books: b
1
,
b
2
, b
3
, b
4
. This example is a relation with three at-
tributes: The first one is ID, it represents the identifier
of a specific reader. The second attribute is BookRate
where the reader expresses its preferred books using
the evidence theory
1
. The uncertainty here deals with
the attribute level. The third attribute is CL, it stores
the interval of confidence about the user, and thus
about its given appreciations. Here we deal with un-
certainty at the tuple level. This example will be used
among this paper.
Table 1: Books Appreciations’ Table: BAT.
ID BookRate CL
1 b
1
0.3 [0.5;1]
{b
2
,b
3
} 0.7
2 b
2
0.5 [0.3;0.8]
b
4
0.5
3 {b
1
,b
2
,b
3
} 1 [1;1]
4 b
3
1 [0.5;0.9]
This paper is organized as follows: we recall, in
section 2, some definitions and concepts of the top-k
Queries, the evidence theory, the evidential databases
and some approaches of ranking intervals. In section
3, we present our main contribution about the evi-
dential top-k queries. The conclusion and the future
works are held in section 4.
2 BACKGROUND MATERIALS
In this section, some notions about top-k queries
and several comparing intervals approaches in the
literature are briefly presented. Other fundamental
1
The literature is abundant in term of methods of prefer-
ence elicitation using the evidence theory. We cite two main
works (Ben Yaghlane et al., 2008; Ennaceur et al., 2014)
concepts like the evidence theory and the evidential
databases are also exposed in this section.
2.1 Top-k Queries
Top-k queries are also known as Ranking queries.
They represent a powerful tool when we want to
order queries’ results in order to only give the most
interesting answers. Top-k queries were introduced
in the multimedia systems (Fagin, 1996; Fagin,
1998). Generally, top-k queries are ranked using a
defined score function where only the k (k ≥ 1) most
important answers are returned. In other words, only
answers with the highest scores are returned.
Ranking queries are needed in real worlds appli-
cations. For example movies can be ordered by the
most watched ones, music can be ranked by the most
listened songs, researchers can be ranked by their
H-index, athletes by their race time, etc.
Several top-k processing techniques exist in the
literature. Based on data uncertainty, they can be clas-
sified into three categories (Ilyas et al., 2008):
• Exact methods over certain data, where top-k
queries and data are deterministic. The majority
of top-k processing techniques are based on exact
methods and certain data.
• Approximate methods over certain data, where
processing top-k queries over certain data pro-
duces approximate results (Amato et al., 2003;
Theobald et al., 2005).
• Methods over uncertain data, where top-k pro-
cessing techniques deal with imperfect data. The
top-k queries are based on different uncertainty
models. At the best of our knowledge, only top-
k queries’ approaches that deal with probabilities
exist in the literature (Re et al., 2007; Soliman
et al., 2007) but there is no work that deal with
other types of imperfect data. Our contribution
copes mainly with this category.
2.2 Evidence Theory and Evidential
Databases
Evidence theory, also called the Dempster-Shafer the-
ory or the belief functions theory, was introduced by
(Dempster, 1967; Dempster, 1968) and was mathe-
matically formalized by (Shafer, 1976). Evidence the-
ory is a powerful tool for the representation of impre-
cise, inconsistent and uncertain data.
A frame of discernment or universe of discourse is
a set Θ = {θ
1
,θ
2
,..., θ
n
}. It is a finite, non empty and
On Top-k Queries over Evidential Data
107

exhaustive set of n elementary and mutually exclusive
hypotheses for a given problem. The power set 2
θ
=
{∅,θ
1
,θ
2
,..., θ
n
,{θ
1
,θ
2
},.., {θ
1
,θ
2
,..., θ
n
}} is a set
of all subsets of Θ.
A mass function, noted m, is a mapping from 2
Θ
to the interval [0,1]. The basic belief mass of an hy-
pothesis x is noted m(x), it represents the belief on
the truth of that hypothesis x. A mass function is also
called basic belief assignment (bba). It is formalized
such that:
∑
x⊆Θ
m
Θ
(x) = 1 (1)
If m
Θ
(x) > 0, x is called focal element. The set of
all focal elements is denoted F and the couple {F,m}
is called body of evidence.
The belief function noted bel is the minimal
amount of belief committed exactly to x and it is cal-
culated as follows:
bel(x) =
∑
y⊆x;y6=∅
m
Θ
(y) (2)
The plausibility function noted pl is the maximal
amount of belief on x and it is computed as follows:
pl(x) =
∑
y⊆Θ;x∩y6=∅
m
Θ
(y) (3)
An Evidential database (EDB), also named
Dempster-Shafer database is a database that stores
both perfect and imperfect data. In such databases
imperfection is modeled using the theory of belief
functions.
Definition 1. An EDB has N objects and A attributes.
An evidential value, noted V
la
, is the value of an at-
tribute a (1 ≤ a ≤ A) for an object l (1 ≤ l ≤ N) that
represents a basic belief assignment.
V
la
: 2
Θ
a
→ [0, 1] (4)
with m
Θ
a
la
(∅) = 0 and
∑
x⊆Θ
a
m
Θ
a
a
(x) = 1 (5)
The set of focal elements relative to the bba V
la
is
noted F
la
such that:
F
la
= {x ⊆ Θ
a
/m
la
(x) > 0} (6)
A confidence level, denoted CL, is a specific at-
tribute used to represent an interval of confidence for
each object l in the evidential database. It is a pair of
belief and plausibility [bel; pl], that reflects the pes-
simistic and optimistic believes of the existence of the
object in the database (Lee, 1992b; Lee, 1992a; Bell
et al., 1996).
An Evidential Database stores various types of
data:
• Perfect data: When the focal element is a sin-
gleton and its mass is equal to one then the bba
is called certain. In table 1, the BookRate of the
fourth object is a certain bba.
• Probabilistic data: When focal elements are
singletons then the bba is called bayesian. In
table 1, the BookRate of the second object is a
bayesian bba.
• Possibilistic data: When focal elements are
nested, then the bba is called consonant.
• Evidential data: When none of the previous cases
is present the bba is called evidential. This is
the case of the first and the third objects of the
BookRate as shown in table 1.
In relational databases, data are stored to be fur-
ther queried using the relational operators: selection,
projection, join, etc. Evidential databases are also in-
terrogated using the extended relational operators like
the extended selection, the extended projection, etc.
(Bell et al., 1996; Lee, 1992a; Lee, 1992b).
Applying a query Q over an evidential database
EDB gives a set of evidential results R. For each an-
swer R
i
, a new confidence level is computed denoted
CL=[bel
c
; pl
c
] that quantifies the degree of faith about
that answer.
The extended select operator consists on extract-
ing from EDB the objects whose values satisfy the
condition of a query Q. The result is a relation that
obey to a threshold of belief and plausibility. For
each object l in the evidential database, the belief b
and plausibility p of the condition are multiplied with
the confidence level CL[bel; pl] of that object. The
computed CL of each resulting object in the relation
is defined as follows:
CL = [b ∗ bel; p ∗ pl] (7)
Example 1. We take an example of a select query that
we process over the evidential table 1, the query is the
following:
Q
1
: SELECT * FROM BAT WHERE (BookRate
= {b
1
})
Book b
1
appears only in tuples t
1
and t
3
with two con-
fidence levels t
1
.CL
b
1
and t
3
.CL
b
1
, summarized in ta-
ble 2 and computed as follows:
ICEIS 2017 - 19th International Conference on Enterprise Information Systems
108

• t
1
.CL
b
1
= [0.3*0.5 ; 0.3*1] = [0.15 ; 0.3].
with

bel(b
1
)=0.3 and pl(b
1
)=0.3 for tuple t
1
bel(t
1
)=0.5 and pl(t
1
)=1
• t
3
.CL
b
1
= [0*1 ; 1*1] = [0 ; 1].
with

bel(b
1
)=0 and pl(b
1
)=1 for tuple t
3
bel(t
3
)=1 and pl(t
3
)=1
Table 2: Result of query Q
1
.
ID BookRate CL
1 b
1
[0.15 ; 0.3]
{b
2
,b
3
}
3 {b
1
,b
2
,b
3
} [0 ; 1]
The book b
1
appears in the first object with a de-
gree of confidence of [0.15 ; 0.3] and it appears also
in the third object with a confidence level of [0 ; 1].
2.3 Methods of Ranking Intervals
Many approaches were introduced to compare or rank
intervals: First, (Borda, 1781), managed the ordinal
ranking problem and proposed a method to rank can-
didates in election. Then, (Kendall, 1990) proposed
a statistical framework for the ranking problem based
on summing ranks assigned to each candidate by the
voters. (Arrow, 2012) and (Inada, 1964; Inada, 1969)
solved the problem of ranking using the majority rule.
(Kemeny and Snell, 1962) used distance measures for
the ranking.
(Salo and H
¨
am
¨
al
¨
ainen, 1992) introduced the de-
cision maker (DM) method that compares intervals in
order to get one interval that dominates the other inter-
vals but this method is not always feasible. (Ishibuchi
and Tanaka, 1990) used a comparison rule on inter-
val numbers to define an order, however this approach
fails when intervals are nested. (Kundu, 1997) defined
a fuzzy method that calculates the degree that an inter-
val is superior or inferior to another one. This method
requires that all interval numbers are independent and
uniformly distributed.
(Wang et al., 2005) developed a ranking method of
interval numbers. They proposed a preference aggre-
gation method by combining individual preferences
which is a typical group decision making problem
like committee decision, voting systems, etc. The fi-
nal ranking, in preference aggregation, is based on the
comparison and the ranking interval numbers. To do
so, they used a simple interval ranking approach. That
approach will be used and adopted in this paper but
other fitted approach can be also used.
3 EVIDENTIAL Top-k QUERIES
Processing queries over evidential databases gives an-
swers, each one quantified with a degree of confi-
dence. That degree reflects the lower and the upper
bounds of trust in that response which is calculated
from the database. In this case, these given answers
are not ranked which don’t allow the user to choose
the most interesting ones from the set of results ac-
cording to a defined criteria. In order to give the de-
cision maker the responses that satisfy its request, we
need to introduce a new top-k approach specific for
evidential databases.
In this section, we present a new formalism to rank
evidential results based on a score function. First, we
process a query Q over the evidential database EDB.
Then, for each generated response an Evidential Score
is computed. That score is an interval of belief and
plausibility, defined as follows:
Definition 2. Evidential Score: Let R
i
be a response
generated from processing a query Q over an eviden-
tial database EDB, S(R
i
) is the score function of that
answer R
i
and bel(R
i
) and pl(R
i
) are respectively its
belief and plausibility in the table, such that:
S(R
i
) = [bel(R
i
); pl(R
i
)] (8)
Where bel(R
i
) =
∑
N
l=1
bel
l
(R
i
) ∗ bel
l
N
pl(R
i
) =
∑
N
l=1
pl
l
(R
i
) ∗ pl
l
N
The belief of an answer, bel(R
i
), is a disjunction of
the response’s beliefs in each object of the database.
The belief of a response in one object l is the product
of its belief in the attribute and the belief of that ob-
ject. Same for the plausibility of an answer, pl(R
i
).
It is the disjunction of the response’s plausibilities in
each object of the database where the plausibility of a
response in one object l is the product of its plausibil-
ity in the attribute and the plausibility of that object
(Bell et al., 1996; Lee, 1992a).
Example 2. Let us process the query Q
2
over the
evidential table 1 in order to get the top-2 answers.
Q
2
: T he most appreciated books f rom
table BAT .
The score of each item in the relation that may be
a response to the query Q
2
is computed as follows:
• The first possible response is book b
1
, it appears
in objects l
1
and l
3
. Therefore:
(
bel(b
1
) =
(0.3∗0.5)+(0∗0.3)+(0∗1)+(0∗0.5)
4
pl(b
1
) =
(0.3∗1)+(0∗0.8)+(1∗1)+(0∗0.9)
4
On Top-k Queries over Evidential Data
109

Thus:
S(b
1
) = [bel(b
1
); pl(b
1
)] = [0.0375; 0.325]
• The second possible response is book b
2
, it
appears in objects l
1
, l
2
and l
3
. Therefore:
(
bel(b
2
) =
(0∗0.5)+(0.5∗0.3)+(0∗1)+(0∗0.5)
4
pl(b
2
) =
(0.7∗1)+(0.5∗0.8)+(1∗1)+(0∗0.9)
4
Thus:
S(b
2
) = [bel(b
2
); pl(b
2
)] = [0.0375; 0.525]
• The third possible response is book b
3
, it appears
in objects l
1
, l
3
and l
4
. Therefore:
(
bel(b
3
) =
(0∗0.5)+(0∗0.3)+(0∗1)+(1∗0.5)
4
pl(b
3
) =
(0.7∗1)+(0∗0.8)+(1∗1)+(1∗0.9)
4
Thus:
S(b
3
) = [bel(b
3
); pl(b
3
)] = [0.125; 0.65]
• The final response is book b
4
, it appears only in
object l
2
. Therefore:
(
bel(b
4
) =
(0∗0.5)+(0.5∗0.3)+(0∗1)+(0∗0.5)
4
pl(b
4
) =
(0∗1)+(0.5∗0.8)+(0∗1)+(0∗0.9)
4
Thus:
S(b
4
) = [bel(b
4
); pl(b
4
)] = [0.0375; 0.1]
The computed evidential scores are shown in table 3.
Table 3: Evidential Score per Item.
item EvidentialScore
b
1
R
1
= [0.0375 ; 0.325]
b
2
R
2
= [0.0375 ; 0.525]
b
3
R
3
= [0.125 ; 0.65]
b
4
R
4
= [0.0375 ; 0.1]
Top-k queries are based on a defined score func-
tion. That function produces precise values, in con-
trary to the evidential top-k queries whose score func-
tion produces intervals bounded by belief and plausi-
bility values. (Wang et al., 2005) introduced an ap-
proach of ranking intervals based on preference de-
grees. Their method is the one that we will adopt to
rank scores previously generated.
Definition 3. Preference Degree: Let
S(R
i
)=[bel
i
; pl
i
] and S(R
j
)=[bel
j
; pl
j
] be two
evidential scores. Each one is an interval composed
of a degree of belief and a degree of plausibility. The
degree of one interval to be greater than the other
one is called a degree of preference and denoted P.
The degree of preference that S(R
i
) > S(R
j
) is de-
fined such that:
P(S(R
i
) > S(R
j
)) =
max(0, pl
i
− bel
j
) − max(0,bel
i
− pl
j
)
(pl
i
− bel
i
) + (pl
j
− bel
j
)
(9)
The degree of preference that S(R
i
) < S(R
j
) is de-
fined such that:
P(S(R
i
) < S(R
j
)) =
max(0, pl
j
− bel
i
) − max(0,bel
j
− pl
i
)
(pl
i
− bel
i
) + (pl
j
− bel
j
)
(10)
The different cases of comparing intervals S(R
i
)
and S(R
j
) are as follows:
• If P(S(R
i
) > S(R
j
)) > P(S(R
j
) > S(R
i
)), then
S(R
i
) is said to be superior to S(R
j
), denoted by
S(R
i
)  S(R
j
).
• If P(S(R
i
) > S(R
j
)) = P(S(R
j
) > S(R
i
)) = 0.5,
then S(R
i
) is said to be indifferent to S(R
j
),
denoted by S(R
i
) ∼ S(R
j
).
• If P(S(R
j
) > S(R
i
)) > P(S(R
i
) > S(R
j
)), then
S(R
i
) is said to be inferior to S(R
j
), denoted by
S(R
i
) ≺ S(R
j
).
Theorem 1. Let S(R
i
)=[bel
i
; pl
i
] and
S(R
j
)=[bel
j
; pl
j
] be two evidential scores such
that:
• Shortcut 1: if S(R
i
) = S(R
j
) then
P(S(R
i
)) > P(S(R
j
)) = P(S(R
i
)) < P(S(R
j
)) = 0.5.
• Shortcut 2: if bel
i
≥ pl
j
then P(S(R
i
) > S(R
j
)) = 1.
• Shortcut 3: if bel
i
≥ bel
j
and pl
i
≥ pl
j
then
P(S(R
i
) > S(R
j
)) ≥ 0.5 and P(S(R
j
) > S(R
i
)) ≤ 0.5.
In order to detect the dominant interval between
the score of relation R
i
denoted S(R
i
) and the score of
relation R
j
denoted S(R
j
), we need to compute the de-
gree of preference that S(R
i
) > S(R
j
) and the degree
of preference that S(R
i
) < S(R
j
). The complexity of
this computation can be reduced thanks to the comple-
mentarity of P(S(R
i
) > S(R
j
)) and P(S(R
i
) < S(R
j
)).
The complementarity is only feasible when:

S(R
i
) 6= S(R
j
)
bel
i
< pl
j
(11)
ICEIS 2017 - 19th International Conference on Enterprise Information Systems
110

Proof. Complementarity:
P(S(R
i
) < S(R
j
)) =
max(0, pl
j
− bel
i
) − max(0,bel
j
− pl
i
)
(pl
i
− bel
i
) + (pl
j
− bel
j
)
P(S(R
j
) < S(R
i
)) =
max(0, pl
i
− bel
j
) − max(0,bel
i
− pl
j
)
(pl
i
− bel
i
) + (pl
j
− bel
j
)
P(S(R
i
) < S(R
j
)) + P(S(R
j
) < S(R
i
))
=
max(0, pl
j
− bel
i
) − max(0,bel
j
− pl
i
)
(pl
i
− bel
i
) + (pl
j
− bel
j
)
+
max(0, pl
i
− bel
j
) − max(0,bel
i
− pl
j
)
(pl
i
− bel
i
) + (pl
j
− bel
j
)
=
max(0, pl
j
− bel
i
) − 0 + max(0, pl
i
− bel
j
) − 0
(pl
i
− bel
i
) + (pl
j
− bel
j
)
=
pl
j
− bel
i
+ pl
i
− bel
j
pl
i
− bel
i
+ pl
j
− bel
j
= 1
P(S(R
i
) < S(R
j
)) + P(S(R
j
) < S(R
i
)) = 1
Figure 1 summarizes the different cases of eviden-
tial scores intervals. It represents also which property
from the presented ones to use for each case.
The transitivity property is helpful to achieve a
complete ranking order for scores. In (Wang et al.,
2005), authors proved that preference relations are
transitive.
Property 1. Transitivity
Let S(R
i
) = [bel
i
; pl
i
], S(R
j
) = [bel
j
; pl
j
] and S(R
k
) =
[bel
k
; pl
k
] be three intervals. If S(R
i
)  S(R
j
) and
S(R
j
)  S(R
k
) then S(R
i
)  S(R
k
).
Previous definitions provide a total ranking of
answers that respond to the proposed top-k query.
But how to interpret any evidential answer ?
The top-k queries in deterministic databases are
semantically clear. However, the interpretation of
top-k queries in imperfect databases are challenging.
(Soliman et al., 2007) introduced new semantics rela-
tive to probabilistic top-k queries. They defined them
as the most probable query answers. Their work is
based on the possible worlds’ model and they pro-
posed interpretations like: (i) The top-k tuples in the
most probable world. (ii)The most probable top-k tu-
ples that belongs to valid possible world.
The interpretations of probabilistic top-k queries
can not be considered for evidential top-k queries.
Thus, a new specific semantic for Evidential Top-k
Queries is defined as follows:
(a) Shortcut1
(b) Shortcut2
(c) Shortcut3
(d) Evidential Score / Complemen-
tarity
Figure 1: Comparison of Evidential Scores.
Definition 4. E-Top-k:
Let EDB be an evidential database with N objects and
A attributes; CL is an attribute where the intervals
associated to objects reflects the degrees of confidence
about these objects. Let S(R
i
) be a score function that
maximizes both CL and the interval of belief on each
result. Responses R are ordered according to scores.
An E-topk returns the k most credible answers
from the set of answers such that:
S(R
i
) = Argmax
R
i
∈R
([bel(R
i
); pl(R
i
)]) (12)
Example 3. We carry on with the same example of
table 1 and we give a total ranking of the resulting
evidential scores. We also deduce the top-2 answers
and their interpretation.
(i) Since bel
b
1
= bel
b
2
and pl
b
2
> pl
b
1
> pl
b
4
On Top-k Queries over Evidential Data
111

then b
2
 b
1
 b
4
(ii) Since bel
b
3
> bel
b
2
and pl
b
3
> pl
b
2
then b
3
 b
2
The final ranking deduced from (i) and (ii) is:
b
3
 b
2
 b
1
 b
4
.
The Top-2 appreciated books are:
• b
3
with a confidence level [0.125 ; 0.65]
• b
2
with a confidence level [0.0375 ; 0.525]
Books b
3
and b
2
are the most appreciated credible
answers from the set of results.
4 CONCLUSION AND FUTURE
WORKS
In this paper, we presented a new imperfect top-k
query called the evidential top-k query. It consists
in processing top-k query over evidential data (data
modeled using the theory of belief functions). First,
we introduced a new score function that computes
an interval of belief and plausibility relative to each
answer responding a given top-k query. Then, we
adopted a preference approach of comparing intervals
(Wang et al., 2005). We also presented the proof of
complementarity relative to that approach, in order to
reduce the complexity of computations while calcu-
lating the evidential score. Finally, we introduced a
new semantics relative to evidential top-k.
As future works, top-k queries may be imple-
mented and other types of such a query (like the ag-
gregation, the project and the join uncertain queries
for the evidential databases) may be also detailed.
REFERENCES
Amato, G., Rabitti, F., Savino, P., and Zezula, P. (2003).
Region proximity in metric spaces and its use for ap-
proximate similarity search. ACM Transactions on In-
formation Systems (TOIS), 21(2):192–227.
Andritsos, P., Fuxman, A., and Miller, R. J. (2006). Clean
answers over dirty databases: A probabilistic ap-
proach. In 22nd International Conference on Data
Engineering (ICDE’06), pages 30–30. IEEE.
Arrow, K. J. (2012). Social choice and individual values,
volume 12. Yale university press.
Bell, D. A., Guan, J. W., and Lee, S. K. (1996). Generalized
union and project operations for pooling uncertain and
imprecise information. Data & Knowledge Engineer-
ing (DKE), 18:89–117.
Ben Yaghlane, A., Denœux, T., and Mellouli, K. (2008).
Elicitation of expert opinions for constructing belief
functions. Uncertainty and Intelligent Information
Systems, pages 75–88.
Borda, J. C. (1781). M
´
emoire sur les
´
elections au scrutin.
Translated in the political theory of condorcet. Som-
merlad F, Mclean I. Social studies, Oxford, 1989.
Bousnina, F. E., Bach Tobji, M. A., Chebbah, M., Li
´
etard,
L., and Ben Yaghlane, B. (2015). A new formalism
for evidential databases. In 22nd International Sym-
posium on Methodologies for Intelligent Systems (IS-
MIS), Foundations of Intelligent Systems, pages 31–
40. Springer.
Cheng, R., Kalashnikov, D. V., and Prabhakar, S. (2004a).
Querying imprecise data in moving object environ-
ments. IEEE Transactions on Knowledge and Data
Engineering (TKDE), 16(9):1112–1127.
Cheng, R., Xia, Y., Prabhakar, S., Shah, R., and Vitter, J. S.
(2004b). Efficient indexing methods for probabilistic
threshold queries over uncertain data. In 13th Interna-
tional Conference on Very Large Data Bases (VLDB),
pages 876–887. VLDB Endowment.
Chung, B. S., Lee, W.-C., and Chen, A. L. (2009). Process-
ing probabilistic spatio-temporal range queries over
moving objects with uncertainty. In 12th International
Conference on Extending Database Technology, Ad-
vances in Database Technology, pages 60–71. ACM.
Considine, J., Li, F., Kollios, G., and Byers, J.
(2004). Approximate aggregation techniques for sen-
sor databases. In 20th International Conference on
Data Engineering (ICDE), pages 449–460. IEEE.
Dempster, A. P. (1967). Upper and lower probabilities in-
duced by a multiple valued mapping. The Annals of
Mathematical Statistics, 38(2):325–339.
Dempster, A. P. (1968). A generalization of bayesian infer-
ence. Journal of the Royal Statistical Society, Series
B, 30:205–247.
Ding, X. and Jin, H. (2012). Efficient and progressive algo-
rithms for distributed skyline queries over uncertain
data. IEEE Transactions on Knowledge and Data En-
gineering (TKDE), 24(8):1448–1462.
Elmi, S., Benouaret, K., HadjAli, A., Bach Tobji, M. A.,
and Ben Yaghlane, B. (2014). Computing skyline
from evidential data. In 8th International Conference
on Scalable Uncertainty Management (SUM), pages
148–161, Oxford, UK.
Elmi, S., Benouaret, K., HadjAli, A., Bach Tobji, M. A.,
and Ben Yaghlane, B. (2015). Requ
ˆ
etes skyline en
pr
´
esence des donn
´
ees
´
evidentielles. In Extraction et
Gestion des Connaissances (EGC), pages 215–220.
Ennaceur, A., Elouedi, Z., and Lefevre, E. (2014). Multi-
criteria decision making method with belief pref-
erence relations. International Journal of Un-
certainty, Fuzziness and Knowledge-Based Systems,
22(04):573–590.
Fagin, R. (1996). Combining fuzzy information from multi-
ple systems. In 15th ACM SIGACT-SIGMOD-SIGART
ICEIS 2017 - 19th International Conference on Enterprise Information Systems
112

symposium on Principles of database systems, pages
216–226. ACM.
Fagin, R. (1998). Fuzzy queries in multimedia database sys-
tems. In 17th ACM SIGACT-SIGMOD-SIGART sym-
posium on Principles of database systems, pages 1–
10. ACM.
Ilyas, I. F., Beskales, G., and Soliman, M. A. (2008). A sur-
vey of top-k query processing techniques in relational
database systems. ACM Computing Surveys (CSUR),
40(4):11.
Inada, K.-i. (1964). A note on the simple majority deci-
sion rule. Econometrica, Journal of the Econometric
Society, pages 525–531.
Inada, K.-i. (1969). The simple majority decision rule.
Econometrica, Journal of the Econometric Society,
pages 490–506.
Ishibuchi, H. and Tanaka, H. (1990). Multiobjective pro-
gramming in optimization of the interval objective
function. European Journal of Operational Research
(EJOR), 48(2):219–225.
Kemeny, J. G. and Snell, L. (1962). Preference ranking:
an axiomatic approach. Mathematical models in the
social sciences, pages 9–23.
Kendall, M. (1990). Rank correlation methods. Oxford Uni-
versity Press, 5th edition.
Kundu, S. (1997). Min-transitivity of fuzzy leftness rela-
tionship and its application to decision making. Fuzzy
sets and systems, 86(3):357–367.
Laplace, P. S. d. (1812). Th
´
eorie analytique des proba-
bilit
´
es. Courcier, Paris.
Lee, S. K. (1992a). An extended relational database model
for uncertain and imprecise information. In 18th Con-
ference on Very Large Data Bases (VLDB), pages
211–220, Canada.
Lee, S. K. (1992b). Imprecise and uncertain information
in databases : an evidential approach. In 8th In-
ternational Conference on Data Engineering (ICDE),
pages 614–621.
Re, C., Dalvi, N., and Suciu, D. (2007). Efficient top-
k query evaluation on probabilistic data. In IEEE
23rd International Conference on Data Engineering
(ICDE), pages 886–895. IEEE.
Salo, A. and H
¨
am
¨
al
¨
ainen, R. (1992). Processing interval
judgments in the analytic hierarchy process.
Shafer, G. (1976). A mathematical theory of evidence.
Princeton University Press.
Silberstein, A., Braynard, R., Ellis, C., Munagala, K., and
Yang, J. (2006). A sampling-based approach to opti-
mizing top-k queries in sensor networks. In 22nd In-
ternational Conference on Data Engineering (ICDE),
pages 68–68. IEEE.
Soliman, M. A., Ilyas, I. F., and Chang, K. C.-C. (2007).
Top-k query processing in uncertain databases. In
23rd International Conference on Data Engineering
(ICDE), pages 896–905. IEEE.
Tao, Y., Xiao, X., and Cheng, R. (2007). Range search on
multidimensional uncertain data. ACM Transactions
on Database Systems (TODS), 32(3):15.
Theobald, M., Schenkel, R., and Weikum, G. (2005). An
efficient and versatile query engine for topx search.
In 31st International Conference on Very large Data
Bases (VLDB), pages 625–636. VLDB Endowment.
Wang, Y.-M., Yang, J.-B., and Xu, D.-L. (2005). A pref-
erence aggregation method through the estimation of
utility intervals. Computers & Operations Research,
32(8):2027–2049.
Zadeh, L. A. (1965). Fuzzy sets. Information and Control,
8:338–353.
Zadeh, L. A. (1978). Fuzzy sets as a basis for a theory of
possibility. Fuzzy Sets and Systems, 1:3–28.
On Top-k Queries over Evidential Data
113