Database Reference
In-Depth Information
several efforts have been made, such as: RankSQL (Li, Chen-chuan, Ihab, Ilyas & Song, 2005) retrieving
the top-k ranked answers; SKYLINE (Börzsönyi, Kossmann, & Stocker, 2001) selection of best rows
or all non-dominated based on a crisp multi criteria comparison; SQLf (Bosc & Pivert, 1995) allowing
fuzzy conditions anywhere SQL expects Boolean ones; Soft-SQL (Bordogna & Psaila, 2008) allowing
customizable fuzzy terms definitions for querying; FSQL (Galindo, 2005) using Fuzzy Sets for imperfect
data representation; MayBMS (Koch, 2009), MystiQ (Boulos, Dalvi, Mandhani, Mathur, Re & Suciu,
2009) and Trio (Widom, 2009) are proposals to implement probabilistic uncertain databases; there are
also proposals for fuzzy queries combination on multiple data sources (Fagin, 2002).
Fuzzy Set based is the more general approach to solve the problem of database rigidity (Bosc & Pi-
vert, 2007; Goncalves & Tineo, 2008). Nevertheless, Fuzzy Set handling adds extra processing costs to
database systems that must be controlled (Bosc & Pivert, 2000; Tineo, 2006). We need efficient evalua-
tion mechanisms in order to make the use of fuzzy queries possible in real world applications (Lopez &
Tineo, 2006). Moreover, it would be appreciated to enhance existing RDBMS with native fuzzy query
capability in order to improve performance and scalability; this is the focus and contribution of present
chapter.
Fuzzy Sets
Zadeh (1965) introduced Fuzzy Sets in order to model fuzzy classes in Control Systems, since then
Fuzzy Set has been infiltrating into many branches of pure and applied mathematics that are set theory
based. A Fuzzy Set is defined as a subset
F
of a domain
X
characterized by a membership function
μ
F
ranked on the real interval [0,1]. Some correspondence operators between a Fuzzy Set
F
and regular
ones are defined:
support(F)
is the set of elements with
μ
F
(x)>0
;
core(F)
is the set of elements with
μ
F
(x)=1
;
border(F)
is the set of elements with
μ
F
(x)
∧
{0,1}
;
α-cut(F)
is the set of elements with
μ
F
(x)≥α
.
In the numeric domain, trapezoidal shape functions are often used, they are described by
μ
F
=
(
x
1
,
x
2
, x
3
, x
4
), where the range [
x
2
, x
3
] is the
core
, the interval ]
x
1,
x
4
[ is the
support
, the interval ]
x
1,
x
2
[ is the
increasing part or the
border
where the membership function is given by the line segment from (
x
1,
0)
to
(x
2
,1)
and the interval ]
x
3,
x
4
[ is the decreasing side or the
border
characterized by the segment from
(
x
3,
1)
to
(x
4
,0).
A trapezoidal shape Fuzzy Set is said to be
Monotonous
if it is has only an increasing or
a decreasing side but not both (
x
1
=x
2
or x
3
=x
4
). We say that is
unimodal
when it has both increasing and
decreasing sides (
x
1
≠x
2
and x
3
≠x
4
.)
Fuzzy Sets give meaning to linguistic terms (predicates, modifiers, comparators, connectors and
quantifiers), giving rise to a fuzzy logic where sentence
S
truth-value,
μ(S)
is in [0,1], being 0 completely
false, and 1 completely true. Conjunction and disjunction are extended by means of operators t-norm and
t-conorm (s-norm) respectively, satisfying the properties: boundary in [0,1], monotonicity, commutativ-
ity, associativity and neutral element (1 and 0 respectively). Most common used t-norm and t-conorm
couple are
minimum
and
maximum
operators.
SQLf Fuzzy Querying Language
There are several different proposals of Fuzzy Set based flexible querying languages; SQLf is one that
has been proposed by Bosc and Pivert (1995) from IRISA-ENSSAT. SQLf is a fuzzy extension of the
standard SQL that allows the use of fuzzy conditions in any place where SQL allows a classical or
boolean. Goncalves and Tineo (2001a; 2001b) have proposed extensions to SQLf.
