Thuyết trình cơ sở dữ liệu nâng cao fuzzy orderings in flexible query answering systems

Advanced Database Systems
Fuzzy Orderings in Flexible Query
Answering Systems
Lê Hồng Dũng – 7140819
Âu Mậu Dương – 7140820
Lê Nguyên Dũng – 7140224
Ngô Đình Dũng – 1570203


Content
Introduction
VQS
Conditions based on fuzzy orderings
The aggregation issue
Summary


1/ The vague query system (VQS)
VQS is an add-on to conventional relational databases which acts as a
proxy between the user and the database
Since VQS communicates with the underlying database only on the
basis of standard SQL which allows easy integration into existing
applications.
In case of numeric attributes, considering Euclidean distances is most
often sufficient.
For non-numeric attributes, VQS using a so-called NCR table (numeric
coordinate representation)


1/ The vague query system (VQS)
The syntax of VQL

The operator ‘‘IS’’ should be understood in the sense of ‘‘is
similar to’’


1/ The vague query system (VQS)
There is one single ‘‘IS’’ condition in the query, VQS retrieves all records
from the data source and ranks them according to the distance from the
query value.
In case that the column contains numeric values, the distance between two values x;
y can easily be computed as the absolute value of the difference |x - y| (Euclidean
norm for the one-dimensional real space R).
If the column under consideration is non-numeric, the distance is computed as the
distance of the associated values in the corresponding NCR table.

VQS works with normalized distances. Every condition, therefore, is
assigned a distance value normalized to the unit interal [0;1].


1/ The vague query system (VQS)
In case that two or more ‘‘IS’’ conditions are combined with ‘‘AND’’,
a weighted average of the distances in the different columns is used
to rank the results (equal weights are used by default, which can be
overridden using the optional ‘‘WEIGHTED BY’’ expression).


VÍ DỤ 1
Xét bảng dữ liệu

VÍ DỤ 1
Bảng 2. Ma trận khoảng cách của Location trong bảng 1 (bảng NCR)


VÍ DỤ 1
Consider the following query:
SELECT FROM HotelTable
WHERE Location IS ‘Salzburg Center’
AND Price IS 70
AND Category IS 4
INTO ResultSet
Assume that the distances between locations are given as in Table 2 (as the result of
computing a distance measure for corresponding values in an NCR table).
To compute the result set for this query which means that the distance of any two
locations is divided by 147.8, each discrepancy in the price is divided by 60, and
each discrepancy in the category is divided by 2


VÍ DỤ 1
Using equal weights, we obtain the result set shown in Table 3 sorted
by the closeness to the query


NHƯỢC ĐIỂM CỦA VQS
Disadvantages: Example 1 clearly demonstrates two severe shortcomings of
VQS:
1. VQS is restricted to ‘‘IS’’ queries that are interpreted with a certain tolerance for
imprecision. For the price column, however, this is a painful limitation, as the user is not
necessarily interested in a price that is as close to 70 as possible, but more likely in a
price that exceeds 70 as little as possible.

2. The normalization of distances is done for all columns independently solely on the
basis of the largest distance between two values in the column. The result is that two
distance values for different columns may be difficultly comparable.

This paper is to tackle the first problem.


Ngôn ngữ oVQL (Ordering-enriched vague query language)
Ngôn ngữ oVQL (Ordering-enriched vague query language)


Ngôn ngữ oVQL (Ordering-enriched vague query language)
It is obvious that oVQL differs from VQL in the respect that there is
an explicit distinction between numeric and non-numeric attributes.
For non-numeric ones, only the ‘‘IS’’ condition is defined like in VQL
For numeric ones, three new types of conditions ‘‘IS AT LEAST’’, ‘‘IS AT
MOST’’, and ‘‘IS WITHIN’’ are added.


Content
Introduction
VQS
Conditions based on fuzzy orderings
The aggregation issue
Summary


Conditions based on fuzzy orderings
The single conditions ‘‘IS’’, ‘‘IS AT LEAST’’, ‘‘IS AT MOST’’, and ‘‘IS
WITHIN’’ are modeled.

Example: Price IS AT MOST 70


Conditions based on fuzzy orderings

Definition 1: Fuzzy equivalence relation


A binary fuzzy relation E: X2 → [0,1] is called fuzzy equivalence relation with respect to a t-norm T, for
brevity T-equivalence, if and only if the following three axioms are fulfilled for all x, y, z X:
1. Reflexivity: E(x,x) = 1.
2. Symmetry: E(x,y) = E (y,x).
3. T- transitivity: T(E(x,y), E(y,z)) E(x,z)


Conditions based on fuzzy orderings

Definition 2: fuzzy ordering



A fuzzy relation L: X2 → [0,1] is called fuzzy ordering with respect to a t-norm T and a T-equivalence E, for
brevity T-E-ordering, if and only if it is T-transitive and fulfills the following two axioms for all x, y
1. E-Reflexivity: E(x,y) L(x,y).
2. T-E-antisymmetry: T(L(x,y),L(y,x))


Conditions based on fuzzy orderings
Definition 3:
A crisp ordering on a domain X and a T-equivalence E: X2 → [0,1] are

called compatible if and only if the following holds for all x, y, z X:
x

y

z => E(x,z)


Conditions based on fuzzy orderings
Theorem 1: [1, 2] Consider a fuzzy relation L on a domain X and a T-equivalence E. Then
the following two statements are equivalent:
L is a strongly complete T-E-ordering.
There exists a linear ordering
follows:
• L(x,y) =

the relation E is compatible with such that L can be represented as


Conditions based on fuzzy orderings

Theorem 2: Consider a continuous Archimedean t-norm T with additive generator f.

For any pseudo-metric d: X2 → [0,] the mapping Ed: X2 → [0,] defined as
Ed(x,y) = f-1 ( min (d(x,y), f(0) ) (1)
Provided that E: X2 → [0,] is a T-equivalence, we can define a pseudo-metric
dE: X2 → [0,] as dE (x,y) = f (E(x,y))

(2)

Conditions based on fuzzy orderings

Proposition 1: Let T be a continuous Archimedean t-norm with an additive
generator f and let be an ordering of the domain X.
If a pseudo-metric d: X2 → [0,], fulfills
x

y

z => d(x,z) (3)

If a fuzzy equivalence relation E: X2 → [0,1] is compatible with , , its induced pseudo-metric
dE defined as in (2), fulfills property (3).


Conditions based on fuzzy orderings

Some important formula be deduced from theorem 1&2:
EC(x,y) = max(1 ) (TL-equivalence)
E’C(x,y) = exp() (TP-equivalence)
The value C is obviously the maximal distance of two objects x and y.


Conditions based on fuzzy orderings
Some important formula be deduced from theorem 1&2:
LC(x,y) =

(TL-Ed,C-ordering)

L’C(x,y) =

(TP-Ed,C-ordering)


Conditions based on fuzzy orderings
Formula is applied following as:
t(“x IS q” | x0) = EC(x0,q)
t(“x IS AT LEAST q” | x0) = LC(q, x0)
t(“x IS AT MOST q” | x0) = LC(x0, q)
t(“x IS WITHIN (a,b)” | x0) = min(LC(min(a, b), x0), LC(x0, max(a, b)))


Content
Introduction
VQS
Conditions based on fuzzy orderings
The aggregation issue
Summary