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Information in databases can be imperfect and this imperfection has several forms and causes. In some cases, a single value should be stored, but it is (partially) unknown. The uncertainty about which value to store leads to the aforementioned imperfection. In temporal databases, uncertainty can arise, concerning which temporal notion needs to be stored. Because in temporal databases, temporal notions influence the consistency with which the database models the reality, this uncertainty has a direct impact on the consistency of the model. To represent this temporal uncertainty, previous works have adapted fuzzy sets with conjunctive interpretation, an approach that might prove misleading. This work presents a model that represents the uncertainty using possibility and necessity measures, which are fuzzy sets with disjunctive interpretations.
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A Possibilistic Valid-timeModel
Jose Enrique Pons1 Christophe Billiet2 Olga Pons Capote1
Guy De Tre2
1 Department of Computer Science and Artificial IntelligenceUniversity of Granada, Spain{jpons,opc}@decsai.ugr.es
2 Department of Telecommunications and Information ProcessingGhent University, Belgium
{Christophe.Billiet,Guy.De.Tre}@telin.ugent.be
June 29, 2012
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1. Contents
The structure of the presentation is:
2 Motivation.
3 Context:
3.1 Temporal databases.
3.2 Possibilistic variables and fuzzy numbers.
4 Proposal: Interval evaluation by ill-known con-straints.
5 Analysis of proposed transformations.
6 Conclusions and future work.
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2. Motivation
• The study of fuzzy intervals is of particular in-terest in temporal database research.
• To optimize the storage of fuzzy temporal inter-vals, some transformations have been proposed.⇒ Information Lost.
• The proposal is a framework to deal with theevaluation of ill-known temporal intervals.
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I Before J I
J
I Equal J
-Time
J
I Meets J J
I Overlaps J J
I During J J
I Starts J
I Finishes J
J
J
Relations
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3. Context
3.1 Temporal databases
3.2 Possibilistic variables and fuzzy numbers
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3.1. Temporal Databases:
A temporal database is a database that managesthe time in its schema.
• The time is usually represented as an interval inthe database.
X Y
• The user provides a crisp temporal interval in thequery specification.
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Example:
ID Name Year Works for Start Year End Year1 Peter 1987 John 2010 -2 Maria 1978 John 2001 -3 John 1954 - 1999 -4 Sarah 1982 Maria 2005 2009
Consider that ID is the primary key.Problem: If Sarah is hired in 2010, we can notinsert the new tuple.
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Example:
New primary key:
{ ID ∪[ Start Year, End Year ]}
ID Name Year Works for Start Year End Year1 Peter 1987 John 2010 -2 Maria 1978 John 2001 -3 John 1954 - 1999 -4 Sarah 1982 Maria 2005 20094 Sarah 29 Maria 2010 -
Also a consistence mechanism must be defined...
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Example:
Some spurious values might be inserted:
ID Name Year Works for Start Year End Year1 Peter 1987 John 2010 -2 Maria 1978 John 2001 -3 John 1954 - 1999 -4 Sarah 1982 Maria 2005 20094 Sarah 29 Maria 2001 20074 Sarah 1982 Maria 2010 -
Usually, DML sentences (insert, update, delete) arere-defined to ensure consistency.
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3.1.1. Valid-time DML
ID Entity Start End3 E.U. 15/3/2012 -
• Modify.
• Insert.
• Delete.
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Modify:
ID Entity Start End3 E.U. 15/3/2012 30/3/20123 E.U. 4/4/2012 UC
• Insert new information about an existing entity.
• This operation does not remove any previous
value for the entity.
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Insert:
ID Entity Start End3 E.U. 15/3/2012 30/3/20124 N.A.T.O. 25/3/2012 4/4/20123 E.U. 4/4/2012 UC
There are two main cases:
1. The entity is not in the relation. E.g., entity with
ID = 4.
2. The entity is already in the relation. E.g. entity
with ID = 3.
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Insert:
ID Entity Start End3 E.U. 15/3/2012 30/3/20124 N.A.T.O. 25/3/2012 4/4/20123 E.U. 4/4/2012 11/6/20123 E.U. 12/6/2012 -
If the entity is already in the relation, then:
• Insert the new version for the entity if it does not
overlap any other version.
• Modify and close the current version of the entity
and insert the new version.
• Reject the insertion if the time interval for the
entity does overlap any existing valid-time for the
entity.
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Delete:
ID Entity Start End4 N.A.T.O. 25/3/2012 4/4/2012
Removes all the versions for a given entity. For ex-
ample consider the deletion of the entity with ID =
3.
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3.1.2. Temporal models
• Valid-Time: The time when the fact is true inthe modelled reality.
• Transaction time: The time when the fact isstored in the database.
• Decision-time: The time when an event wasdecided to happen.
• Bi-temporal: Valid and transaction time.
• Tri-temporal: Valid, transaction and decisiontime.
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3.1.3. Temporal operators:
Allen’s temporal relation between intervals:Operator Inverse RepresentationX Before Y After XXX YYY
X Equal Y Equal XXXY Y Y
X Meets Y Meet by XXXYYY
X Overlaps Y Overlap by XXXY Y Y
X During Y Contains XXXY Y Y Y Y Y
X Starts Y Start by XXXY Y Y Y Y Y
X Finishes Y Finished by XXXY Y Y Y Y Y
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3.2. Possibilistic variables and fuzzy
numbers
Two different natures for a fuzzy set:
• Conjunctive nature: The fuzzyfication of aregular set. This interpretation corresponds withthe following two semantics: Degree of prefer-ence and degree of similarity.
• Disjunctive nature: In this case, the disjunc-tive nature indicates a description of incompleteknowledge. This interpretation corresponds withthe semantics for the degree of uncertainty.
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Possibilistic Variable:A possibilistic variable X over a universe U is de-fined as a variable taking exactly one value in U ,but for which this value is (partially) unknown.The possibility distribution πX gives the availableknowledge about the value that X takes. For eachu ∈ U , πX(u) represents the possibility that Xtakes the value u.
-
6
1
N1 2 3 4
r
r
πX
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It is important to understand the difference betweenthe following two concepts:
• A possibilistic variable X is bounded to takeonly one value , but this value is not known dueto incomplete knowledge.
• An ill-known set : a possibilistic variable definedover the universe P(U).
Note that while a possibilistic variable refers to one(partially) unknown value, an ill-known set is a crispset but, for some reason, (partially) unknown.
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Fuzzy numbers and fuzzy intervalsA fuzzy interval is a fuzzy set M on the set of realnumbers R such that:
∀(u, v) ∈ R2 :
∀w ∈ [u, v] : µM(w) ≥ min(µM(u), µM(v))
∃m ∈ R : µM(m) = 1
If m is unique, then M is referred to as a fuzzynumber, instead of a fuzzy interval.
1
0
possibility
values
D-a D D+b
1
0
possibility
α β γ δ
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4. Interval evaluation by ill-known constraints
4.1 Constraint
4.2 Ill-known constraint
4.3 Example
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4.1. Constraint:
Given a universe U , a constraint C on a set A ⊆ Uis specified by means of the binary relation R ⊆ R2
and a fixed value x ∈ U :
C4= (R, x)
It is said that a set A satisfies the constraint C ifand only if:
∀a ∈ A : (a, x) ∈ R.
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4.2.
Ill-known constraint:
Given a universe U , an ill-known constraint C on a set A ⊆ U is specified bymeans of a binary relation R ⊆ U2 and an ill-known value X, i.e.:
C4= (R,X) .
The uncertainty that a set A ⊆ U satisfies C is given by:
Pos(C(A)) = mina∈A
(Pos(a,X) ∈ R
)= min
a∈A
(sup
(a,w)∈R
πX(w)
)Nec(C(A)) = min
a∈A
(Nec(a,X) ∈ R
)= min
a∈A
(inf
(a,w)/∈R1− πX(w)
)
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4.3.
Consider the two ill-known values X and Y .
X Y
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Allen Relation Constraints B(C1(I), ..., Cn(I)
)I before J C1
4= (<,X) C1(I)
I equal J
C14= (≥, X) C1(I) ∧ ¬C2(I) ∧ C3(I) ∧ ¬C4(I)
C24= (6=, X)
C34= (≤, Y )
C44= (6=, Y )
I meets JC1
4= (≤, X) C1(I) ∧ ¬C2(I)
C24= (6=, X)
I overlaps JC1
4= (<, Y ) C1(I) ∧ ¬C2(I) ∧ ¬C3(I)
C24= (≤, X)
C34= (≥, X)
I during J
C14= (>,X)
(C1(I) ∧ C2(I)
)∨(C3(I) ∧ C4(I)
)C2
4= (≤, Y )
C34= (≥, X)
C44= (<, Y )
I starts JC1
4= (≥, X) C1(I) ∧ ¬C2(I)
C24= (6=, X)
I finishes JC1
4= (≤, Y ) C1(I) ∧ ¬C2(I)
C24= (6=, Y )
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5. Analysis of proposedtransformations
Optimize storage⇒ Transformation from two fuzzynumbers to a fuzzy interval.
2 main proposals:
• Transf. Preserving the imprecision.
• Transf. based on the convex-hull.
Drawbacks:
• (Dubois and Prade): the fuzzy interval is apossibility distribution on R while the twofuzzy numbers are a set that belong to P(R).
• The lack of the necessity measure, used forranking purposes.
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5.1. Transformation that preserves the
imprecision
1
0
possibility
ds-as ds
1
0
possibilityds+bs
de-ae de de+be
S1
S2
S3
S4
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5.2. Transformation based on the convex
hull
1
0
possibility
1
0
possibility
ds deds-as ds+bs de-ae de+be
ds-as ds de de+be
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ComparativeConsider two ill-known points representing a timeinterval: X = [3, 2, 1] and Y = [7, 2, 3]The value for I = [a, b] is [3, 6]The relation R: I is inside X :
Method Possibility NecessityIll-known constraint 1 0.5
Preserving the imprecision 0.667 -Convex hull 1 -
Nec (C (A)) > 0⇐⇒ Pos (C (A)) = 1
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Pos+Nec
0
1
2
Poss Nec
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6. Conclusions
• The necessity measure is lost when dealing witha transformation.
• The possibility measure in the transformations is(w.r.t. the ill-known evaluation):
– Convex hull returns the same value as possi-bility.
– The preserving the imprecision approach re-turns a different value.
• If the support for the ill-known values do overlap,it is not possible to compute any transformations.
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Future work:
• A new theoretical model for valid-time databases.
• Extension of the Allen’s relations for the compar-ison between two ill-known values.
• Implementation of the theoretical model in a re-lational database.
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Thank you!
Questions?
Contact:
http://decsai.ugr.es/˜ jpons
