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About the family of Closure Systems preserving non unit implications in the Guigues-Duquenne Base

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Alain Gély & Lhouari Nourine

LIMOS – Clermont-Ferrand - France

ICFCA’06 - Dresden

Definitions & Problematic

Incremental Approach

Implications in the Guigues-Duquenne base

Results about the family of closure systems preserving non unit implications in the Guigues-Duquenne Base.

Conclusions & Perspectives

Definitions & Problematic

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F

Closure system

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134

M(F)

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234

Meet-irreducible elements

Implicational base

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134

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234

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F

Closure system

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4 1

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M(F)

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Meet-irreducible elements

Implicational base

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234

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1

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F

Closure system

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4 1

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13 2

M(F)

14

23

Meet-irreducible elements

Implicational base

1234

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134

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234

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4

1

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F

Closure system

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4 1

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13 2

M(F)

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34 2

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Meet-irreducible elements

Implicational base

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234

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F

Closure system

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4 1

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13 2

M(F)

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34 2

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Meet-irreducible elements

Minimal Implicational base

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Polynomial

Polynomial

Polynomial

Polynomial

?

?

F

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4 1

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13 2

M(F)

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23

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F

Polynomial

Polynomial

Process 2n operations

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M(F)

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Output size : 0

Input size : n

?

?

A question arise :

What happen if the set of implications is modified…

For the implicational base

For the meet-irreducible elements

Incremental Approach

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4 1

124

13 2

M(F)

14

+/- x y

23

What about Implications in the Guigues-Duquenne base

To study changes in an implicationnal base,

We need to choose a canonical minimal base :

The Guigues-Duquenne base

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Size of premise > 1

Premise is singleton

=

J

Unit implications

Non unit implications

Canonical minimum base(Guigues-Duquenne base) [Guigues & Duquenne 86]

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Let F be a closure system

= { P P | P a pseudo-closed set of F }is a minimum implicational base for F.

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1

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3 13

12 1234

4 1234

Unit Implications

- Implications in J are
- Easy to compute from M(F)
- In polynomial number relative to M(F)

Non Unit Implications

Implications in may be

- Not easy to compute from M(F)
- In exponential number relative to M(F)

Interesting if

J

Modify Jwithout modify

We look for - equivalent closure systems

Example of application :

- |F’| ≤ |F|
- |M(F’) | ≤ |M(F)|

F’

F

M(F’)

M(F)

{a b}

Modification of

Add an implication a b :

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J

2 23

{a b} shall not be a Guigues-Duquenne Base

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Add Unit Implications

12 123

J

2 3

1. Closure of P may change

Premise is not anymore a pseudo-closed set because…

Three cases of problem

P P

2. It is not anymore a quasi-closed set

3. It remain a quasi-closed set, but not minimal

Result

Keep conclusion

Remains a quasi-closed set

Remains a minimal quasi-closed set

Characterization : -equivalence addinga b

a b may be added without modification of iff

For all P P

(i) if a P then b P

(ii)if a P then b P

(iii)if a j , j P, then (jb) ≠P

123

123

12

12

1

123

cover relation in C(F)

= {}

J = {3 123, 2 12}

= {}

J = {3 123, 2 12, 112}

= {}

J = {3 123, 2 123, 1123}

Result

Characterization : cover relation in C(F)

a b may be added without modification of ,

andF’ covers FinC(F)iff

(i’)For all P P , P≠ a , if a P then b P

(ii)For all P P if a P then b P

(iii’)For all P P , if a P then (ab) ≠P

3 1

1 2

3 2

123

12 123

13

23

1

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3 2

3 1

123

123

12 123

12 123

13

23

3 23

3 13

1

2

1

2

3 1

3 2

123

12 123

3 123

1

2

1234

3 13

4 24

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12 1234

1

2

Family of J - equivalent closure systems is a closure system

[Nation & Pogel 97]

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3 13

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F’’ is not

- equivalent

to F

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3 123

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2

1

F’’

Family of - equivalent closure systems is not a closure system

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124 1234

3 123

3 123

4 14

4 24

F

F’

Result

Conditions on implications

Characterization : cover relation in C(F)

a b may be added without modification of ,

andF’ covers FinC(F)iff

(i’)For all P P , P≠ a , if a P then b P

(ii)For all P P if a P then b P

(iii’)For all P P , if a P then (ab) ≠P

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(i’) et (ii)

Isomorphism between A and A*

A

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23

12

A*

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(iii’)

A (A* B) F F

Detection : can I add the implication a b ? (using only M(F) )

Athe closure of a, B the closure of b in F

example

3 1

A*immediate predecessor ofA in F

A family of sets F such that a F and b F

A* family of closed sets F such that A* F, a F and b F

A (A* B)

B =

A =

A* =

23

A*

3

Reduction from F to F’

A family of sets F such that a F and b F

example

3 1

123

A

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1

2

A

A*

Reduction from F to F’

Evolution of meet-irreducible elements

Sufficient and necessary conditions in polynomial time

Transformation of the data in polynomial time

New data is smaller than the original one

Closure system is smaller than the original one

Interesting method to reduce data

Conclusion

we can add an implicationa b

Perspectives

- Links between implications in and J
- What happen with other bases ?
- Structural Properties of C(F)
- Efficient algorithms to add an implication