About the family of closure systems preserving non unit implications in the guigues duquenne base
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About the family of Closure Systems preserving non unit implications in the Guigues-Duquenne Base. Alain Gély & Lhouari Nourine. LIMOS – Clermont-Ferrand - France. ICFCA’06 - Dresden. Definitions & Problematic. Incremental Approach. Implications in the Guigues-Duquenne base.

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About the family of closure systems preserving non unit implications in the guigues duquenne base
About the family of Closure Systems preserving non unit implications in the Guigues-Duquenne Base

Alain Gély & Lhouari Nourine

LIMOS – Clermont-Ferrand - France

ICFCA’06 - Dresden


Definitions & Problematic implications in the Guigues-Duquenne Base

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 implications in the Guigues-Duquenne Base

1234

123

134

124

234

12

13

14

23

24

34

1

2

3

4

F

Closure system

123

134

M(F)

124

234

Meet-irreducible elements

Implicational base


1234 implications in the Guigues-Duquenne Base

123

134

124

234

12

13

14

23

24

34

1

2

3

4

F

Closure system

123

4 1

134

M(F)

124

23

Meet-irreducible elements

Implicational base


1234 implications in the Guigues-Duquenne Base

123

134

124

234

12

13

14

23

24

34

4

1

2

3

F

Closure system

123

4 1

124

13  2

M(F)

14

23

Meet-irreducible elements

Implicational base


1234 implications in the Guigues-Duquenne Base

123

134

124

234

12

13

14

23

24

34

4

1

2

3

F

Closure system

123

4 1

124

13  2

M(F)

14

34 2

23

Meet-irreducible elements

Implicational base


1234 implications in the Guigues-Duquenne Base

123

134

124

234

12

13

14

23

24

34

4

1

2

3

F

Closure system

123

4 1

124

13  2

M(F)

14

34 2

23

Meet-irreducible elements

Minimal Implicational base


1234 implications in the Guigues-Duquenne Base

123

134

124

234

12

13

14

23

24

34

4

1

2

3

Polynomial

Polynomial

Polynomial

Polynomial

?

?

F

123

4 1

124

13  2

M(F)

14

23


1234 implications in the Guigues-Duquenne Base

123

134

124

234

12

13

14

23

24

34

1

2

3

4

F

Polynomial

Polynomial

Process 2n operations

123

124

M(F)

134

234

Output size : 0

Input size : n


? implications in the Guigues-Duquenne Base

?

A question arise :

What happen if the set of implications is modified…

For the implicational base

For the meet-irreducible elements

Incremental Approach

123

4 1

124

13  2

M(F)

14

+/- x  y

23


What about Implications in the Guigues-Duquenne base 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


1234 implications in the Guigues-Duquenne Base

123

124

134

13

14

24

12

4

3

1234

1234

123

134

124

234

13

12

13

14

23

24

34

1

2

3

4

1

2

4  1234

3  13

12  1234


Size of premise > 1 implications in the Guigues-Duquenne Base

Premise is singleton

 =

J



Unit implications

Non unit implications

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

1234

Let F be a closure system

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

13

1

2

3  13

12  1234

4  1234


Unit Implications implications in the Guigues-Duquenne Base

  • 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 implications in the Guigues-Duquenne Base

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 implications in the Guigues-Duquenne Base 



Add an implication a b :

123

J

2  23

{a b} shall not be a Guigues-Duquenne Base

123

13

23

12

13

23

12

1

2

3

1

2

3

Add Unit Implications



12  123

J

2  3


1. Closure of P may change implications in the Guigues-Duquenne Base

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 implications in the Guigues-Duquenne Base

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 implications in the Guigues-Duquenne Base

123

12

12

1

123

cover relation in C(F)

 = {}

J = {3  123, 2  12}

 = {}

J = {3  123, 2  12, 112}

 = {}

J = {3  123, 2  123, 1123}


Result implications in the Guigues-Duquenne Base

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 implications in the Guigues-Duquenne Base  1

1 2

3 2

123

12 123

13

23

1

2

3

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 implications in the Guigues-Duquenne Base

3  13

4  24

13

24

12 1234

1

2

Family of J - equivalent closure systems is a closure system

[Nation & Pogel 97]

1234

1234

3  13

3  13

123

4  24

124

4  24

13

13

24

12 123

24

12 124

124 1234

123 1234

2

1

2

1


1234 implications in the Guigues-Duquenne Base

1234

123

123

14

12

24

12

2

1

1

2

1234

F’’ is not

 - equivalent

to F

123

3 123

12

4 1234

2

1

F’’

Family of  - equivalent closure systems is not a closure system

124  1234

124  1234

3 123

3 123

4 14

4 24

F

F’


Result implications in the Guigues-Duquenne Base

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


123 implications in the Guigues-Duquenne Base

(i’) et (ii)

Isomorphism between A and A*

A

13

23

12

A*

1

2

3

(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 implications in the Guigues-Duquenne Base

A*

3

Reduction from F to F’

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

example

3  1

123

A

13

12

1

2


A implications in the Guigues-Duquenne Base

A*

Reduction from F to F’

Evolution of meet-irreducible elements


Sufficient and necessary conditions in polynomial time implications in the Guigues-Duquenne Base

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 implications in the Guigues-Duquenne Base

  • Links between implications in  and J

  • What happen with other bases ?

  • Structural Properties of C(F)

  • Efficient algorithms to add an implication


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