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


About the family of closure systems preserving non unit implications in the guigues duquenne base

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


About the family of closure systems preserving non unit implications in the guigues duquenne base

Definitions & Problematic

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


About the family of closure systems preserving non unit 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

4 1

134

M(F)

124

23

Meet-irreducible elements

Implicational base


About the family of closure systems preserving non unit implications in the guigues duquenne base

1234

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


About the family of closure systems preserving non unit implications in the guigues duquenne base

1234

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


About the family of closure systems preserving non unit implications in the guigues duquenne base

1234

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


About the family of closure systems preserving non unit implications in the guigues duquenne base

1234

123

134

124

234

12

13

14

23

24

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4

1

2

3

Polynomial

Polynomial

Polynomial

Polynomial

?

?

F

123

4 1

124

13  2

M(F)

14

23


About the family of closure systems preserving non unit implications in the guigues duquenne base

1234

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


About the family of closure systems preserving non unit 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


About the family of closure systems preserving non unit implications in the guigues duquenne base

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


About the family of closure systems preserving non unit implications in the guigues duquenne base

1234

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


About the family of closure systems preserving non unit implications in the guigues duquenne base

Size of premise > 1

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


About the family of closure systems preserving non unit implications in the guigues duquenne base

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)


About the family of closure systems preserving non unit implications in the guigues duquenne base

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}


About the family of closure systems preserving non unit implications in the guigues duquenne base

Modification of 



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


About the family of closure systems preserving non unit implications in the guigues duquenne base

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


About the family of closure systems preserving non unit implications in the guigues duquenne base

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


About the family of closure systems preserving non unit implications in the guigues duquenne base

123

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}


About the family of closure systems preserving non unit implications in the guigues duquenne base

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


About the family of closure systems preserving non unit implications in the guigues duquenne base

3 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


About the family of closure systems preserving non unit implications in the guigues duquenne base

1234

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


About the family of closure systems preserving non unit implications in the guigues duquenne base

1234

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’


About the family of closure systems preserving non unit implications in the guigues duquenne base

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


About the family of closure systems preserving non unit implications in the guigues duquenne base

123

(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* =


About the family of closure systems preserving non unit implications in the guigues duquenne base

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

13

12

1

2


About the family of closure systems preserving non unit implications in the guigues duquenne base

A

A*

Reduction from F to F’

Evolution of meet-irreducible elements


About the family of closure systems preserving non unit implications in the guigues duquenne base

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


About the family of closure systems preserving non unit implications in the guigues duquenne base

Perspectives

  • 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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