Constraint Satisfaction Problems and Consistency Restoration

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Constraint Satisfaction Problems and Consistency
Restoration

• Florent Launay
• Undergraduate student at FIT

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Summary

• Focused interest in time point-sequencing
  problem.
• What are CSPs ?
• Crossword Problem.
• Dealing with consistency.
• 1. Finding and storing consistent relations
• 2. Finding a minimal set to remove
  inconsistency
• Linking the algorithms
• Questions, comments.
• The Program

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Focus interest

• Relation from lt, gt, , ltgt, lt, gt, ltgt

• Here, we can see that some constraints between
  the set of points P1, P2, P3 exist
• P1 lt P2
• P2 lt P3

P1
P2
P3
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What are CSPs ?

• Constraints

X gt 5 X ltgt 9 X lt 12
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What are CSPs ?

• Example
• Common names from English vocabulary beginning
  with a c, ending with a t, but that doesnt
  contain any e

Valid set count, cart, chat, court,
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What are CSPs ?

• Valid regions and Box

X gt 5
X ltgt 9
X lt 12
X lt 15
X gt 3
5
9
12
15
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Valid regions 5,5, 5,9, 9,12
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What are CSPs ?

• Inconsistencies

X gt 5
X ltgt 9
X lt 12
X gt 13
5
9
12
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No valid region gt The system is inconsistent
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The crossword problem

• Constraints

Definition, Size of the world, across or down,
eventual existing letters. Definition 13 Part
of the human body, beginning with an
H (ambiguous)
W
O
E A D ?
A N D ?
A I R
H
N O T
A N D

• Valid Set

HEAD, HAND, HAIR

• Inconsistency

Definition 14 Negation
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Dealing with consistency

• Within a set of relations, some relation can
  conflict with other relations in a system, making
  the whole system inconsistent.

Lets have a look to the following example.
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A simple example

• Let s consider a point P1 on a time line basis

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A simple example

• P2gtP1

P3ltP1 and P3gtP2
Valid region
P2
P1
No valid region can be found for P3 !!!
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Dealing with consistency

• With that kind of inconsistency, two interesting
  questions arise
• Which are the relations that make the system
  inconsistent ?
• What would be a minimal set to remove to make the
  system consistent ?

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1. Finding and storing inconsistent relations
System of relations
Consistent
NO
Store inconsistent relations
YES
Finish
Finish (Call the 2nd algorithm)
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Inconsistent relations

• FIND MULTIPLE EQUALITY
• gtStore inconsistent relations

• VERIFY CONSISTENCY IN CASE OF UNIQUE EQUALITY
• gtStore inconsistent relations

• FIND THE BOX PRE-PROCESSING
• gtStore inconsistent relations

• FIND VALID REGIONS
• gtIf no inconsistency has been found

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Inconsistent relations

• FIND MULTIPLE EQUALITY

If Pnew P3
And Pnew P6
The system is inconsistent since P3ltgtP6
P2
P1
P3
P4
P5
P6
Pnew
Pnew
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Inconsistent relations

• VERIFY CONSISTENCY IN CASE OF UNIQUE EQUALITY

If Pnew P3
And Pnew lt P2
The system is inconsistent
P2
P1
P3
P4
P5
P6
Pnew
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Inconsistent relations

• FIND THE BOX PRE-PROCESSING

If If we found that a valid region for Pnew was
P3,P4, P4,P5
Then, no other relation should forbid that ex
Pnew gt P6, Or the system becomes inconsistent
P2
P1
P3
P4
P5
P6
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Inconsistent relations

• FIND VALID REGIONS
• gtIf no inconsistency has been found

A valid region region could be P2,P3,
P3,P3, P3,P4, P4,P5, P5,P5, P5,P6
P2
P1
P3
P4
P5
P6
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Inconsistent relations
End of the first algorithm
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2. Finding a minimal set to remove to make the
system consistent
SET OF RELATIONS MAKING THE SYSTEM INCONSISTENT
Algorithm
MINIMUM SET OF RELATIONS MAKING THE SYSTEM
INCONSISTENT
MAXIXMUM SET OF CONSISTENT RELATIONS
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2. Finding a minimal set to remove to make the
system consistent

• This short algorithm finds a minimal set. We
  will illustrate this with an other example. But
  first, we have to introduce the notion of degree
  of inconsistency.

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Degree of inconsistency

• Here, if we want to add a new point Pnew such
  that
• Pnew lt P1, Pnew gt P5
• Pnew lt P2, Pnew gt P6
• Pnew lt P3,
• Pnew lt P4,

P2
P1
P3
P4
P5
P6
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Degree of inconsistency

• From here, we can see that
• Pnew lt P1 conflicts with Pnew gt P5 Pnew lt P3
  conflicts with Pnew gt P5
• Pnew lt P1 conflicts with Pnew gt P6 Pnew lt P3
  conflicts with Pnew gt P6
• Pnew lt P2 conflicts with Pnew gt P5 Pnew lt P4
  conflicts with Pnew gt P5
• Pnew lt P2 conflicts with Pnew gt P6 Pnew lt P4
  conflicts with Pnew gt P6

P2
P1
P3
P4
P5
P6
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• Questions before we move to the Algorithm ?

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Algorithm

• Var
• S set of all relations making the system
  inconsistent
• nS number of relations in S
• R Temporary relation with highest degree of
  inconsistency in S
• mR degree of inconsistency of R
• I a set of minimal relation to remove in
  order to make the system consistent

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Algorithm

• Initialize I to be empty
• While (nS ! 0)
• Find the relation R with highest inconsistent
  degree m
• nS nS mR
• Remove 1 from every inconsistent degree from
  every relation conflicting with R
• Record R in I
• Remove every inconsistent relation
  conflicting with R from S
• Remove R from S

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Example

• Let us consider the last example. Here are the
  variables of the algorithm.

S
2 Pinc lt P1 Pinc gt P5 4
S P1,P2, P3, P4, P5, P6
2 Pinc lt P1 Pinc gt P6 4
2 Pinc lt P2 Pinc gt P5 4
mR 0
2 Pinc lt P2 Pinc gt P6 4
R
nS relations
2 Pinc lt P3 Pinc gt P5 4
nS 8
2 Pinc lt P3 Pinc gt P6 4
I
2 Pinc lt P4 Pinc gt P5 4
2 Pinc lt P4 Pinc gt P6 4
Degree of inconsistency
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Example

• One relation with highest inconsistency degree is
  
• Pinc gt P5.

S
2 Pinc lt P1 Pinc gt P5 4
1
S P1,P2, P3, P4, P5, P6
2 Pinc lt P1 Pinc gt P6 4
1
2 Pinc lt P2 Pinc gt P5 4
mR 0
4
1
PincgtP5
2 Pinc lt P2 Pinc gt P6 4
R
1
2 Pinc lt P3 Pinc gt P5 4
nS 8
8 4 4
1
2 Pinc lt P3 Pinc gt P6 4
I
PincgtP5
1
2 Pinc lt P4 Pinc gt P5 4
1
2 Pinc lt P4 Pinc gt P6 4
1
Remove 1 from every inconsistent degree from
every relation conflicting with R
Find the relation R with highest inconsistency
degree
nS nS mR
Record R in I
Remove every inconsistent relation conflicting
with R from S
Remove R from S
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Example

• The variables are now

The relation with highest inconsistency degree is
now PincgtP6.
S P1,P2, P3, P4, P6
S
mR 4
4
PincgtP6
R PincgtP5
1 Pinc lt P1 Pinc gt P6 4
0
nS 4
4 4 0
1 Pinc lt P2 Pinc gt P6 4
0
I PincgtP5
PincgtP5 , PincgtP6
1 Pinc lt P3 Pinc gt P6 4
0
1 Pinc lt P4 Pinc gt P6 4
0
n 0, the system is now consistent, and a
minimal set to remove is found I PincgtP5,
PincgtP6
Remove 1 from every inconsistent degree from
every relation conflicting with R
nS nS mR
Find the relation R with highest inconsistency
degree
Record R in I
Remove every inconsistent relation conflicting
with R from S
Remove R from S
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Linking the algorithms
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Linking the algorithms
Choose input
Set constraints
Perform algorithm
Consistent ?
Store Input
Yes
No
Find Set of inconsistent relations
S
Find minimal set to remove to make the system
consistent
I
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• Constraint programming represents one of the
  closest approaches computer science has yet made
  to the Holy Grail of programming the user states
  the problem, the computer solves it.
• Eugene C. Freuder, CONSTRAINTS, April 1997

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• Questions, comments ?

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• The Program