G??sse?%20pe?????s

1
G??sse? pe?????sµ??

• ?? e??a? ?? pe?????sµ??(constraints)?
• ???te??p???s? pe?????sµ??
• ?p???s? pe?????sµ??
• ???d?a pe?????sµ??
• ???e? pe?????? pe?????sµ??
• Properties of constraint solving

2
?e?????sµ??
?etaß??t?? ap????e?s? t?µ??
S??a?t?s?a?? s?µß??a a?t?st????s? t?µ?? se t?µ??
S?es?a?? s?µß??a s??se?? µeta?? t?µ??
3
?e?????sµ??
St???e??d?? pe?????sµ?? s??se?? pe?????sµ?? µe
???sµata
unary
binary
?e?????sµ?? s?????s? st???e??d?? pe?????sµ??
4
??a??p???s?µ?t?ta
?p?t?µ?s? a???es? t?µ?? st?? µetaß??t??
?p???s? ap?t?µ?s? p?? ??a??p??e? t???
pe?????sµ???
5
??a??p???s?µ?t?ta
??a??p???s?µ? ? pe?????sµ?? ??e? ??s? ??
??a??p???s?µ? ? pe?????sµ?? de? ??e? ??s?
??a??p???s?µ?
?? ??a??p???s?µ?
6
S??ta?? pe?????sµ??

• ?? pe?????sµ?? e??a? s?µß???se????
• ?? a????e? de? ep??e????? (de? t??
  ???s?µ?p????µe)
• ? se??? ep??e??e?
• ?e????? a??????µ?? e?a?t??ta? ap? t?? se???

7
?s?d??aµ?a pe?????sµ??
??? d?af??et???? pe?????sµ?? µp??e? ?a
a?apa??st??? t?? ?d?a p????f???a
??? pe?????µ?? e??a? ?s?d??aµ?? ?ta? ????? t?
?d?? s????? ??se??
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???te??p???s? ???s?µ?p????ta? pe?????sµ???

• ?? pe?????sµ?? pe?????f??? ?deat?? s?µpe??f????
  a?t??e?µ???? st?? p?a?µat??? ??sµ?

9
??a??p???s? pe?????sµ??

• ?????t?? e??? pe?????sµ?? C ????µe d?? e??t?se??
• ??a??p???s?µ? ??e? ??s?
• ep???s? d?se µ?? µ?a ??s?, e?? ?p???e?
• ?? p??t? e??a? p?? ßas???
• ??a s?st?µa ep???s?? pe?????sµ?? apa?t?e? t?
  e??t?µa t?? ??a??p???s?µ?t?ta?

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??a??p???s? pe?????sµ??

• ??? apa?t?µe t?? e??t?s?
• ?p?? p??s????s? ap?t?µ?s? ???? t?? p???a???
  t?µ??(genarate test)

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Generate Test

• ???a??? ? p?? ?e???? µ???d?? ep???s??
• ???????µ??
• generate labelling
• test ??a??p???s?µ?t?ta?
• ?e???e?t?µata ?e?t??se??
• blind generator smart generator --gt local
  search
• a??? e?a???? ??e???? ?at? t?? e?a????
• as??epe??? --gt backtracking

12
Backtracking (BT)
Systematic Search Methods

• incrementally extends a partial solution towards
  a complete solution
• Algorithm
• assign value to variable
• check consistency
• until all variables labelled
• Drawbacks
• thrashing
• redundant work
• late detection of conflict

A
1
B
2
1
C
2
1
1
D
2
2
1
1
1
?
A D, B ? D, AC lt 4
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GT BT - Example
Systematic Search Methods

• Problem X1,2, Y1,2, Z1,2 X Y, X
  ? Z, Y gt Z
• generate test backtracking

14
Consistency Techniques

• removing inconsistent values from variables
  domains
• graph representation of the CSP
• binary and unary constraints only (no problem!)
• nodes variables
• edges constraints
• node consistency (NC)
• arc consistency (AC)
• path consistency (PC)
• (strong) k-consistency

Agt5
A
AltC
A?B
C
B
BC
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Arc Consistency (AC)
Consistency Techniques

• the most widely used consistency technique (good
  simplification/performance ratio)
• deals with individual binary constraints
• repeated revisions of arcs
• AC-3, AC-4, Directional AC

a b c
a b c
a b c
Y
X
Z
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AC - Example
Consistency Techniques

• Problem X1,2, Y1,2, Z1,2 X Y, X
  ? Z, Y gt Z

X
X
1 2
1 2
1 2
1 2
Y
Y
1 2
1 2
Z
Z
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Is AC enough?
Consistency Techniques

• empty domain gt no solution
• cardinality of all domains is 1 gt solution
• Problem X1,2, Y1,2, Z1,2 X ? Y, X
  ? Z, Y ? Z

X
1 2
1 2
Y
Z
1 2
18
Path Consistency (PC)
Consistency Techniques
V2

• consistency along the path only
• checking paths of length 2 is enough
• Plus/Minus
• detects more inconsistencies than AC
• - extensional representation of constraints
• - changes in graph connectivity
• Directional PC, Restricted PC

V4
V3
V5
V0
V1
???
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K -consistency
Consistency Techniques

• K-consistency
• consistent valuation o (K-1) variables can be
  extended to K-th variable
• strong K-consistency ? J-consistency for each
  J?K
• NC ? strong 1-consistency
• AC ? strong 2-consistency
• PC ? strong 3-consistency

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??a??p???s? pe?????sµ??

• ? ap?? p??s????s? de? d???e?e? ??a p??ß??µata t??
  p?a?µat???? ??sµ??
• ??a p?? ???p?? ??d?s? ???s?µ?p??e?ta? ??a
  pe?????sµ??? µe pepe?asµ??? ped?? ???sµ??
• Gauss-Jordan s??µa apa???f??

21
Gauss-Jordan s??µa apa???f??

• ?p????? e??s?s?? c ap? C
• ?a?a????e c st? µ??f? x e
• ??t??at?st?se t? x pa?t?? st? C µe e
• ?pa???aße µ????
• ??e? ?? e??s?se?? ?a e??a? t?? µ??f?? x e
• ? µ?a e??s?s? e??a? ?s?d??aµ? µe d 0 (d ! 0)
• ?p?st?e?e True st?? p??t? pe??pt?s? a????? False

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Gauss-Jordan ?a??de??µa 1
??t??at?st?se X µe 2YZ-1
??t??at?st?se Y µe -1
?p?st?e?e false
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Gauss-Jordan Example 2
??t??at?st?se X µe 2YZ-1
??t??at?st?se Y µe -1
?p???s? µ??f?? pe?????sµ?? a?t?? t?? µ??f??
e??a? ??a??p???s?µ??
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?p???s? µ??f??

• ?? pa?aµet????? µetaß??t?? eµfa?????ta? sta
  a??ste?? µ?a? e??s?s??.
• ?a?aµet????? µetaß??t?? eµfa?????ta? sta de???
  µ?a? e??s?s??.
• ?p???s? ep????? pa?aµe????? t?µ?? ?a? ?a????sµ??
  t?? µ? pa?aµet?????

25
?????

• ?e???t? ???µ?s???
• Scene Labelling
• ??????? ?????aµµat?sµ??
• unification --gt constraint solving
• G?af???? a????ep?d??se??
• Sketchpad
• ThingLab
• ?p??e???s?a?? ??e??a
• NP-hard combinatorial problems

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?fa?µ????

• ???ß??µata ????es??
• t?p??es?a ?atas?e??? ae??d??µ???
• a?????ß???? p?????
• p??s?p??? a???es?
• ?at?????? ??s???µ??
• a???es? p????µ?t?? pt?se??
• d?a?e???s? ?a? d?aµ??f?s? d??t???
• s?ed?asµ?? ?a??d??s?? se t??ep????????a?? d??t?a
• ß??t?st? t?p???t?ts? sta?µ?? se as??µata d??t?a
• µ???a?? ß??????a
• DNA a???????a?
• a?a??????? ?a? ??f?a??? s?ed?asµ?? ?????µ?t??

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?fa?µ???? se ?????d?a???µµata

• ? p?? ep?t???? pe????? efa?µ????
• ?????d????aµµa pa?a?????
• s?ed?asµ?? pa?a????? ae??p????? (Dassault
  Aviation)

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?e???e?t?µata

• NP-hard p??ß??µata
• µ? p??ß????µ? s?µpe??f???
• sta?e??t?ta µ??te??p???s??
• p??? ???? ep?ped? (???? pe?????sµ??, ep???t??)
• p??? ?aµ??? ep?ped? (µ??te??p???s?)
• p??? t?p???
• µ? a???t???? (rescheduling)
• ad??aµ?a s??e??as?a? ep???t??

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??e??e?t?µata

• epe????µat??? f?s?
• ep????t??s? t?? e?d?af???? st? pe????af? t??
  p??ß??µat?? ?a? ??? st?? ep???s? t??
• ap?te?esµat??? ep???s? p??ß??µ?t??
• e??p???s? a??????µ?? p?? ???s?µ?p?????ta? ??a
  e?d????? s??p???
• ?eµe???d?? s?µas??????a
• ?a?a??? ?a? ??µ??? ???sse?
• ßas??eta? st? ?????? p????aµµat?sµ?
• p??????e? efa?µ????
• a??et? ep?t???a se d??f????? t?µe??

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?ate????se??

• µ??te??p???s?
• ?a???????? pe?????sµ??? (???? d?af??et????)
• µ??te??p???s? ???ss?? (Numerica, VisOpt)
• ?ata???t? µe??t?
• ?pt???p???s?, ap?te?esµat??? debugging
• ?ß??d???? a??????µ??
• s??e??as?a ep???t??
• pa?a?????sµ??
• multi-agent te???????a