1. Consistency algorithms
Chapter 3
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2. Consistency methods Approximation of inference:
Arc, path and i-consistecy
Methods that transform the original network into a tighter and tighter representations
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3. Arc-consistency   =  X Y 1, 2, 3 1, 2, 3 1  X, Y, Z, T  3 X  Y
T  Z
X  T  = 1, 2, 3 1, 2, 3  T Z
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4. Arc-consistency   =  X Y 1 3 2 1  X, Y, Z, T  3 X  Y Y = Z T  Z
X  T  =  T Z
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5. Arc-consistency
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6. Revise for arc-consistency
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7. A matching diagram describing a network of constraints that is not arc-consistent (b) An arc-consistent equivalent network.
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8. AC-1 Complexity (Mackworth and Freuder, 1986):
e = number of arcs, n variables, k values
(ek^2, each loop, nk number of loops), best-case = ek,
Arc-consistency is:
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9. AC-3
Complexity:
Best case O(ek), since each arc may be processed in O(2k)
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10. Example: A 3 variables network with 2 constraints: z divides x and z divides y (a) before and (b) after AC-3 is applied.
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11. AC-4
Complexity:
(Counter is the number of supports to ai in xi from xj. S_(xi,ai) is the set of pairs that (xi,ai) supports)
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12. Example applying AC-4
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13. Distributed arc-consistency (Constraint propagation)
Implement AC-1 distributedly.
Node x_j sends the message to node x_i
Node x_i updates its domain:
Messages can be sent asynchronously or scheduled in a topological order
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14. Exercise: make the following network arc-consistent
Draw the network’s primal and dual constraint graph
Network =
Domains {1,2,3,4}
Constraints: y < x, z < y, t < z, f<t, x<=t+1, Y<f+2
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15. Arc-consistency Algorithms
AC-1: brute-force, distributed
AC-3, queue-based
AC-4, context-based, optimal
AC-5,6,7,…. Good in special cases
Important: applied at every node of search
(n number of variables, e=#constraints, k=domain size)
Mackworth and Freuder (1977,1983), Mohr and Anderson, (1985)…
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16. Using constraint tightness in analysis t = number of tuples bounding a constraint
AC-1: brute-force,
AC-3, queue-based
AC-4, context-based, optimal
AC-5,6,7,…. Good in special cases
Important: applied at every node of search
(n number of variables, e=#constraints, k=domain size)
Mackworth and Freuder (1977,1983), Mohr and Anderson, (1985)…
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17. Constraint checking Arc-consistency B C: [ 6 .. 15 ] 1- B: [ 5 .. 14 ]13
C: [ ]
1- B: [ ]
A < B 14 A
[ ] 2
2- A: [ ]
C: [ ]
B < C
[ ]
2 < C - A < 5
3- B: [ ] 6 14
[ ] C
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18. Is arc-consistency enough?
Example: a triangle graph-coloring with 2 values.
Is it arc-consistent?
Is it consistent?
It is not path, or 3-consistent.
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19. Path-consistency
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20. Path-consistency
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21. Revise-3
Complexity: O(k^3)
Best-case: O(t)
Worst-case O(tk)
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22. PC-1 Complexity: O(n^3) triplets, each take O(k^3) steps  O(n^3 k^3)
Max number of loops: O(n^2 k^2) .
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23. PC-2 Complexity: Optimal PC-4:
(each pair deleted may add: 2n-1 triplets, number of pairs: O(n^2 k^2)  size of Q is O(n^3 k^2), processing is O(k^3))
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24. Example: before and after path-consistency
PC-1 requires 2 processings of each arc while PC-2 may not
Can we do path-consistency distributedly?
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25. Path-consistency Algorithms
Apply Revise-3 (O(k^3)) until no change
Path-consistency (3-consistency) adds binary constraints.
PC-1:
PC-2:
PC-4 optimal:
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26. I-consistency
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27. Higher levels of consistency, global-consistency
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28. Revise-i Complexity: for binary constraints For arbitrary constraints:
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29. 4-queen example
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30. I-consistency
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31. Arc-consistency for non-binary constraints: Generalized arc-consistency
Complexity: O(t k), t bounds number of tuples.
Relational arc-consistency:
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32. Examples of generalized arc-consistency
x+y+z <= 15 and z >= 13 implies
x<=2, y<=2
Example of relational arc-consistency
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33. More arc-based consistency
Global constraints: e.g., all-different constraints
Special semantic constraints that appears often in practice and a specialized constraint propagation. Used in constraint programming.
Bounds-consistency: pruning the boundaries of domains
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34. Example for alldiff A = {3,4,5,6} B = {3,4} C= {2,3,4,5} D= {2,3,4}
Alldiff (A,B,C,D,E}
Arc-consistency does nothing
Apply GAC to sol(A,B,C,D,E)?
 A = {6}, F = {1}….
Alg: bipartite matching kn^1.5
(Lopez-Ortiz, et. Al, IJCAI-03 pp 245 (A fast and simple algorithm for bounds consistency of alldifferent constraint)
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35. Global constraints Alldifferent
Sum constraint (variable equal the sum of others)
Global cardinality constraint (a value can be assigned a bounded number of times to a set of variables)
The cummulative constraint (related to scheduling tasks)
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36. Bounds consistency
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37. Bounds consistency for Alldifferent constraints
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38. Boolean constraint propagation
(A V ~B) and (B)
B is arc-consistent relative to A but not vice-versa
Arc-consistency by resolution:
res((A V ~B),B) = A
Given also (B V C), path-consistency:
Res((A V ~B),(B V C) = (A V C)
What will generalized arc-consistency can do to cnfs?
Relational arc-consistency rule = unit-resolution
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39. Boolean constraint propagation
Example: party problem
If Alex goes, then Becky goes:
If Chris goes, then Alex goes:
Query:
Is it possible that Chris goes to the party but Becky does not?
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40. Gausian and Boolean propagation
Linear inequalities
Boolean constraint propagation
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41. Constraint propagation for Boolean constraints: Unit propagation
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42. Consistency for numeric constraints
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43. Tractable classes
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44. Changes in the network graph as a result of arc-consistency, path-consistency and 4-consistency.
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45. Distributed arc-consistency (Constraint propagation)
Implement AC-1 distributedly.
Node x_j sends the message to node x_i
Node x_i updates its domain:
Generalized arc-consistency can be implemented distributedly: sending messages between constraints over the dual graph:
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46. Distributed Arc-Consistency
Arc-consistency can be formulated as a distributed algorithm: A B C D F G
a Constraint network
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47. Relational Arc-consistencyB 1 2 3 A 1 2 3 A
The message that R2 sends to R1 is
R1 updates its relation and domains and sends messages to neighbors A C 1 2 3 B C D F B C F 1 2 3 A B D 1 2 3 G D F G 1 2 3
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48. DRAC on the dual join-graph1 2 3 A B 1 2 3 A AB AC
ABD
BCF
DFG B 4 5 3 6 2 D F C 1 A C 1 2 3 A B D 1 2 3 B C F 1 2 3 D F G 1 2 3
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49. Distributed Relational Arc-Consistency
DRAC can be applied to the dual problem of any constraint network:
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50. Iteration 1 Node 1 sends messages Node 3 sends messages2 1 3 A 1 3 A 2 1 3 A
Node 1 sends messages
Node 3 sends messages
Node 6 sends messages
Node 5 sends messages
Node 4 sends messages
Node 2 sends messages A 1 2 3 1 3 B 1 2 3 B A 1 3 A 2 1 3 A A B 1 2 3 1 2 1 3 A 2 1 3 A 2 1 3 A 2 C A A C 1 2 3 2 A A 3 AB A AC A A AB C 1 2 D 1 3 B 1 3 A 1 2 3 B A 2 1 3 A B 2 1 3 B 2 C 2 1 3 B 1 3 F 4 5 A B D 1 2 3
ABD B
BCF B C F 1 2 3 6 D F
DFG 2 1 3 D 1 3 F D F G 1 2 3
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51. Iteration 1 A 1 3 A B 1 3 2 4 5 3 6 2 1 A C 1 2 3 A B D 1 3 2 B C F 1
DFG B 4 5 3 6 2 D F C 1 A C 1 2 3 A B D 1 3 2 B C F 1 2 3 D F G 2 1 3
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52. Iteration 2 A 1 3 A 1 2 3 A 1 3 A 1 2 3 1 3 B B 1 2 3 A 1 3 A B 1 3 2 A AB AC
ABD
BCF
DFG B 4 5 3 6 2 D F C 1 A C 1 2 3 A 1 3 C 2 D 2 B 1 3 A 1 3 A B D 1 3 2 B C F 1 2 3 B 1 3 C 2 F 1 D F G 2 1 3 D 1 2 F 1 3
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53. Iteration 2 A 1 3 A B 1 3 4 5 3 6 2 1 A C 1 2 3 A B D 1 3 2 B C F 3 2
DFG B 4 5 3 6 2 D F C 1 A C 1 2 3 A B D 1 3 2 B C F 3 2 1 D F G 2 1 3
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54. Iteration 3 A 1 3 A 1 3 A 1 3 A 1 3 B 3 B 1 3 A 1 3 A B 1 3 A AB AC
ABD
BCF
DFG B 4 5 3 6 2 D F C 1 A C 1 2 3 A 1 3 C 2 D 2 B 1 3 A 1 3 A B D 1 3 2 B C F 3 2 1 B 1 3 C 2 F 1 D F G 2 1 3 D 2 F 1
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55. Iteration 3 A 1 3 A B 1 3 4 5 3 6 2 1 A C 1 2 3 A B D 1 3 2 B C F 3 2
DFG B 4 5 3 6 2 D F C 1 A C 1 2 3 A B D 1 3 2 B C F 3 2 1 D F G 2 1 3
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56. Iteration 4 A 1 3 A 1 A 1 3 A 1 3 B 3 B 1 3 A 1 3 A B 1 3 A AB AC
ABD
BCF
DFG B 4 5 3 6 2 D F C 1 A C 1 2 3 A 1 3 C 2 D 2 B 1 3 A 1 3 A B D 1 3 2 B C F 3 2 1 B 3 C 2 F 1 D F G 2 1 3 D 2 F 1
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57. Iteration 4 A 1 A B 1 3 4 5 3 6 2 1 A C 1 2 3 A B D 1 3 2 B C F 3 2 1
DFG B 4 5 3 6 2 D F C 1 A C 1 2 3 A B D 1 3 2 B C F 3 2 1 D F G 2 1 3
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58. Iteration 5 A 1 A 1 A 1 A 1 B 3 B 3 A 1 A B 1 3 4 5 3 6 2 1 A C 1 2 3
ABD
BCF
DFG B 4 5 3 6 2 D F C 1 A C 1 2 3 A 1 C 2 D 2 B 3 A 1 A B D 1 3 2 B C F 3 2 1 B 3 C 2 F 1 D F G 2 1 3 D 2 F 1
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59. Iteration 5 A 1 A B 1 3 4 5 3 6 2 1 A C 1 2 A B D 1 3 2 B C F 3 2 1 D
DFG B 4 5 3 6 2 D F C 1 A C 1 2 A B D 1 3 2 B C F 3 2 1 D F G 2 1 3
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60. What is SAT? Given a sentence:
Sentence: conjunction of clauses
Clause: disjunction of literals
Literal: a term or its negation
Term: Boolean variable
Question: Find an assignment of truth values to the Boolean variables such the sentence is satisfied.
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61. CSP is NP-Complete
Verifying that an assignment for all variables is a solution
Provided constraints can be checked in polynomial time
Reduction from 3SAT to CSP
Many such reductions exist in the literature (perhaps 7 of them)
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62. Problem reduction
Example: CSP into SAT (proves nothing, just an exercise)
Notation: variable-value pair = vvp
vvp  term
V1 = {a, b, c, d} yields x1 = (V1, a), x2 = (V1, b), x3 = (V1, c), x4 = (V1, d),
V2 = {a, b, c} yields x5 = (V2, a), x6 = (V2, b), x7 = (V2,c).
The vvp’s of a variable  disjunction of terms
V1 = {a, b, c, d} yields
(Optional) At most one VVP per variable
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63. CSP into SAT (cont.) Constraint:
Way 1: Each inconsistent tuple  one disjunctive clause
For example: how many?
Way 2:
Consistent tuple  conjunction of terms
Each constraint  disjunction of these conjunctions
 transform into conjunctive normal form (CNF)
Question: find a truth assignment of the Boolean variables such that the sentence is satisfied
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