Constraint Propagation

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Constraint Propagation
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Constraint Propagation

• is the process of determining how the
  possible values of one variable affect the
  possible values of other variables

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Forward Checking

• After a variable X is assigned a value v, look
  at each unassigned variable Y that is connected
  to X by a constraint and deletes from Ys domain
  any value that is inconsistent with v

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Map Coloring
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Map Coloring
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Map Coloring
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Map Coloring
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? constraint propagation
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Edge Labeling in Computer Vision
Russell and Norvig Chapter 24, pages 745-749
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Edge Labeling
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Edge Labeling
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Edge Labeling
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Edge Labeling
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Junction Label Sets
(Waltz, 1975 Mackworth, 1977)
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Edge Labeling as a CSP

• A variable is associated with each junction
• The domain of a variable is the label set of the
  corresponding junction
• Each constraint imposes that the values given to
  two adjacent junctions give the same label to the
  joining edge

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Edge Labeling
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Edge Labeling
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Edge Labeling


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Edge Labeling


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Removal of Arc Inconsistencies

• REMOVE-ARC-INCONSISTENCIES(J,K)
• removed ? false
• X ? label set of J
• Y ? label set of K
• For every label y in Y do
• If there exists no label x in X such that the
  constraint (x,y) is satisfied then
• Remove y from Y
• If Y is empty then contradiction ? true
• removed ? true
• Label set of K ? Y
• Return removed

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CP Algorithm for Edge Labeling

• Associate with every junction its label set
• contradiction ? false
• Q ? stack of all junctions
• while Q is not empty and not contradiction do
• J ? UNSTACK(Q)
• For every junction K adjacent to J do
• If REMOVE-ARC-INCONSISTENCIES(J,K) then
• STACK(K,Q)

(Waltz, 1975 Mackworth, 1977)
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General CP for Binary Constraints

• Algorithm AC3
• contradiction ? false
• Q ? stack of all variables
• while Q is not empty and not contradiction do
• X ? UNSTACK(Q)
• For every variable Y adjacent to X do
• If REMOVE-ARC-INCONSISTENCIES(X,Y) then
• STACK(Y,Q)

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General CP for Binary Constraints

• Algorithm AC3
• contradiction ? false
• Q ? stack of all variables
• while Q is not empty and not contradiction do
• X ? UNSTACK(Q)
• For every variable Y adjacent to X do
• If REMOVE-ARC-INCONSISTENCY(X,Y) then
• STACK(Y,Q)

• REMOVE-ARC-INCONSISTENCY(X,Y)
• removed ? false
• For every value y in the domain of Y do
• If there exists no value x in the domain of X
  such that the constraints on (x,y) is satisfied
  then
• Remove y from the domain of Y
• If Y is empty then contradiction ? true
• removed ? true
• Return removed

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Complexity Analysis of AC3

• n number of variables
• d number of values per variable
• s maximum number of constraints on a
  pair of variables
• Each variables is inserted in Q up to d times
• REMOVE-ARC-INCONSISTENCY takes O(d2) time
• CP takes O(n s d3) time

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Is AC3 All What is Needed?
NO!
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Solving a CSP

• Interweave constraint propagation, e.g.,
• forward checking
• AC3
• and backtracking
• Take advantage of the CSP structure

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4-Queens Problem
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4-Queens Problem
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4-Queens Problem
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4-Queens Problem
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4-Queens Problem
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4-Queens Problem
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4-Queens Problem
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4-Queens Problem
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4-Queens Problem
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4-Queens Problem
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Structure of CSP

• If the constraint graph contains multiple
  components, then one independent CSP per
  component

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Structure of CSP

• If the constraint graph contains multiple
  components, then one independent CSP per
  component
• If the constraint graph is a tree (no loop),
  then the CSP can be solved efficiently

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Constraint Tree
? (X, Y, Z, U, V, W)
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Constraint Tree

• Order the variables from the root to the
  leaves ? (X1, X2, , Xn)
• For j n, n-1, , 2 do REMOVE-ARC-INCONSIST
  ENCY(Xj, Xi) where Xi is the parent of Xj
• Assign any legal value to X1
• For j 2, , n do
• assign any value to Xj consistent with the value
  assigned to Xi, where Xi is the parent of Xj

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Structure of CSP

• If the constraint graph contains multiple
  components, then one independent CSP per
  component
• If the constraint graph is a tree, then the CSP
  can be solved efficiently
• Whenever a variable is assigned a value by the
  backtracking algorithm, propagate this value and
  remove the variable from the constraint graph

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Structure of CSP

• If the constraint graph contains multiple
  components, then one independent CSP per
  component
• If the constraint graph is a tree, then the CSP
  can be solved in linear time
• Whenever a variable is assigned a value by the
  backtracking algorithm, propagate this value and
  remove the variable from the constraint graph

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Over-Constrained Problems

• Weaken an over-constrained problem by
• Enlarging the domain of a variable
• Loosening a constraint
• Removing a variable
• Removing a constraint

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Non-Binary Constraints

• So far, all constraints have been binary (two
  variables) or unary (one variable)
• Constraints with more than 2 variables would be
  difficult to propagate
• Theoretically, one can reduce a constraint with
  kgt2 variables to a set of binary constraints by
  introducing additional variables

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When to Use CSP Techniques?

• When the problem can be expressed by a set of
  variables with constraints on their values
• When constraints are relatively simple (e.g.,
  binary)
• When constraints propagate well (AC3 eliminates
  many values)
• When the solutions are densely distributed in
  the space of possible assignments

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Summary

• Forward checking
• Constraint propagation
• Edge labeling in Computer Vision
• Interweaving CP and backtracking
• Exploiting CSP structure
• Weakening over-constrained CSP

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Game Playing
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Games as search problems

• Chess, Go
• Simulation of war (war game)
• ??????? ??
• Claude Shannon, Alan Turing ? Chess program
  (1950??)

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Contingency problems

• The opponent introduces uncertainty
• ?????? co-work? ??
• ?????? co-work??? ??
• Hard to solve ? in chess, 35100 possible nodes,
  1040 different legal positions
• Time limits ? how to make the best use of time to
  reach good decisions
• Pruning, heuristic evaluation function

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Perfect decisions in two person games

• The initial state, A set of operators, A terminal
  test, A utility function (payoff function)
• Mini-max algorithm,
• Negmax algorithms

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Mini-max algorithm(AND-OR tree)
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???? ??
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Negmax

• Knuth and Moore (1975)
• F(n) f(n), if n has no successors
• F(n) max-F(n1), , -F(nk),
• if n has successors n1, , nk

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The Negmax formalism
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Imperfect Decisions

• utility function ? evaluation
• terminal test ? cutoff test
• Evaluation function an estimate of the
  utility of the game from a given position
• Chess ? material value (??? ??)
• Weighted linear function ?
• w1f1w2f2.wnfn

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Cutting off search

• To set a fixed depth limit, so that the cutoff
  test succeeds for all nodes at or below depth d ?
  iterative deepening until time runs out ? ??? ??
  ? ??
• Quiescent posiiton unlikely to exhibit wild
  swings in value in near future
• Quiescent search Non-quiescent search ? extra
  search to find quiescent position
• Horizon problem

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Alpha-beta pruning

• Eliminate unnecessary evaluations
• Pruning

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Alpha-beta pruning
Alpha cutoff Beta cutoff
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Negmax representation
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Example
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Games with Chance

• Chance nodes ? expected value
• Backgammon, ???
• Expectimax value

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A backgammon position
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Comparision
MAX
A
A
2
A
1
A
2
1
40.9
1.3
21
2.1
DICE
.9
.1
.9
.1
.9
.1
.9
.1
20
30
1
400
MIN
2
3
1
4
20
20
30
30
1
1
400
400
2
2
3
3
1
1
4
4
64
??

• 5.6, 5.8, 5.11, 5.15, 5.16, 5.17