Constraint Propagation

Constraint PropagationConstraint Propagation

Constraint Propagation 2

Constraint Propagation …Constraint Propagation … … is the process of determining

how the possible values of one variable affect the possible values of other variables

Forward CheckingForward 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 Y’s domain any value that is inconsistent with v

Map ColoringMap Coloring

WA NT Q NSW V SA TRGB RGB RGB RGB RGB RGB RGB

WA NT Q NSW V SA TRGB RGB RGB RGB RGB RGB RGBR GB RGB RGB RGB GB RGB

WA NT Q NSW V SA TRGB RGB RGB RGB RGB RGB RGBR GB RGB RGB RGB GB RGBR B G RB RGB B RGB

WA NT Q NSW V SA TRGB RGB RGB RGB RGB RGB RGBR GB RGB RGB RGB GB RGBR B G RB RGB B RGBR B G R B RGB

Impossible assignments that forward checking do not detect

constraint propagationconstraint propagation

Edge Labeling in Computer Edge Labeling in Computer VisionVision

Russell and Norvig: Chapter 24, pages 745-749

Edge LabelingEdge Labeling

Edge LabelingEdge Labeling+ –

Junction Label SetsJunction Label Sets

+ + --

-- - + +

(Waltz, 1975; Mackworth, 1977)

Edge Labeling as a CSPEdge Labeling as a CSPA variable is associated with each junctionThe domain of a variable is the label set of the corresponding junctionEach constraint imposes that the values given to two adjacent junctions give the same label to the joining edge

+- -++

Edge LabelingEdge Labeling +

-- - + +

+ + --

Removal of Arc Removal of Arc InconsistenciesInconsistencies

REMOVE-ARC-INCONSISTENCIES(J,K)removed falseX label set of JY label set of KFor 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 YReturn removed

CP Algorithm for Edge CP Algorithm for Edge LabelingLabeling

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)

General CP for Binary General CP for Binary ConstraintsConstraints

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)

General CP for Binary General CP for Binary ConstraintsConstraints

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

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)

Complexity Analysis of Complexity Analysis of AC3AC3

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

Is AC3 All What is Is AC3 All What is Needed?Needed?

NO!X Y

X Z Y Z

{1, 2}

{1, 2}{1, 2}

Solving a CSPSolving a CSP

Interweave constraint propagation, e.g.,• forward checking• AC3 and backtracking

+ Take advantage of the CSP structure

4-Queens Problem4-Queens Problem

32 41X1

{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

32 41X1

{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

32 41X1

{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

32 41X1

{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

32 41X1

{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

32 41X1

{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

32 41X1

{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

32 41X1

{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

32 41X1

{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

32 41X1

{1,2,3,4}

X3{1,2,3,4}

X4{1,2,3,4}

X2{1,2,3,4}

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

Structure of CSP 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

Constraint TreeConstraint TreeX

W (X, Y, Z, U, V, W)

Constraint TreeConstraint Tree Order the variables from the root to the leaves (X1, X2, …, Xn) For j = n, n-1, …, 2 do REMOVE-ARC-INCONSISTENCY(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

Structure of CSP 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

Structure of CSP 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

Over-Constrained Over-Constrained ProblemsProblems

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

Non-Binary ConstraintsNon-Binary ConstraintsSo far, all constraints have been binary (two variables) or unary (one variable)Constraints with more than 2 variables would be difficult to propagateTheoretically, one can reduce a constraint with k>2 variables to a set of binary constraints by introducing additional variables

When to Use CSP When to Use CSP Techniques?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

SummarySummary

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

Game PlayingGame Playing

Games as search Games as search problemsproblems

Chess, GoSimulation of war (war game)스타크래프트의 전투Claude Shannon, Alan Turing Chess program (1950 년대 )

Contingency problemsContingency problemsThe opponent introduces uncertainty마이티에서는 co-work 이 필요고스톱에서는 co-work 방지가 필요Hard to solve in chess, 35100 possible nodes, 1040 different legal positionsTime limits how to make the best use of time to reach good decisionsPruning, heuristic evaluation function

Perfect decisions in two Perfect decisions in two person gamesperson games

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

Mini-max algorithmMini-max algorithm(AND-OR tree)(AND-OR tree)

12lose

14lose

15lose

16draw

17draw

19draw

20lose

21draw

상대방의 관점상대방의 관점

10lose

11lose

13lose

16draw

17draw

18lose

19draw

21draw

NegmaxNegmax

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

The Negmax formalismThe Negmax formalism1A

10F=-1

11F=-1

12F=+1

13F=-1

14F=+1

15F=+1

18F=-1

20F=+1

Imperfect DecisionsImperfect Decisions

utility function evaluationterminal test cutoff testEvaluation function ::: an estimate of the utility of the game from a given positionChess material value ( 장기도 유사 )Weighted linear function

w1f1+w2f2+….+wnfn

Cutting off searchCutting off searchTo 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 위험이 있을 수 있다Quiescent posiiton ::: unlikely to exhibit wild swings in value in near futureQuiescent search :: Non-quiescent search extra search to find quiescent positionHorizon problem

Alpha-beta pruning Alpha-beta pruning

Eliminate unnecessary evaluationsPruning

Alpha-beta pruningAlpha-beta pruning1

MINF(2)=15

F(4)=205

F(6)=25

F(4)=10

Alpha cutoff Beta cutoff

Negmax representationNegmax representation

F(4)=10

F(4)=20

F(6)=-25

F(2)=-15

ExampleExample

1 2 2 3 X X 3 X X 2 2 2 X X X X X X 2 2 2 X X X X X X

1 2 2 3 X X 3 X X

Distribution of critical nodes

Games with ChanceGames with Chance

Chance nodes expected value Backgammon, 윷놀이 Expectimax value

A backgammon positionA backgammon position

… … …

………

2 -1 1 -1 1

…… … …

1/361,1

1/181,2

6,5 6,6

1/361,1

1/181,2 6,5 6,6

TREMINAL

ComparisionComparision

2.1 1.3

.9 .1 .9 .1

2 2 3 3 1 1 4 4

2 3 1 4

21 40.9

.9 .1 .9 .1

20 20 30 30 1 1 400 400

20 30 1 400

숙제숙제

5.6, 5.8, 5.11, 5.15, 5.16, 5.17

Constraint Propagation

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