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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 Constraint Propagation
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Constraint Propagation
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 Y’s domain any value that is inconsistent with v Constraint Propagation
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Constraint Propagation
Map Coloring T WA NT SA Q NSW V WA NT Q NSW V SA T RGB Constraint Propagation
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Constraint Propagation
Map Coloring T WA NT SA Q NSW V WA NT Q NSW V SA T RGB R GB Constraint Propagation
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Constraint Propagation
Map Coloring T WA NT SA Q NSW V WA NT Q NSW V SA T RGB R GB B G RB Constraint Propagation
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Constraint Propagation
Map Coloring T WA NT SA Q NSW V Impossible assignments that forward checking do not detect WA NT Q NSW V SA T RGB R GB B G RB Constraint Propagation
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Example: Street Puzzle
1 2 3 4 5 Ni = {English, Spaniard, Japanese, Italian, Norwegian} Ci = {Red, Green, White, Yellow, Blue} Di = {Tea, Coffee, Milk, Fruit-juice, Water} Ji = {Painter, Sculptor, Diplomat, Violonist, Doctor} Ai = {Dog, Snails, Fox, Horse, Zebra}
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Example: Street Puzzle
1 2 3 4 5 Ni = {English, Spaniard, Japanese, Italian, Norwegian} Ci = {Red, Green, White, Yellow, Blue} Di = {Tea, Coffee, Milk, Fruit-juice, Water} Ji = {Painter, Sculptor, Diplomat, Violonist, Doctor} Ai = {Dog, Snails, Fox, Horse, Zebra} The Englishman lives in the Red house The Spaniard has a Dog The Japanese is a Painter The Italian drinks Tea The Norwegian lives in the first house on the left The owner of the Green house drinks Coffee The Green house is on the right of the White house The Sculptor breeds Snails The Diplomat lives in the Yellow house The owner of the middle house drinks Milk The Norwegian lives next door to the Blue house The Violonist drinks Fruit juice The Fox is in the house next to the Doctor’s The Horse is next to the Diplomat’s Who owns the Zebra? Who drinks Water?
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Example: Task Scheduling
T1 must be done during T3 T2 must be achieved before T1 starts T2 must overlap with T3 T4 must start after T1 is complete Are the constraints compatible? Find the temporal relation between every two tasks
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Constraint Propagation
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Edge Labeling in Computer Vision
Russell and Norvig: Chapter 24, pages Constraint Propagation
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Constraint Propagation
Edge Labeling Constraint Propagation
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Constraint Propagation
Edge Labeling Constraint Propagation
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Constraint Propagation
Edge Labeling + – Constraint Propagation
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Constraint Propagation
Edge Labeling + - Constraint Propagation
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Constraint Propagation
Junction Label Sets + - - + + - - + (Waltz, 1975; Mackworth, 1977) Constraint Propagation
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Constraint Propagation
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 Constraint Propagation
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Constraint Propagation
Edge Labeling + - + - Constraint Propagation
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Constraint Propagation
Edge Labeling + + + - Constraint Propagation
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Constraint Propagation
Edge Labeling + + + - + + Constraint Propagation
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Constraint Propagation
Edge Labeling + - + + + - + Constraint Propagation
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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 Constraint Propagation
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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) Constraint Propagation
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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) Constraint Propagation
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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 Constraint Propagation
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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 Constraint Propagation
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Constraint Propagation
Is AC3 All What is Needed? NO! X Y Z X Y X Z Y Z {1, 2} Constraint Propagation
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Constraint Propagation
Solving a CSP Interweave constraint propagation, e.g., forward checking AC3 and backtracking + Take advantage of the CSP structure Constraint Propagation
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Constraint Propagation
4-Queens Problem X1 {1,2,3,4} X3 X4 X2 1 3 2 4 Constraint Propagation
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Constraint Propagation
4-Queens Problem X1 {1,2,3,4} X3 X4 X2 1 3 2 4 Constraint Propagation
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Constraint Propagation
4-Queens Problem X1 {1,2,3,4} X3 X4 X2 1 3 2 4 Constraint Propagation
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Constraint Propagation
4-Queens Problem X1 {1,2,3,4} X3 X4 X2 1 3 2 4 Constraint Propagation
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Constraint Propagation
4-Queens Problem X1 {1,2,3,4} X3 X4 X2 1 3 2 4 Constraint Propagation
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Constraint Propagation
4-Queens Problem X1 {1,2,3,4} X3 X4 X2 1 3 2 4 Constraint Propagation
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Constraint Propagation
4-Queens Problem X1 {1,2,3,4} X3 X4 X2 1 3 2 4 Constraint Propagation
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Constraint Propagation
4-Queens Problem X1 {1,2,3,4} X3 X4 X2 1 3 2 4 Constraint Propagation
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Constraint Propagation
4-Queens Problem X1 {1,2,3,4} X3 X4 X2 1 3 2 4 Constraint Propagation
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Constraint Propagation
4-Queens Problem X1 {1,2,3,4} X3 X4 X2 1 3 2 4 Constraint Propagation
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Constraint Propagation
Structure of CSP If the constraint graph contains multiple components, then one independent CSP per component T WA NT SA Q NSW V Constraint Propagation
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Constraint Propagation
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 Propagation
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Constraint Propagation
Constraint Tree X Y Z U V W (X, Y, Z, U, V, W) Constraint Propagation
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Constraint Propagation
Constraint 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 Constraint Propagation
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Constraint Propagation
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 WA NT SA Q NSW V Constraint Propagation
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Constraint Propagation
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 WA NT Q NSW V Constraint Propagation
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Backtracking Algorithm
partial assignment of variables CSP-BACKTRACKING({}) CSP-BACKTRACKING(a) If a is complete then return a X select unassigned variable D select an ordering for the domain of X For each value v in D do If v is consistent with a then Add (X= v) to a result CSP-BACKTRACKING(a) If result failure then return result Return failure
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Map Coloring {} WA=red WA=green WA=blue NT=green NT=blue Q=red Q=blue
SA Q NSW V T
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Questions Which variable X should be assigned a value next?
In which order should its domain D be sorted?
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Questions Which variable X should be assigned a value next?
In which order should its domain D be sorted? What are the implications of a partial assignment for yet unassigned variables? ( Constraint Propagation -- see next class)
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Choice of Variable Map coloring WA NT WA NT SA Q NSW V T SA
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Choice of Variable Most-constrained-variable heuristic:
Select a variable with the fewest remaining values
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Choice of Variable Most-constraining-variable heuristic:
WA NT SA Q NSW V T SA Most-constraining-variable heuristic: Select the variable that is involved in the largest number of constraints on other unassigned variables
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Choice of Value WA NT WA NT SA Q NSW V T {}
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Choice of Value Least-constraining-value heuristic:
WA NT WA NT SA Q NSW V T {blue} Least-constraining-value heuristic: Prefer the value that leaves the largest subset of legal values for other unassigned variables
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Local Search for CSP Pick initial complete assignment (at random)
1 2 3 2 Pick initial complete assignment (at random) Repeat Pick a conflicted variable var (at random) Set the new value of var to minimize the number of conflicts If the new assignment is not conflicting then return it (min-conflicts heuristics)
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Remark Local search with min-conflict heuristic works extremely well for million-queen problems The reason: Solutions are densely distributed in the O(nn) space, which means that on the average a solution is a few steps away from a randomly picked assignment
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Applications CSP techniques allow solving very complex problems
Numerous applications, e.g.: Crew assignments to flights Management of transportation fleet Flight/rail schedules Task scheduling in port operations Design Brain surgery
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Stereotaxic Brain Surgery
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Stereotaxic Brain Surgery
• < Tumor < 2200 2000 < B2 + B4 < 2200 2000 < B4 < 2200 2000 < B3 + B4 < 2200 2000 < B3 < 2200 2000 < B1 + B3 + B4 < 2200 2000 < B1 + B4 < 2200 2000 < B1 + B2 + B4 < 2200 2000 < B1 < 2200 2000 < B1 + B2 < 2200 • 0 < Critical < 500 0 < B2 < 500 T C B1 B2 B3 B4
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Constraint Programming
“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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Over-Constrained Problems
Weaken an over-constrained problem by: Enlarging the domain of a variable Loosening a constraint Removing a variable Removing a constraint Constraint Propagation
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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 k>2 variables to a set of binary constraints by introducing additional variables Constraint Propagation
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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 Constraint Propagation
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Constraint Propagation
Summary Forward checking Constraint propagation Edge labeling in Computer Vision Interweaving CP and backtracking Exploiting CSP structure Weakening over-constrained CSP Constraint Propagation
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Game Playing
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Games as search problems
Chess, Go Simulation of war (war game) 스타크래프트의 전투 Claude Shannon, Alan Turing Chess program (1950년대) Constraint Propagation
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Constraint Propagation
Contingency problems The opponent introduces uncertainty 마이티에서는 co-work이 필요 고스톱에서는 co-work방지가 필요 Hard to solve in chess, possible nodes, 1040 different legal positions Time limits how to make the best use of time to reach good decisions Pruning, heuristic evaluation function Constraint Propagation
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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 Constraint Propagation
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Mini-max algorithm (AND-OR tree)
Constraint Propagation
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Constraint Propagation
상대방의 관점 Constraint Propagation
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Constraint Propagation
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 Constraint Propagation
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Constraint Propagation
The Negmax formalism Constraint Propagation
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Constraint Propagation
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 (장기도 유사) Weighted linear function w1f1+w2f2+….+wnfn Constraint Propagation
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Constraint Propagation
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 위험이 있을 수 있다 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 Constraint Propagation
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Constraint Propagation
Alpha-beta pruning Eliminate unnecessary evaluations Pruning Constraint Propagation
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Constraint Propagation
Alpha-beta pruning Alpha cutoff Beta cutoff Constraint Propagation
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Negmax representation
Constraint Propagation
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Constraint Propagation
Example Constraint Propagation
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Constraint Propagation
Games with Chance Chance nodes expected value Backgammon, 윷놀이 Expectimax value Constraint Propagation
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Constraint Propagation
A backgammon position Constraint Propagation
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Constraint Propagation
Comparision MAX A A A A 2 1 2 1 1.3 21 40.9 DICE 2.1 .9 .1 .9 .1 .9 .1 .9 .1 MIN 20 30 1 400 2 3 1 4 20 20 30 30 1 1 400 400 2 2 3 3 1 1 4 4 Constraint Propagation
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Constraint Propagation
Monte Carlo Search Constraint Propagation
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Monte Carlo Search (Algo.)
Four steps are applied per search iteration: 1) Selection: Starting at the root node, a child selection policy is recursively applied to descend through the tree until the most urgent expandable node is reached. A node is expandable if it represents a nonterminal state and has unvisited (i.e. unexpanded) children.
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Monte Carlo Search(Algo.)
2) Expansion: One (or more) child nodes are added to expand the tree, according to the available actions. 3) Simulation: A simulation is run from the new node(s) according to the default policy to produce an outcome. 4) Backpropagation: The simulation result is “backed up” (i.e. backpropagated) through the selected nodes to update their statistics.
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Monte Carlo Search (policy)
1) Tree Policy: Select or create a leaf node from the nodes already contained within the search tree (selection and expansion). 2) Play out the domain from a given non-terminal state to produce a value estimate (simulation).
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