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Constraint Propagation

Constraint Propagation. Constraint Propagation …. … is the process of determining how the possible values of one variable affect the possible values of other variables. Forward Checking.

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Constraint Propagation

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  1. Constraint Propagation

  2. Constraint Propagation … … is the process of determining how the possible values of one variable affect the possible values of other variables Constraint Propagation

  3. 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

  4. NT Q WA NSW SA T V Map Coloring Constraint Propagation

  5. NT Q WA NSW SA T V Map Coloring Constraint Propagation

  6. NT Q WA NSW SA T V Map Coloring Constraint Propagation

  7. NT Q WA NSW SA T V Impossible assignments that forward checking do not detect Map Coloring Constraint Propagation

  8.  constraint propagation Constraint Propagation

  9. Edge Labeling in Computer Vision Russell and Norvig: Chapter 24, pages 745-749 Constraint Propagation

  10. Edge Labeling Constraint Propagation

  11. Edge Labeling Constraint Propagation

  12. + – Edge Labeling Constraint Propagation

  13. + + + + + - - + + + + + Edge Labeling Constraint Propagation

  14. - - + + - + + - - + + - - - - + + + - + Junction Label Sets (Waltz, 1975; Mackworth, 1977) Constraint Propagation

  15. 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

  16. - + + - - + + - + Edge Labeling Constraint Propagation

  17. + - - + + - - - - + + Edge Labeling Constraint Propagation

  18. + + + + - + + - - + + Edge Labeling + + Constraint Propagation

  19. - - + + + + + - - Edge Labeling + + Constraint Propagation

  20. 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

  21. 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 contradictiondo • 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

  22. 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

  23. 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

  24. 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

  25. {1, 2} {1, 2} X  Y X Y X  Z Y  Z Z {1, 2} Is AC3 All What is Needed? NO! Constraint Propagation

  26. Solving a CSP • Interweave constraint propagation, e.g., • forward checking • AC3 • and backtracking • + Take advantage of the CSP structure Constraint Propagation

  27. X1 {1,2,3,4} X2 {1,2,3,4} 1 2 3 4 1 2 3 4 X3 {1,2,3,4} X4 {1,2,3,4} 4-Queens Problem Constraint Propagation

  28. X1 {1,2,3,4} X2 {1,2,3,4} 1 2 3 4 1 2 3 4 X3 {1,2,3,4} X4 {1,2,3,4} 4-Queens Problem Constraint Propagation

  29. X1 {1,2,3,4} X2 {1,2,3,4} 1 2 3 4 1 2 3 4 X3 {1,2,3,4} X4 {1,2,3,4} 4-Queens Problem Constraint Propagation

  30. X1 {1,2,3,4} X2 {1,2,3,4} 1 2 3 4 1 2 3 4 X3 {1,2,3,4} X4 {1,2,3,4} 4-Queens Problem Constraint Propagation

  31. X1 {1,2,3,4} X2 {1,2,3,4} 1 2 3 4 1 2 3 4 X3 {1,2,3,4} X4 {1,2,3,4} 4-Queens Problem Constraint Propagation

  32. X1 {1,2,3,4} X2 {1,2,3,4} 1 2 3 4 1 2 3 4 X3 {1,2,3,4} X4 {1,2,3,4} 4-Queens Problem Constraint Propagation

  33. X1 {1,2,3,4} X2 {1,2,3,4} 1 2 3 4 1 2 3 4 X3 {1,2,3,4} X4 {1,2,3,4} 4-Queens Problem Constraint Propagation

  34. X1 {1,2,3,4} X2 {1,2,3,4} 1 2 3 4 1 2 3 4 X3 {1,2,3,4} X4 {1,2,3,4} 4-Queens Problem Constraint Propagation

  35. X1 {1,2,3,4} X2 {1,2,3,4} 1 2 3 4 1 2 3 4 X3 {1,2,3,4} X4 {1,2,3,4} 4-Queens Problem Constraint Propagation

  36. X1 {1,2,3,4} X2 {1,2,3,4} 1 2 3 4 1 2 3 4 X3 {1,2,3,4} X4 {1,2,3,4} 4-Queens Problem Constraint Propagation

  37. NT Q WA NSW SA T V Structure of CSP • If the constraint graph contains multiple components, then one independent CSP per component Constraint Propagation

  38. 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

  39. X Y Z V U W Constraint Tree  (X, Y, Z, U, V, W) Constraint Propagation

  40. Constraint Tree • Order the variables from the root to the leaves  (X1, X2, …, Xn) • For j = n, n-1, …, 2 doREMOVE-ARC-INCONSISTENCY(Xj, Xi) whereXiis the parent ofXj • 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

  41. NT Q WA NSW SA V 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 Constraint Propagation

  42. NT Q WA NSW V 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 Constraint Propagation

  43. 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

  44. 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

  45. 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

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

  47. Game Playing

  48. Games as search problems • Chess, Go • Simulation of war (war game) • 스타크래프트의 전투 • Claude Shannon, Alan Turing  Chess program (1950년대) Constraint Propagation

  49. Contingency problems • The opponent introduces uncertainty • 마이티에서는 co-work이 필요 • 고스톱에서는 co-work방지가 필요 • 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 Constraint Propagation

  50. 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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