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Constraint Satisfaction

Constraint Satisfaction. R & N Chapter 5. Constraint Satisfaction – Early Use. From http://www-2.cs.cmu.edu/~awm/731/constraint07-2x2.pdf. The Waltz Algorithm. More on Waltz Algorithm. The constraints come from the fact that a line must have the same label at both ends. Crossword Puzzles.

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Constraint Satisfaction

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  1. Constraint Satisfaction R & N Chapter 5

  2. Constraint Satisfaction – Early Use From http://www-2.cs.cmu.edu/~awm/731/constraint07-2x2.pdf

  3. The Waltz Algorithm

  4. More on Waltz Algorithm The constraints come from the fact that a line must have the same label at both ends.

  5. Crossword Puzzles

  6. Latin Squares Using only the numbers 1, 2, 3, and 4, arrange four sets of these numbers into a four-by-four array so that no column or row contains the same two numbers. The result is known as a Latin square. Here are two examples of Latin squares of order 4: 1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 1 2 3 4 3 4 1 2 4 3 2 1 2 1 4 3 http://www.sciencenews.org/20000506/mathtrek.asp

  7. Satisfiability • (P  Q  S)  • (A   Q  B)  • (P   B  C)  • ( A  B  C)  • (Q  A  C) • Example applications:

  8. Real Examples Scheduling What else?

  9. Algorithms for CSPs • Branching • State space: usually incremental formulation • Straight line • State space: usually complete-state formulation

  10. Cryptarithmetic SEND +MORE MONEY PEAS description: • State space when formulated as a CSP: variables, domain, constraints • Incremental state space formulation: • Complete state formulation:

  11. Cryptarithmetic SEND +MORE MONEY Representing the constraints: As formulas: As a graph: Using special constraints: Alldiff

  12. Cryptarithmetic – Best First Search SEND +MORE MONEY M = 1 M = 2 M = 2

  13. Cryptarithmetic – Best First Search SEND +MORE MONEY M = 1 M = 2 M = 2 M = 1 S = 2 O =3 …

  14. Cryptarithmetic – Branching Algorithms • Backtracking • Minimum remaining values • Degree heuristic • Least-constraining value SEND +MORE MONEY M = 1 M = 2 M = 2

  15. Cryptarithmetic – Constraint Propagation Forward checking: After assigning a value to a variable X, look at all the variables connected to X and prune inconsistent values. Constraint propagation using arc consistency: The tradeoff: how much time to spend propagating vs. searching. SEND +MORE MONEY

  16. Backtracking – Chronological vs. Dependency Directed

  17. Straight-Line Algorithms: Min-Conflicts function Min-Conflicts(csp, max_steps) returns a solution or failure inputs: csp, a constraint satisfaction problem max_steps, the number of steps allowed before giving up current  an initial assignment for csp for i = 1 to max_steps do if current is a solution for csp then return current var  a randomly chosen, conflicted variable from Variables[csp] value  the value v for var that minimizes Conflicts(var, v, current, csp) set var = value in current return failure

  18. Do Straight-Line Algorithms Work? Does Min-Conflicts work? From: http://www.cs.mu.oz.au/303/slides/week03ah.pdf

  19. Do Straight-Line Algorithms Work?

  20. Another Win of Straight Line Algorithms • They can be used to perturb a solution that ceases to be correct: • Most scheduling problems

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