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## Constraint-based problem solving

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**Constraint-based problem solving**• Model problem • specify in terms of constraints on acceptable solutions • define variables (denotations) and domains • define constraints in some language • Solve model • define search space / choose algorithm • incremental assignment / backtracking search • complete assignments / stochastic search • design/choose heuristics • Verify and analyze solution**Example: n-queens**• Place n-queens on an n n board so that no pair of queens attacks each other • CSP formulation: • variable for each row: x1, …, xn • domains are possible column positions: {1, …, n} • binary constraints between each pair of variables not same column and not same diagonal**Example: 4-queens**1 2 3 4 x1 Constraints: xi xjand | xi - xj | | i- j| x2 x3 x4**Example: 4-queens**One solution: x1 2 x2 4 x3 1 x4 3 1 2 3 4 x1 Q x2 Q x3 Q x4 Q**Search graph for 4-queens**(1,1,1,2) (1,1,4,2) 4 (1,1,1,1) 1 6 (1,1,1,3) 3 (3,1,4,2) 0 (1,1,1,4) 4 (1,4,1,3) 1 (4,1,4,2) 1 (2,4,1,3) 0**Pick queen in conflict**x1 Q x2 2 1 Q 1 x3 Q x4 Q Move to min. conflicts Pick queen in conflict x1 x1 Q Q x2 Q x2 Q x3 Q x3 0 2 2 Q x4 Q x4 Q Stochastic search Initial assignment x1 Q x2 Q x3 Q x4 Q