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Maintaining Arc Consistency We have a constraint graph G of variables X 1,...X n, and constraint relations {X i X j}, and each Xi has a value set V (X i ). An arc from Xi to Xj is consistent if v V (X i ) w V (X j ) v,w is consistent Conversely, an arc is inconsistent if v V (X i ) w V (X j ) v,w is inconsistent. The arc can be made consistent by removing v. Q3 Q2 Q1 Q4

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Arc Consistency (4 queens problem) Example 1 - 59 - 48 - 27 1 36 Q1 Q2 Q3 Q4 Remove a value v of Qx if there is a variable Qy such that v is inconsistent with all remaining values of Qy 43214321 Numbers indicate sequence of values that are deleted. Starting with Q1=1, arc consistency check proves this inconsistent without trying any more assignments.

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Arc Consistency (4 queens problem) Example 2 - 69 - 35 248 - 1 7 Q1 Q2 Q3 Q4 Remove a value v of Qx if there is a variable Qy such that v is inconsistent with all remaining values of Qy 43214321 Numbers indicate sequence of values that are deleted. Starting with Q1=2, arc consistency check finds a solution without trying any more assignments.

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