MIT16_410F10_lec06

# MIT16_410F10_lec06 - Solving Constraint Programs using...

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1 Solving Constraint Programs using Backtrack Search and Forward Checking 9/29/10 1 Slides draw upon material from: Brian C. Williams 6.034 notes, by Tomas Lozano Perez AIMA, by Stuart Russell & Peter Norvig 16.410-13 Constraint Processing, by Rina Dechter September 27 th , 2010 Assignments • Remember: • Problem Set #3: Analysis and Constraint Programming, due this Wed., Sept. 29 th , 2010. • Reading: • Today: [AIMA] Ch. 6.2-5 ; Constraint Satisfaction. • Wednesday: Operator-based Planning [AIMA] Ch. 10 “Graph Plan,” by Blum & Furst, posted on Stellar. • To Learn More: Constraint Processing , by Rina Dechter – Ch. 5: General Search Strategies: Look-Ahead – Ch. 6: General Search Strategies: Look-Back – Ch. 7: Stochastic Greedy Local Search 2 Brian Williams, Fall 10

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Constraint Problems are Everywhere © Source unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse . 3 Constraint Satisfaction Problems (CSP) Input: A Constraint Satisfaction Problem is a triple <V,D,C>, where: V is a set of variables V i D is a set of variable domains , The domain of variable V i is denoted D i C = is a set of constraints on assignments to V Each constraint C i = <S i ,R i > specifies allowed variable assignments. S i the constraint’s scope , is a subset of variables V. R i the constraint’s relation , is a set of assignments to S i . Output: A full assignment to V , from elements of V’s domain, such that all constraints in C are satisfied . Brian Williams, Fall 10 4 2
Constraint Modeling (Programming) Languages Features Declarative speciﬁcation of the problem that separates the formulation and the search strategy. Example : Constraint Model of the Sudoku Puzzle in Number Jack ( http://4c110.ucc.ie/numberjack/home ) matrix = Matrix(N*N,N*N,1,N*N) sudoku = Model( [AllDiff(row) for row in matrix.row], [AllDiff(col) for col in matrix.col], [AllDiff(matrix[x:x+N, y:y+N].flat) for x in range(0,N*N,N) for y in range(0,N*N,N)] ) 5 Constraint Problems are Everywhere 6 3 © Source unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse .

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Analysis of constraint propagation Solving CSPs using Search Brian Williams, Fall 10 7 What is the Complexity of AC-1? AC-1 ( CSP ) Input: A network of constraints CSP = <X, D, C>. Output : CSP’ , the largest arc-consistent subset of CSP. 1. ± repeat 2. ± for every c ij C, 3. Revise( x i , x j ) 4. Revise( x j , x i ) 5. endfor 6. ± until no domain is changed. Assume: There are n variables. Domains are of size at most k . There are
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• Fall '10
• Prof.BrianWilliams
• Constraint satisfaction, Constraint satisfaction problem, forward checking, Perform initial pruning

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MIT16_410F10_lec06 - Solving Constraint Programs using...

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