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l10bwconstraints

# l10bwconstraints - Constraint Satisfaction Problems...

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Constraint Satisfaction Problems: Formulation, Arc Consistency & Propagation Brian C. Williams 16.410-13 Session 10 Slides draw material from: 6.034 notes, by Tomas Lozano Perez AIMA, by Stuart Russell & Peter Norvig Constraint Processing, by Rina Dechter 1 1 Reading Assignments: Constraints Readings: • Lecture Slides (most material in slides only, READ ALL). • AIMA Ch. 5 – Constraint Satisfaction Problems (CSPs ) • AIMA Ch. 24.4 pp. 881-884 – Visual Interpretation of line drawings as solving CSPs . To find out more (optional reading): • Constraint Processing, by Rina Dechter, Morgan-Kauffman, 2003. • Chapter 2 Constraint Networks • Chapter 3 Consistency-Enforcing and Constraint Propagation. 2 Outline • Interpreting line diagrams • Constraint satisfaction problems (CSP) • Solving CSPs • Arc-consistency and propagation • Analysis of constraint propagation • Search (next lecture) • Case study: Scheduling (appendix) 3 4 Line Labeling In Visual Interpretation Problem: Given line drawing, assign consistent types to each edge, in support of object recognition. depth discontinuity surface orientation discontinuity reflectance discontinuity Opaque, trihedral solids. No surface marks. + + + + Convex Edge + Concave Edge Huffman Clowes (1971): 5 Line Labeling In Visual Interpretation depth discontinuity surface orientation discontinuity Opaque, trihedral solids. No surface marks. Constraint: 13 Physically realizable vertex labels + + + + Convex Edge + Concave Edge + + + + Waltz (1972): labeling through local propagation. 6 Labeling must extend to complex objects Huffman Clowes (1971): 1

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Line Interpretation Simplifying Assumptions 1. Limited line interpretations: No shadows or cracks . 2. Three-faced vertices: Intersection of exactly three object faces (e.g., no pyramid tops). 3. General position: Small perturbations of selected viewing points can not lead to a change in junction type. 7 Deriving all possible junction types • Three face vertex divide space into octants. • May not be at right angles. • Consider all possible fillings of octants, viewed from all empty octants. 8 Deriving all possible junction types • Viewing seven filled octants. Deriving all possible junction types • Viewing one filled octant from empty the upper octants. _ _ _ L R + + + L R + L + R 9 10 Deriving all possible junction types • Viewing one filled octant from the empty lower octants. All Junctions L Junctions Fork Junctions Arrow Junctions T Junctions 11 L R + L R R L 12 + + + + + - - - - + + + - - - + - - + 2
13 C B A D C B A D + + + + + - - + + + + + - - - + + + - - - 14 Propagation starting with background borders + + + + + + + + + + + + + - + + - - - - - - - - - + + + Without background borders, interpretations become unstable. 16 Outline • Interpreting line diagrams • Constraint satisfaction problems (CSP) • Case study: Scheduling (appendix) Solving CSPs • Arc-consistency and propagation • Analysis of constraint propagation • Search (next lecture) 17 Constraint Satisfaction Problems Variables Constraints Two positions on a line (vertical, horizontal, diagonal) cannot both be Q Domains Chessboard positions 1 2 3 4 1 2 3 4 Q 4 Queens Problem: Place 4 queens on a 4x4 chessboard so that no queen can attack another.

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• Fall '05
• BrianWilliams
• Arc, Constraint satisfaction, Constraint satisfaction problem, constraint propagation, Propagation Example AC-3, Constraint Propagation Example

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l10bwconstraints - Constraint Satisfaction Problems...

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