CS-INFO372-Module2 - CS-INFO 372: Explorations in...

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CS-INFO 372: Explorations in Artificial Intelligence Prof. Carla P. Gomes gomes@cs.cornell.edu Module 2 Examples of Different Modeling Formalisms http://www.cs.cornell.edu/courses/cs372/2008sp
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Example of a reasoning formalism: Constraint Satisfaction Problems
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Escher: Waterfall, 1961
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Escher: Belvedere, May 1958
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Escher: Ascending and Descending, 1960
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How do we Interpret the Scenes in Escher’s Worlds? Analysis of Polyhedral Scenes origins of Constraint Reasoning researchers in computer vision in the 60s-70s were interested in developing a procedure to assign 3- dimensional interpretations to scenes; They identified Three types of edges Four types of junctions
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Edge Types Hidden – if one of its planes cannot be seen represented with arrows: Convex – from the viewer’s perspective represented with + Concave – from the viewer’s perspective represented with - Huffman-Clowes Labeling
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Types of Junctions Type of junction: L Fork T Arrow
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Scene Interpretation Constraint Reasoning Problem: Variables Edges; Domains {+,-, , } Constraints: 1- The different type junctions define constraints: L, Fork, T, Arrow; L = {( , ) , ( , ), (+, ), ( ,+), (-, ), ( ,-)} Fork = { (+,+,+), (-,-,-), ( , ,-), ( ,-, ),(-, , )} L(A,B) the pair of values assigned to variables A,B has to belong in the set L; Fork(A,B,C) the trio of values assigned to variables A,B,C has to belong in the set Fork;
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Constraint Satisfaction Problem (CSP) T = {( , , ) , ( , , ), ( , ,+), (  ,-)} Arrow = { ( , ,+), (+,+,-), (-,-,+)} T(A,B,C) the trio of values assigned to variables A,B,C has to belong in the set T; Arrow(A,B,C) the trio of values assigned to variables A,B,C has to belong in the set Arrow; 2- For each edge XY its reverse YX has a compatible value Edge = { +,+), (-,-), ( , ),( , )} Edge(A,B) the pair of values assigned to variables A,B has to belong in the set Edge;
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CSP Model - Cube A B C D E F G How to label the cube?
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Constraint Satisfaction Problem (CSP Model) Variables : Edges: AB, BA,AC,CA,AE,EA,CD, DC,BD,DB,DG,GD,GF,FG,EF,FE,AE,EA; Domains {+,-, , } Constraints: L(AC,CD); L(AE,EF); L(DG,GF); Arrow(AC,AE,AB); Arrow(EF,FG,BF); Arrow(CD,DG,DB); Fork(AB,BF,BD); Edge(AB,BA); Edge(AC,CA); Edge(AE,EA); Edge(EF,FE); Edge(BF,FB); Edge(FG,GF); Edge(CD,DC); Edge(BD,DB); Edge(DG,GD); A B C D E F G
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CSP Model Variables : Edges: AB, BA,AC,CA,AE,EA,CD, DC,BD,DB,DG,GD,GF,FG,EF,FE,AE,EA; Domains {+,-, , } Constraints: L(AC,CD); L(AE,EF); L(DG,GF); Arrow(AC,AE,AB); Arrow(EF,FG,BF); Arrow(CD,DG,DB); Fork(AB,BF,BD); Edge(AB,BA); Edge(AC,CA); Edge(AE,EA); Edge(EF,FE); Edge(BF,FB); Edge(FG,GF); Edge(CD,DC); Edge(BD,DB); Edge(DG,GD); A B C D E F G + + + One (out of four) possible labelings (upper right corner)
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The Impossible Objects is Escher’s Worlds Penrose & Penrose Stairs Penrose Triangle
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Impossible Objects: No labeling!
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Other examples using a Constraint Satisfaction formalism
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