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# Lecture06 - CS440/ECE448 Intro to Articial Intelligence...

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Lecture 6: More on constraint satisfaction problems Prof. Julia Hockenmaier [email protected] http://cs.illinois.edu/fa11/cs440 CS440/ECE448: Intro to Artificial Intelligence

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Tuesday ʼ s key concepts Constraint satisfaction problems: Given: a set of n variables X 1 ..X n , with domains (sets of possible values) D 1 …D n , and a set of m constraints C 1 … C m Task: assign a value from D i for each X i subjects to the constraints.
CSP 1: Map coloring (Binary constraints)

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Map coloring: a solution for N=3 4 CS440/ECE448: Intro AI
Constraint satisfaction problems are defined by… - a set of variables X : {WA, NT, QLD, NSW, VA, SA, TA} - a set of domains D i (possible values for variable x i ): D WA = {red, blue, green} - a set of constraints C : { (WA,NT) , WA NT , (WA,QLD), WA QLD ,…} scope relation 5 CS440/ECE448: Intro AI

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States and solutions Each state is a complete or partial assignment of values to variables: state35 = {WA=red, NT=blue, QLD= green, NSW= red, VA= green, SA= blue, TA= red}; state23 = {WA = red} Legal assignments don ʼ t violate any constraints. Solutions are complete legal assignments 6 CS440/ECE448: Intro AI
Binary constraints: constraint graph QLD NT NSW VA TA SA WA CS440/ECE448: Intro AI 7

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Consistency Node consistency: X is node-consistent iff each element in D X satisfies unary constraints on X Arc consistency: X is arc-consistent iff for each C(X, Y) and for each x D X there is a y D Y such that the assignment {X=x, Y=y} satisfies C(X,Y). Path consistency: {X,Y} are path consistent wrt. Z iff for every x D X and y D X there is a z D Z such that the assignment {X=x,Y=y,Z=z} satisfies C(X,Z) and C(Y,Z) 8 CS440/ECE448: Intro AI
AC-3 // Is the CSP c arc-consistent? function AC3(CSP c) input: CSP c = (X,D,C) local: queue q all arcs C(X,Y) in c while q () do: // Can C(X,Y) be satisfied?

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