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Lecture-09-Constraint_Satisfaction

Lecture-09-Constraint_Satisfaction - CS 561 Artificial...

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CS 561: Artificial Intelligence Instructor: Sofus A. Macskassy, [email protected] TA: Harris Chiu ( [email protected] ), Wed 2:45-4:45pm, PHE 328 Penny Pan ( [email protected] ), Fri 10am-noon, PHE 328 Lectures: MW 5:00-6:20pm, ZHS 159 Office hours: By appointment Class page: http://www-rcf.usc.edu/~macskass/CS561-Fall2010/ This class will use https://blackboard.usc.edu/webapps/login/ and class webpage - Up to date information - Lecture notes - Relevant dates, links, etc. Course material: [AIMA] Artificial Intelligence: A Modern Approach, by Stuart Russell and Peter Norvig. (3rd ed)
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CS561 - Lecture 9 - Macskassy - Fall 2010 2 Constraint Satisfaction [AIMA Ch 6] CSP examples Backtracking search for CSPs Problem structure and problem decomposition Local search for CSPs
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Standard search problems: State is a “black box”: arbitrary data structure Goal test: any function over states Successor function can be anything Constraint satisfaction problems (CSPs): A special subset of search problems State is defined by variables X i with values from a domain D (sometimes D depends on i ) Goal test is a set of constraints specifying allowable combinations of values for subsets of variables Simple example of a formal representation language Allows useful general-purpose algorithms with more power than standard search algorithms Constraint satisfaction problems (CSPs) 3 CS561 - Lecture 9 - Macskassy - Fall 2010
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Example: N-Queens Formulation 1: Variables: Domains: Constraints 4 CS561 - Lecture 9 - Macskassy - Fall 2010
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Example: N-Queens Formulation 2: Variables: Domains: Constraints: Implicit: Explicit: -or- 5 CS561 - Lecture 9 - Macskassy - Fall 2010
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CS561 - Lecture 9 - Macskassy - Fall 2010 6 Example: Map-Coloring
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CS561 - Lecture 9 - Macskassy - Fall 2010 7 Example: Map-Coloring contd.
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CS561 - Lecture 9 - Macskassy - Fall 2010 8 Constraint graph Binary CSP : each constraint relates at most two variables Constraint graph : nodes are variables, arcs show constraints General-purpose CSP algorithms use the graph structure to speed up search. E.g., Tasmania is an independent subproblem!
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CS561 - Lecture 9 - Macskassy - Fall 2010 9 Example: Cryptarithmetic Variables: F T U W R O X1 X2 X3 Domains: { 0 ; 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 } Constraints alldiff ( F; T;U;W;R;O ) O + O = R + 10 · X1 X1 + W + W = U + 10 · X2 X2 + T + T = O + 10 · X3 X3 = F Every higher-order finite constraint can be broken into n binary constraints, given enough auxiliary constraints
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Example: Sudoku Variables: Each (open) square Domains: {1,2,…,9} Constraints: 9-way alldiff for each row 9-way alldiff for each column 9-way alldiff for each region 10 CS561 - Lecture 9 - Macskassy - Fall 2010
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Example: Boolean Satisfiability Given a Boolean expression, is it satisfiable?
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