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# CSP - Constraint Satisfaction Problems Chapter 5 Chapter 5...

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Constraint Satisfaction Problems Chapter 5 Chapter 5 1

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Victor has been murdered, and Arthur, Bertram, and Carleton are suspects. Arthur says he did not do it. He says that Bertram was the victim’s friend but that Carleton hated the victim. Bertram says he was out of town the day of the murder, and besides he didn’t even know the guy. Carleton says he is innocent and he saw Arthur and Bertram in town just before the murder.
Example: Map-Coloring Western Australia Northern Territory South Australia Queensland New South Wales Victoria Tasmania Variables WA , NT , Q , NSW , V , SA , T Domains D i = { red, green, blue } Constraints : adjacent regions must have different colors e.g., WA 6 = NT (if the language allows this), or ( WA, NT ) ∈ { ( red, green ) , ( red, blue ) , ( green, red ) , ( green, blue ) , . . . } Chapter 5 4

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Constraint satisfaction problems (CSPs) Standard search problem: state is a “black box”—any old data structure that supports goal test, eval, successor CSP: state is defined by variables X i with values from domain D 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 Chapter 5 3
Example: Map-Coloring contd. Western Australia Northern Territory South Australia Queensland New South Wales Victoria Tasmania Solutions are assignments satisfying all constraints, e.g., { WA = red, NT = green, Q = red, NSW = green, V = red, SA = blue, T = green } Chapter 5 5

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Constraint graph Binary CSP : each constraint relates at most two variables Constraint graph : nodes are variables, arcs show constraints Victoria WA NT SA Q NSW V T General-purpose CSP algorithms use the graph structure to speed up search. E.g., Tasmania is an independent subproblem! Chapter 5 6
Varieties of CSPs Discrete variables finite domains; size d O ( d n ) complete assignments e.g., Boolean CSPs, incl. Boolean satisfiability (NP-complete) infinite domains (integers, strings, etc.) e.g., job scheduling, variables are start/end days for each job need a constraint language , e.g., StartJob 1 + 5 StartJob 3 linear constraints solvable, nonlinear undecidable Continuous variables e.g., start/end times for Hubble Telescope observations linear constraints solvable in poly time by LP methods Chapter 5 7

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Standard search formulation (incremental) Let’s start with the straightforward, dumb approach, then fix it States are defined by the values assigned so far Initial state : the empty assignment, { } Successor function : assign a value to an unassigned variable that does not conflict with current assignment.
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