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# ch06 - Constraint satisfaction problems(CSPs Standard...

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Constraint Satisfaction Problems Chapter 6 Chapter 6 1 Outline CSP examples Backtracking search for CSPs Problem structure and problem decomposition Local search for CSPs Chapter 6 2 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 6 3 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 negationslash = NT (if the language allows this), or ( WA,NT ) ∈{ ( red,green ) , ( red,blue ) , ( green,red ) , ( green,blue ) ,... } Chapter 6 4

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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 6 5 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 6 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 6 7 Varieties of constraints Unary constraints involve a single variable, e.g., SA negationslash = green Binary constraints involve pairs of variables, e.g., SA negationslash = WA Higher-order constraints involve 3 or more variables, e.g., cryptarithmetic column constraints Preferences (soft constraints), e.g., red is better than green often representable by a cost for each variable assignment constrained optimization problems Chapter 6 8
Example: Cryptarithmetic O W T F U R + O W T O W T F O U R X 2 X 1 X 3 Variables : F T U W R O X 1 X 2 X 3 Domains : { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } Constraints alldiff ( F,T,U,W,R,O ) O + O = R + 10 · X 1 , etc.

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