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159.3021 Lecture 11 Last Time: Local Search, Constraint Satisfaction Problems Today: More on CSPs

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159.3022 What is a CSP? A special kind of search problem in which states are defined by the values of a set of variables and the goal test specifies a set of constraints that the values must obey. e.g. 8 queens – assign positions to queens timetabling – assign rooms to classes scheduling – assign times to classes circuit layout – assign position to components sudoko – assign numbers to boxes

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159.3023 Example – Map colouring assign each region a colour, neighbouring regions must be different colours

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159.3024 How are the constraints described? 2 ways: As rules, e.g. WA != NT, WA != SA, NT != SA, NT !=Q, SA !=Q, SA != NSW, Q != NSW, NSW != V, SA !=V As possible values, e.g. {(WA,NT)}={(red,green),(red,blue),(green,red),(green,blue),(blue,re d),(blue,red)}

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159.3025 What types of constraints are there? Constraints can be unary, binary, or higher-order: Unary: concern the value of a single variable Binary: relate pairs of variables (in 8-Queens all constraints are binary) Higher-order: triples, 4-tuples,...

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159.3026 Backtracking Search Variable assignments are commutative, i.e., [WA=red then NT =green] same as [NT =green then WA=red] Only need to consider assignments to a single variable at each node b=d and there are d n leaves Depth-first search for CSPs with single-variable assignments is called backtracking search Backtracking search is the basic uninformed algorithm for CSPs Can solve n-queens for n=25

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159.3027 Backtracking Example

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159.3028 Can backtracking be improved? Three methods 1. Which variable should be assigned next? 2. In what order should its values be tried? 3. Can we detect inevitable failure early?

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159.3029 Which variable to try next? Minimum remaining values (MRV) Choose the variable with the fewest legal values (most constrained variable)

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159.30210 What if variables have the same MRV? Use the Degree Heuristic choose the variable with the most constraints on remaining variables

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159.30211 Which value to try next? Least constraining value Given a variable, choose the least constraining value: The one that rules out the fewest values in the remaining variables Combining these heuristics makes 1000 queens feasible

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159.30212 What is forward checking? Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values

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159.30213 Example: 4 Queens 1 3 2 4 3241 X1 {1,2,3,4} X3 {1,2,3,4} X4 {1,2,3,4} X2 {1,2,3,4}

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159.30214 4 Queens 1 3 2 4 3241 X1 {1,2,3,4} X3 {,2,,4} X4 {,2,3, } X2 {,,3,4}

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159.30215 4 Queens 1 3 2 4 3241 X1 {1,2,3,4} X3 {,,, } X4 {,2,3, } X2 {,,3,4}

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159.30216 4 Queens 1 3 2 4 3241 X1 {1,2,3,4} X3 {,2,, } X4 {,,3, } X2 {,,,4}

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159.30217 4 Queens 1 3 2 4 3241 X1 {1,2,3,4} X3 {,2,, } X4 {,,, } X2 {,,,4}

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159.30218 4 Queens 1 3 2 4 3241 X1 {,2,3,4} X3 {1,,3, } X4 {1,,3,4} X2 {,,,4}

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159.30219 4 Queens 1 3 2 4 3241 X1 {,2,3,4} X3 {1,,, } X4 {1,,3, } X2 {,,,4}

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159.30220 4 Queens 1 3 2 4 3241 X1 {,2,3,4} X3 {1,,, } X4 {,,3, } X2 {,,,4}

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159.30221 4 Queens 1 3 2 4 3241 X1 {,2,3,4} X3 {1,,, } X4 {,,3, } X2 {,,,4}

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159.30222 What is Constraint Propagation? Forward checking propagates information from assigned to unassigned variables, but doesn't provide early detection for all failures: NT and SA cannot both be blue! Constraint propagation repeatedly enforces constraints locally

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Constraint Satisfaction Problems

Constraint Satisfaction Problems

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