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CMPT 463

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What will be covered A* search Local search Game tree Constraint satisfaction problems (CSP)

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Types of Questions T or F Short answers Homework questions (A*, MiniMax, pruning..)

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A * search Best-known form of best-first search. Idea: avoid expanding paths that are already expensive Evaluation function f(n) = g(n) + h(n) g(n) = cost so far to reach n h(n) = estimated cost from n to goal f(n) = estimated total cost of path through n to goal

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Admissible heuristics A heuristic h(n) is admissible if for every node n, h(n) ≤ h * (n), where h * (n) is the true cost to reach the goal state from n. An admissible heuristic never overestimates the cost to reach the goal Example: h SLD (n) (never overestimates the actual road distance)

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A* Search Complete Yes, unless there are infinitely many nodes with f <= f(G) Time Exponential in [relative error of h x length of soln] The better the heuristic, the better the time Best case h is perfect, O(d) Worst case h = 0, O(b d ) same as BFS Space Keeps all nodes in memory and save in case of repetition This is O(b d ) or worse A* usually runs out of space before it runs out of time Optimal Yes, cannot expand f i+1 unless f i is finished

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Manhattan distances

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Heuristic Function Function h(N) that estimate the cost of the cheapest path from node N to goal node. Example: 8-puzzle 123 456 78 12 3 4 5 67 8 N goal h 2 (N) = sum of the distances of every tile to its goal position = 2 + 3 + 0 + 1 + 3 + 0 + 3 + 1 = 13

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Local search algorithms In many optimization problems, the path to the goal is irrelevant; the goal state itself is the solution. e.g., n-queens. We can use local search algorithms: keep a single "current" state, try to improve it generally move to neighbors The paths are not retained 9

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Hill-climbing search Problem: depending on initial state, can get stuck in local maxima.

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Some solutions allow a sideways move shoulder flat local maximum, that is not a shoulder 11

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Some solutions Solution: a limit on the number of consecutive sideway moves E.g., 100 consecutive sideways movies in the 8-queens problem successful rate: raises from14% to 94% cost: 21 steps on average for each successful instance, 64 for each failure 12

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Some more solutions (Variants of hill climbing ) Stochastic hill climbing chooses at random from among the uphill moves converge more slowly, but finds better solutions Random-restart hill climbing “If you don’t succeed, try, try again.” Keep restarting from randomly generated initial states, stopping when goal is found

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Simulated Annealing Picks a random move (instead of the best) If “good move” accepted; else accepted with some probability The probability decreases exponentially with the “badness” of the move It also decreases as temperature “T” goes down

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Local Beam Search Idea: keep k states instead of 1; choose top k of all their successors Not the same as k searches run in parallel! Searches that find good states recruit other searches to join them moves the resources to where the most progress is being made

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Minimax The minimax value of a node is the utility (for Max) of being in the corresponding state, assuming that both players play optimally. Minimax(s) = Utility (s)if Terminal-test(s) max of Minimax(Result(s,a)) if Player(s) = Max min of Minimax(Result(s,a))if Player(s) = Min

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Optimal Play MAX MIN This is the optimal play 271 8 271 8 2 271 8 21 271 8 21 2 271 8

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Alpha-Beta Example [-∞, +∞] Range of possible values Do DFS until first leaf

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Alpha-Beta Example (continued) [-∞,3] [-∞,+∞]

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Alpha-Beta Example (continued) [-∞,3] [-∞,+∞]

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Alpha-Beta Example (continued) [3,+∞] [3,3]

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Alpha-Beta Example (continued) [-∞,2] [3,+∞] [3,3] This node is worse for MAX

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Alpha-Beta Example (continued) [-∞,2] [3,14] [3,3][-∞,14],

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Alpha-Beta Example (continued) [−∞,2] [3,5] [3,3][-∞,5],

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Alpha-Beta Example (continued) [2,2] [−∞,2] [3,3]

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Alpha-Beta Example (continued) [2,2] [-∞,2] [3,3]

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27 Example: Map-Coloring Variables WA, NT, Q, NSW, V, SA, T Domains D i = {red, green, blue} Constraints: adjacent regions must have different colors e.g., WA ≠ NT, or (WA,NT) in {(red,green),(red,blue),(green,red), (green,blue),(blue,red),(blue,green)}

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28 Example: Map-Coloring Solutions are complete and consistent assignments, e.g., WA = red, NT = green, Q = red, NSW = green, V = red, SA = blue, T = green

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29 Varieties of constraints Unary constraints involve a single variable, e.g., SA ≠ green Binary constraints involve pairs of variables, e.g., SA ≠ WA Higher-order constraints involve 3 or more variables, e.g., cryptarithmetic column constraints

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30 Example: Cryptarithmetic 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 X 1 + W + W = U + 10 · X 2 X 2 + T + T = O + 10 · X 3 X 3 = F, T ≠ 0, F ≠ 0

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Sudoku 93 26159 35824 948 6291 523 97153 38741 58

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Example: Sudoku CSP variables: X 11, …, X 99 domains: {1,…,9} constraints: row constraint: X 11 X 12, …, X 11 X 19 col constraint: X 11 X 12, …, X 11 X 19 block constraint: X 11 X 12, …, X 11 X 33 Goal: Assign a value to every variable such that all constraints are satisfied

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33 Most constrained variable Most constrained variable: choose the variable with the fewest legal values a.k.a. minimum remaining values (MRV) heuristic

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34 Most constraining variable Tie-breaker among most constrained variables Most constraining variable (degree heuristics): choose the variable with the most constraints on remaining variables

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35 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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36 Forward checking Idea: Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values

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37 Forward checking Idea: Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values

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38 Forward checking Idea: Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values

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39 Forward checking Idea: Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values

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Constraint Satisfaction Problems

Constraint Satisfaction Problems

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