Chuong 2 - B2 - Cac Chien Luoc Tim Kiem Khac

8/21/2019 Chuong 2 - B2 - Cac Chien Luoc Tim Kiem Khac

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CHI N L CẾ ƯỢ

TÌM KI M NÂNG CAOẾ


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MINI-MAX TERMINOLOGY Utility function: the function applied to leaf

nodes Backed-up value

of a max-position: the value of its largestsuccessor

of a min-position: the value of its smallestsuccessor

Minimax procedure: search down severallevels; at the bottom level apply the utilityfunction, back-up values all the way up to theroot node, and that node selects the move


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MINIMAX !erfect play for deterministic games "dea: choose move to position with highest

minimax value # best achievable payoff

against best play $g, %-ply game:


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MINIMAX STRATEGY &hy do we take the min value every other

level of the tree'

(hese nodes represent the opponent)s choiceof move

(he computer assumes that the human will

choose that move that is of least value to thecomputer


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MINIMAX ALGORITHM


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MINIMAX VỚI ĐỘ SÂU GIỚI HẠN

Minimax nh *+ xt bu c ph i c to.n b kh/ngư ộ ả ộgian tr ng th0i *+ * c tri n khai * c th g0nạ ượ ể ể ểtr cho c0c n1t l0 v. t2nh ng c l iị ượ ạ 3h/ng khảthi v i c0c b.i to0n l n v4 kh/ng gian tr ng th0i l.ớ ớ ạ

5u0 l nớ 6i i h n kh/ng gian tr ng th0i l i theo m t *ớ ạ ạ ạ ộ ộs7u n.o * v. gi i h n c0c node con theo m t 5uiớ ạ ột c n.o *ắ

87y l. chi n l c th/ng th ng c a c0c ng iế ượ ườ ủ ườ ch i c : kh n9ng t2nh tr c bao nhiu n cơ ờ ả ướ ướ

6/5/15 6


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MINIMAX VỚI ĐỘ SÂU GIỚI HẠN

3hi * ta ch tri n khai c0c n1t con * n * s7uỉ ể ế ộgi i h nớ ạ 80nh gi0 cho c0c n1t n.y nh l. n1t l0 b ng m tư ằ ộ

h.m l ng gi0 euristicượ 0p d ng chi n l c minimax cho vi c *0nh gi0ụ ế ượ ệ

c0c n1t c p trnấ 3 thu t n.y g i l. nh4n tr c 3 b c v i 3 la ỹ ậ ọ ướ ướ ớ


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TIC TAC TOE =et p be a position in the game >efine the utility function f?p@ by

f?p@ #

largest positive number if p is a win forcomputersmallest negative number if p is a win for

opponentA> C A>D

where A> is number of rows, columns anddiagonals in which computer could still win

and A>D is number of rows, columns anddiagonals in which opponent could still win


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SAMPLE EVALUATIONS E # omputer; D # Dpponent

OX

X O

rowscolsdiags

O OX

X X

X O

rowscolsdiags


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MINIMAX IS DONE DEPTH-FIRST

max

min

max

leaf

2 5 1


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PROPERTIES OF MINIMAX omplete' Fes ?if tree is finite@ Dptimal' Fes ?against an optimal opponent@ (ime complexity' D?bm@

Gpace complexity' D?bm@ ?depth-firstexploration@

Hor chess, bIJK, mIL for NreasonableNgames exact solution completely infeasible

Need to speed it up.


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ALPHA-BETA PROCEDURE (he alpha-beta procedure can speed up a

depth-first minimax search Olpha: a lower bound on the value that a

max node may ultimately be assignedv P α

Beta: an upper bound on the value that aminimiQing node may ultimately be assigned

v R β


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Α-Β PRUNING EXAMPLE


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alpha cutoff

α = 3


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ALPHA CUTOFF≥3

3

8 10

α 3

!"at "appens "ere# $s t"ere an alp"a cuto%#


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BETA CUTOFF

&'

'

β '

≥8

8 β cuto%


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ALPHA-BETA PRUNING

5 2 10 11 1 2 2 8 ( 5 12 ' 3 25

max

min

max

e)al


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PROPERTIES OF Α-Β !runing does not affect final result (his

means that it gets the exact same result asdoes full minimax

6ood move ordering improves effectivenessof pruning &ith Nperfect ordering,N time complexity #

D?bmS%@ doubles depth of search

O simple example of the value of reasoningabout which computations are relevant ?aform of metareasoning@


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THE Α-Β ALGORITHM


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THE Α-Β ALGORITHM


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WHEN DO WE GET ALPHA

CUTOFFS?

...100

* 100 * 100


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ASSIGNMENT

• Using Minimax Olgorithm to solve the (ic-tac-toe

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Chuong 2 - B2 - Cac Chien Luoc Tim Kiem Khac