# CS代考计算机代写 algorithm CS 561a: Introduction to Artificial Intelligence

CS 561a: Introduction to Artificial Intelligence

CS 561, Sessions 6-7

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This time: Outline

Game playing

The minimax algorithm

Resource limitations

alpha-beta pruning

Elements of chance

CS 561, Sessions 6-7

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What kind of games?

Abstraction: To describe a game we must capture every relevant aspect of the game. Such as:

Chess

Tic-tac-toe

…

Accessible environments: Such games are characterized by perfect information

Search: game-playing then consists of a search through possible game positions

Unpredictable opponent: introduces uncertainty thus game-playing must deal with contingency problems

CS 561, Sessions 6-7

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Searching for the next move

Complexity: many games have a huge search space

Chess: b = 35, m=100 nodes = 35 100

if each node takes about 1 ns to explore

then each move will take about 10 50 millennia

to calculate.

Resource (e.g., time, memory) limit: optimal solution not feasible/possible, thus must approximate

Pruning: makes the search more efficient by discarding portions of the search tree that cannot improve quality result.

Evaluation functions: heuristics to evaluate utility of a state without exhaustive search.

CS 561, Sessions 6-7

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Two-player games

A game formulated as a search problem:

Initial state: ?

Operators: ?

Terminal state: ?

Utility function: ?

CS 561, Sessions 6-7

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Two-player games

A game formulated as a search problem:

Initial state: board position and turn

Operators: definition of legal moves

Terminal state: conditions for when game is over

Utility function: a numeric value that describes the outcome of the

game. E.g., -1, 0, 1 for loss, draw, win.

(AKA payoff function)

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Game vs. search problem

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Example: Tic-Tac-Toe

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Type of games

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Type of games

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The minimax algorithm

Perfect play for deterministic environments with perfect information

Basic idea: choose move with highest minimax value

= best achievable payoff against best play

Algorithm:

Generate game tree completely

Determine utility of each terminal state

Propagate the utility values upward in the three by applying MIN and MAX operators on the nodes in the current level

At the root node use minimax decision to select the move with the max (of the min) utility value

Steps 2 and 3 in the algorithm assume that the opponent will play perfectly.

CS 561, Sessions 6-7

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Generate Game Tree

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Generate Game Tree

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Generate Game Tree

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Generate Game Tree

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1 ply

1 move

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A subtree

win

lose

draw

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CS 561, Sessions 6-7

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What is a good move?

win

lose

draw

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CS 561, Sessions 6-7

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Minimax

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Minimize opponent’s chance

Maximize your chance

CS 561, Sessions 6-7

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Minimax

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MIN

Minimize opponent’s chance

Maximize your chance

CS 561, Sessions 6-7

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Minimax

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MAX

MIN

Minimize opponent’s chance

Maximize your chance

CS 561, Sessions 6-7

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Minimax

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MAX

MIN

Minimize opponent’s chance

Maximize your chance

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minimax = maximum of the minimum

1st ply

2nd ply

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Minimax: Recursive implementation

Complete: ?

Optimal: ?

Time complexity: ?

Space complexity: ?

CS 561, Sessions 6-7

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Minimax: Recursive implementation

Complete: Yes, for finite state-space

Optimal: Yes

Time complexity: O(bm)

Space complexity: O(bm) (= DFS

Does not keep all nodes in memory.)

CS 561, Sessions 6-7

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1. Move evaluation without complete search

Complete search is too complex and impractical

Evaluation function: evaluates value of state using heuristics and cuts off search

New MINIMAX:

CUTOFF-TEST: cutoff test to replace the termination condition (e.g., deadline, depth-limit, etc.)

EVAL: evaluation function to replace utility function (e.g., number of chess pieces taken)

CS 561, Sessions 6-7

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Evaluation functions

Weighted linear evaluation function: to combine n heuristics

f = w1f1 + w2f2 + … + wnfn

E.g, w’s could be the values of pieces (1 for prawn, 3 for bishop etc.)

f’s could be the number of type of pieces on the board

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Note: exact values do not matter

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Minimax with cutoff: viable algorithm?

Assume we have 100 seconds, evaluate 104 nodes/s; can evaluate 106 nodes/move

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2. - pruning: search cutoff

Pruning: eliminating a branch of the search tree from consideration without exhaustive examination of each node

- pruning: the basic idea is to prune portions of the search tree that cannot improve the utility value of the max or min node, by just considering the values of nodes seen so far.

Does it work? Yes, in roughly cuts the branching factor from b to b resulting in double as far look-ahead than pure minimax

CS 561, Sessions 6-7

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Demo

http://homepage.ufp.pt/jtorres/ensino/ia/alfabeta.html

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- pruning: example

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MAX

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MIN

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- pruning: example

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MAX

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MIN

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- pruning: example

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MAX

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MIN

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- pruning: example

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MAX

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MIN

Selected move

Interactive demo:

https://www.yosenspace.com/posts/computer-science-game-trees.html

CS 561, Sessions 6-7

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- pruning: general principle

Player

Player

Opponent

Opponent

m

n

v

If > v then MAX will chose m so prune tree under n

Similar for for MIN

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Properties of -

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The - algorithm

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More on the - algorithm

Same basic idea as minimax, but prune (cut away) branches of the tree that we know will not contain the solution.

We know a branch will not contain a solution once we know a better outcome has already been discovered in a previously explored branch.

CS 561, Sessions 6-7

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Remember: Minimax: Recursive implementation

Complete: Yes, for finite state-space

Optimal: Yes

Time complexity: O(bm)

Space complexity: O(bm) (= DFS

Does not keep all nodes in memory.)

CS 561, Sessions 6-7

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The - algorithm

CS 561, Sessions 6-7

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More on the - algorithm

Same basic idea as minimax, but prune (cut away) branches of the tree that we know will not contain the solution.

Because minimax is depth-first, let’s consider nodes along a given path in the tree. Then, as we go along this path, we keep track of:

: Best choice so far for MAX

: Best choice so far for MIN

CS 561, Sessions 6-7

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The - algorithm

Note: and are both

Local variables. At the

Start of the algorithm,

We initialize them to

= - and = +

CS 561, Sessions 6-7

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More on the - algorithm

…

MAX

MIN

= -

= +

5 10 6 2 8 7

Min-Value loops

over these

In Min-Value:

= -

= 5

= -

= 5

= -

= 5

Max-Value loops

over these

Return v=5

CS 561, Sessions 6-7

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More on the - algorithm

…

MAX

MIN

MAX

5 10 6 2 8 7

In Max-Value:

= -

= 5

= -

= 5

= -

= 5

= 5

= +

Max-Value loops

over these

Return v=5

CS 561, Sessions 6-7

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In Min-Value:

More on the - algorithm

…

MAX

MIN

MAX

5 10 6 2 8 7

= -

= 5

= -

= 5

= -

= 5

= 5

= +

= 5

= +

End loop and return 2

Min-Value loops

over these

CS 561, Sessions 6-7

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In Max-Value:

More on the - algorithm

…

MAX

MIN

MAX

5 10 6 2 x x

= -

= 5

= -

= 5

= -

= 5

= 5

= +

= 5

= +

= 5

= +

Max-Value loops

over these

Return v=2

CS 561, Sessions 6-7

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Another way to understand the algorithm

For a given node N,

is the value of N to MAX

is the value of N to MIN

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Example

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- algorithm: slight variant (from earlier version of textbook)

Is this wrong compared to latest version of textbook?

Please always use latest version of the algorithm as in 3rd or 4th edition of textbook.

CS 561, Sessions 6-7

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Solution

NODE TYPE ALPHA BETA SCORE

A MAX -Inf Inf

B MIN -Inf Inf

C MAX -Inf Inf

D MIN -Inf Inf

E MAX 10 10 10

D MIN -Inf 10

F MAX 11 11 11

D MIN -Inf 10 10

C MAX 10 Inf

G MIN 10 Inf

H MAX 9 9 9

G MIN 10 9 9

C MAX 10 Inf 10

B MIN -Inf 10

J MAX -Inf 10

K MIN -Inf 10

L MAX 14 14 14

K MIN -Inf 10

M MAX 15 15 15

K MIN -Inf 10 10

…

NODE TYPE ALPHA BETA SCORE

…

J MAX 10 10 10

B MIN -Inf 10 10

A MAX 10 Inf

Q MIN 10 Inf

R MAX 10 Inf

S MIN 10 Inf

T MAX 15 15 15

S MIN 10 15

U MAX 2 2 2

S MIN 10 2 2

R MAX 10 Inf

V MIN 10 Inf

W MAX 4 4 4

V MIN 10 4 4

R MAX 10 Inf 10

Q MIN 10 10 10

A MAX 10 Inf 10

CS 561, Sessions 6-7

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State-of-the-art for deterministic games

CS 561, Sessions 6-7

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Nondeterministic games

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Algorithm for nondeterministic games

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Remember: Minimax algorithm

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Nondeterministic games: the element of chance

3

?

0.5

0.5

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17

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CHANCE

?

expectimax and expectimin, expected values over all possible outcomes

CS 561, Sessions 6-7

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Nondeterministic games: the element of chance

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0.5

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CHANCE

4 = 0.5*3 + 0.5*5

Expectimax

Expectimin

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Evaluation functions: Exact values DO matter

Order-preserving transformations do not necessarily behave the same!

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State-of-the-art for nondeterministic games

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Summary

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Exercise: Game Playing

(a) Compute the backed-up values computed by the minimax algorithm. Show your answer by writing values at the appropriate nodes in the above tree.

(b) Compute the backed-up values computed by the alpha-beta algorithm. What nodes will not be examined by the alpha-beta pruning algorithm?

(c) What move should Max choose once the values have been backed-up all the way?

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Max

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Consider the following game tree in which the evaluation function values are shown below each leaf node. Assume that the root node corresponds to the maximizing player. Assume the search always visits children left-to-right.

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