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CS 5368: Artificial Intelligence Fall 2010. Lecture 5: Search Part 4 9/9/2010. Mohan Sridharan Slides adapted from Dan Klein. Announcements. Search assignment will be up shortly. Notice change in assignment content and deadlines. Include a few questions/comments in all future responses.

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Cs 5368 artificial intelligence fall 2010

CS 5368: Artificial IntelligenceFall 2010

Lecture 5: Search Part 4

9/9/2010

Mohan Sridharan

Slides adapted from Dan Klein


Announcements
Announcements

  • Search assignment will be up shortly.

  • Notice change in assignment content and deadlines.

  • Include a few questions/comments in all future responses.


Today
Today

  • Begin adversarial search!


Adversarial search
Adversarial Search

  • Pacman adversarial by definition 

  • Agents have to contend with ghosts.


Game playing
Game Playing

  • Many different kinds of games!

  • Axes:

    • Deterministic or stochastic?

    • One, two, or more players?

    • Perfect information (can you see the state)?

  • Need algorithms for computing a strategy (policy) which suggests a move in each state.


Deterministic games
Deterministic Games

  • Many possible formalizations, one is:

    • States: S (start at s0)

    • Players: P={1...N} (usually take turns)

    • Actions: A (may depend on player / state)

    • Transition Function: SxA  S

    • Terminal Test: S  {t,f}

    • Terminal Utilities: SxP  R

  • Solution for a player is a policy: S  A


Deterministic single player

lose

win

lose

Deterministic Single-Player?

  • Deterministic, single player, perfect information:

    • Know the rules and action results.

    • Know when you win.

    • Free-cell, 8-Puzzle, Rubik’s cube.

  • Just search!

  • Slight reinterpretation:

    • Each node stores a value: the best outcome it can reach.

    • This is the maximal outcome of its children.

    • No path sums (utilities at end).

  • After search, can pick move that leads to best node.


Deterministic two player
Deterministic Two-Player

  • Tic-tac-toe, chess, checkers.

  • Zero-sum games:

    • One player maximizes result.

    • The other minimizes result.

    • Chess!

  • Minimax search:

    • A state-space search tree.

    • Players take turns.

    • Each layer, or ply, consists of a round of moves*.

    • Choose move to position with highest minimax value = best achievable utility against best play.

max

min

8

2

5

6

* Slightly different from the book definition




Minimax basics
Minimax Basics

  • Minimax(n) is the utility for MAX in being in that state, assuming both players play optimally from there to the end of the game!

  • Minimax of terminal state is the utility.

  • Simple recursive search computation and value backups (Figure 5.3).



Minimax properties
Minimax Properties

  • Optimal against a perfect player. Otherwise?

  • Time complexity?

    • O(bm)

  • Space complexity?

    • O(bm)

  • For chess, b  35, m  100

    • Exact solution is completely infeasible.

    • Do we need to explore the whole tree?

max

min

10

10

9

100


Resource limits
Resource Limits

  • Cannot search to leaves!

  • Depth-limited search:

    • Search a limited depth of tree.

    • Replace utilities with an evaluation function for non-terminal positions.

  • Guarantee of optimal play is gone!

  • More plies makes a BIG difference 

  • Example:

    • Suppose we have 100 seconds, can explore 10K nodes / sec.

    • So can check 1M nodes per move.

    • - reaches about depth 8 – decent chess program.

max

4

min

min

-2

4

-1

-2

4

9

?

?

?

?


Evaluation functions
Evaluation Functions

  • Function which scores non-terminals.

  • Ideal function: returns the utility of the position.

  • In practice: typically weighted linear sum of features:

  • E.g. f1(s) = (num white queens – num black queens).



Can pacman starve
Can Pacman Starve?

  • Knows score will go up by eating the dot now.

  • Knows score will go up just as much by eating the dot later on.

  • No point-scoring opportunities after eating the dot.

  • Therefore, waiting seems just as good as eating!


Iterative deepening
Iterative Deepening

Iterative deepening uses DFS as a subroutine:

  • Do a DFS which only searches for paths of length 1 or less. (DFS gives up on any path of length 2).

  • If “1” failed, do a DFS which only searches paths of length 2 or less.

  • If “2” failed, do a DFS which only searches paths of length 3 or less.

    ….and so on.

    Do we want to do this for multiplayer games?

b


Pruning
- Pruning

  • At node n: if player has better choice m at parent node of n or further up, n will never be reached!


Pruning1
- Pruning

  • General configuration:

    •  is the best value that MAX can get at any choice point along the current path.

    • If n becomes worse than , MAX will avoid it, so can stop considering n’s other children.

    • Define  similarly for MIN.

    • Figure 5.7 in textbook.

Player

Opponent

Player

Opponent

n


Pruning pseudocode

v

- Pruning Pseudocode


Pruning properties
- Pruning Properties

  • Pruning has no effect on final result! 

  • Good move ordering improves effectiveness of pruning. (Section 5.3.1).

  • With “perfect ordering”:

    • Time complexity drops to O(bm/2)

    • Doubles solvable depth.

    • Full search of, e.g. chess, is still hopeless 

  • A simple example of meta-reasoning: reasoning about which computations are relevant.


Non zero sum games
Non-Zero-Sum Games

  • Similar to minimax:

    • Utilities are now tuples.

    • Each player maximizes their own entry at each node.

    • Propagate (or back up) nodes from children.

1,2,6

4,3,2

6,1,2

7,4,1

5,1,1

1,5,2

7,7,1

5,4,5


Advance notice
Advance Notice…

  • Get ready for:

    • Probabilities.

    • Expectations.

  • Upcoming topics (in a week):

    • Dealing with uncertainty.

    • How to learn evaluation functions.

    • Markov Decision Processes.


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