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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 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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