CS451CS551EE565 ARTIFICIAL INTELLIGENCE

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CS451/CS551/EE565ARTIFICIAL INTELLIGENCE

• Adversary Search
• 9-29-2006
• Prof. Janice T. Searleman
• jets_at_clarkson.edu, jetsza

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Outline

• Adversary Search
• Optimal decisions Minimax
• AlphaBeta
• Reading Assignment AIMA Chapter 6

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Game Tree Search

• What are games?
• Optimal decisions in games
• Which strategy leads to success?
• ?-? pruning
• Games of imperfect information
• Games that include an element of chance

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What are and why study games?

• Games are a form of multi-agent environment
• What do other agents do and how do they affect
  our success?
• Cooperative vs. competitive multi-agent
  environments.
• Competitive multi-agent environments give rise to
  adversarial search a.k.a. games
• Why study games?
• Fun historically entertaining
• Interesting subject of study because they are
  hard
• Easy to represent and agents restricted to small
  number of actions

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Relation of Games to Search

• Search no adversary
• Solution is (heuristic) method for finding goal
• Heuristics CSP techniques can find optimal
  solution
• Evaluation function estimate of cost from start
  to goal through given node
• Examples path planning, scheduling activities
• Games adversary
• Solution is strategy (strategy specifies move for
  every possible opponent reply).
• Time limits force an approximate solution
• Evaluation function evaluate goodness of game
  position
• Examples chess, checkers, Othello, backgammon

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Games vs. search problems

• "Unpredictable" opponent ? specifying a move for
  every possible opponent reply
• Time limits ? unlikely to find goal, must
  approximate
• Plan of attack
• Computer considers possible lines of play
  (Babbage, 1846)
• Algorithm for perfect play (Zermelo, 1912 Von
  Neumann, 1944)
• Finite horizon, approximate evaluation (Zuse,
  1945 Wiener, 1948 Shannon, 1950)
• First chess program (Turing, 1951)
• Machine learning to improve evaluation accuracy
  (Samuel, 1952-57)
• Pruning to allow deeper search (McCarthy, 1956)

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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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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 tree
  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 assume that the opponent plays
  perfectly.

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Minimax algorithm
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Properties of minimax

• Complete?
• Yes (if tree is finite)
• Optimal?
• Yes (against an optimal opponent)
• Time complexity?
• O(bm)
• Space complexity?
• O(bm) (depth-first exploration)
• For chess, b 35, m 100 for "reasonable"
  games? exact solution completely infeasible

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Resource limits

• Suppose we have 100 secs, explore 104 nodes/sec?
  106 nodes per move
• Standard approach
• cutoff test
• e.g., depth limit (perhaps add quiescence search)
• (static) evaluation function
• estimated desirability of position

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Evaluation functions

• Types of evaluation functions
• - Weighted linear function sum of weights times
  features
• - Non-linear e.g. neural nets for backgammon
• Characteristics of a good evaluation function
• - should agree with the real utility function
• - should be quick to compute
• - should reflect the ability to win
• Often a linear weighted sum of features
• Eval(s) w1 f1(s) w2 f2(s) wn fn(s)

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Evaluation functions

• Weighted linear evaluation function to combine n
  heuristics f w1f1 w2f2 wnfn
• e.g, ws could be the values of pieces (1 for
  pawn, 3 for bishop etc.), fs could be the number
  of type of pieces on the board

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Heuristic EVAL example
Eval(s) w1 f1(s) w2 f2(s) wnfn(s)
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Static Evaluation
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Cutting off search

• MinimaxCutoff is identical to MinimaxValue except
• Terminal? is replaced by Cutoff?
• Utility is replaced by Eval
• Does it work in practice?
• bm 106, b35 ? m4
• 4-ply lookahead is a hopeless chess player!
• 4-ply human novice
• 8-ply typical PC, human master
• 12-ply Deep Blue, Kasparov

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A subtree
win
lose
draw
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What is a good move?
win
lose
draw
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TicTacToe
Static evaluation function for state s 8 if s
is a win for X eval(s) -8 if s is a win for
O ( of rows, cols diags open for X) (
of rows, cols diags open for O)
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First move
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Second move
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Next move
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When to cut off search

• fixed depth limit (static or dynamic)
• Iterative deepening
• cutoff at depth d
• problem horizon effect may cut off before some
  really good move for opponent
• solution add quiescence
• another variation add secondary search

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What if MIN does not play optimally?

• Definition of optimal play for MAX assumes MIN
  plays optimally maximizes worst-case outcome for
  MAX.
• But if MIN does not play optimally, MAX will do
  even better. proven.

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Multiplayer games

• Games allow more than two players
• Single minimax values become vectors

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Problem of minimax search

• Number of games states is exponential to the
  number of moves.
• Solution Do not examine every node
• gt Alpha-beta pruning
• Remove branches that do not influence final
  decision
• Revisit example next time