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CSCE 580 Artificial Intelligence Ch.6: Adversarial Search

CSCE 580 Artificial Intelligence Ch.6: Adversarial Search. Fall 2008 Marco Valtorta mgv@cse.sc.edu. Acknowledgment. The slides are based on the textbook [AIMA] and other sources, including other fine textbooks and the accompanying slide sets The other textbooks I considered are:

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CSCE 580 Artificial Intelligence Ch.6: Adversarial Search

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  1. CSCE 580Artificial IntelligenceCh.6: Adversarial Search Fall 2008 Marco Valtorta mgv@cse.sc.edu

  2. Acknowledgment • The slides are based on the textbook [AIMA] and other sources, including other fine textbooks and the accompanying slide sets • The other textbooks I considered are: • David Poole, Alan Mackworth, and Randy Goebel. Computational Intelligence: A Logical Approach. Oxford, 1998 • A second edition (by Poole and Mackworth) is under development. Dr. Poole allowed us to use a draft of it in this course • Ivan Bratko. Prolog Programming for Artificial Intelligence, Third Edition. Addison-Wesley, 2001 • The fourth edition is under development • George F. Luger. Artificial Intelligence: Structures and Strategies for Complex Problem Solving, Sixth Edition. Addison-Welsey, 2009

  3. Outline • Optimal decisions • α-β pruning • Imperfect, real-time decisions

  4. Games vs. search problems • "Unpredictable" opponent  specifying a move for every possible opponent reply • Time limits  unlikely to find goal, must approximate

  5. Game tree (2-player, deterministic, turns)

  6. Minimax • Perfect play for deterministic games • Idea: choose move to position with highest minimax value = best achievable payoff against optimal play • E.g., 2-ply (=1-move) game:

  7. Minimax algorithm

  8. 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) (O(m) if successors are generated one-at-a-time, as in backtracking) • For chess, b ≈ 35, m ≈100 for “reasonable” games exact solution completely infeasible

  9. α-β pruning example

  10. α-β pruning example

  11. α-β pruning example

  12. α-β pruning example

  13. α-β pruning example

  14. Alpha-Beta Example Using Intervals Do DF-search until first leaf Range of possible values [-∞,+∞] [-∞, +∞]

  15. Alpha-Beta Example (continued) [-∞,+∞] [-∞,3]

  16. Alpha-Beta Example (continued) [-∞,+∞] [-∞,3]

  17. Alpha-Beta Example (continued) [3,+∞] [3,3]

  18. Alpha-Beta Example (continued) [3,+∞] This node is worse for MAX [3,3] [-∞,2]

  19. Alpha-Beta Example (continued) , [3,14] [3,3] [-∞,2] [-∞,14]

  20. Alpha-Beta Example (continued) , [3,5] [3,3] [−∞,2] [-∞,5]

  21. Alpha-Beta Example (continued) [3,3] [2,2] [3,3] [−∞,2]

  22. Alpha-Beta Example (continued) [3,3] [2,2] [3,3] [-∞,2]

  23. Properties of α-β • Pruning does not affect final result • Good move ordering improves effectiveness of pruning • With "perfect ordering," time complexity = O(bm/2) doubles depth of search • A simple example of the value of reasoning about which computations are relevant (a form of metareasoning)

  24. α is the value of the best (i.e., highest-value) choice found so far at any choice point along the path for max If v is worse than α, max will avoid it  prune that branch Define β similarly for min Why is it called α-β?

  25. The α-β algorithm

  26. The α-β algorithm

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

  28. Evaluation functions • For chess, typically linear weighted sum of features Eval(s) = w1 f1(s) + w2 f2(s) + … + wn fn(s) • e.g., w1 = 9 with f1(s) = (number of white queens) – (number of black queens), etc.

  29. 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, b=35  m=4 4-ply lookahead is a hopeless chess player! • 4-ply ≈ human novice • 8-ply ≈ typical PC, human master • 12-ply ≈ Deep Blue, Kasparov

  30. Deterministic games in practice • Checkers: Chinook ended 40-year-reign of human world champion Marion Tinsley in 1994. Used a precomputed endgame database defining perfect play for all positions involving 8 or fewer pieces on the board, a total of 444 billion positions • Jonathan Schaeffer at the department of CS of the University of Alberta showed that checkers is a forced draw: perfect players cannot defeat each other (http://www.cs.ualberta.ca/~chinook/) • Chess: Deep Blue defeated human world champion Garry Kasparov in a six-game match in 1997. Deep Blue searches 200 million positions per second, uses very sophisticated evaluation, and undisclosed methods for extending some lines of search up to 40 ply • Othello: human champions refuse to compete against computers, who are too good • Go: human champions refuse to compete against computers, who are too bad. In go, b > 300, so most programs use pattern knowledge bases to suggest plausible moves

  31. Summary • Games are fun to work on! • They illustrate several important points about AI • perfection is unattainable  must approximate • good idea to think about what to think about

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