More on Adversial Games

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More on Adversial Games
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Tic-Tac-Toe

• Can do minimax
• And alpha-beta pruning
• And improved representation
• Can we use logical inference? ? ??

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Logical approach

• X(n), O(n) represent board position.
• O(1), X(5), O(6), X(9)
• X(n) ? O(n) ? full(n)
• full(n) ? empty(n)

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Some basic facts

• ol(1,2,3), ol(4,5,6), ol(7,8,9)
• ol(1,4,7), ol(2,5,8), ol(3,6,9)
• ol(1,5,9), ol(3,5,7)
• ol(a,b,c) ? line(a,b,c), ol(a,b,c) ? line(a,c,b),
  etc. for the permutations of a,b,c.
• Hmmpermutations bad

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How to move

• empty(a) ? goodMove(a) ? move(a)

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Whats a good move?

• Winning is good.
• X(a) ? X(b) ? line(a,b,c) ? win(c)
• win(n) ? goodMove(n)

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Whats a good move?

• Blocking a win is good.
• O(a) ? O(b) ? line(a,b,c) ? blockwin(c)
• blockwin(n) ? goodMove(n)

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Whats a good move?

• Forking/splitting is good
• X(b) ? X(c) ? b?c line(a,b,d) ? line(a,c,e) ?
  empty(d) ? empty(d) ? fork/split(c)
• fork/split(n) ? goodMove(n)

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Whats a good move?

• Blocking forking/splitting is good
• O(b) ? O(c) ? b?c line(a,b,d) ? line(a,c,e) ?
  empty(d) ? empty(d) ? blockfork/split(c)
• blockfork/split(n) ? goodMove(n)

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Whats a good move?

• Just moving somewhere is good (base facts).
• goodMove(1), goodMove(2), goodMove(3), etc.
• Remember empty(a) ? goodMove(a) ? move(a)

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Questions

• How to order the inference application?
• How to order the goodMove rule application?
• How does this differ from minimax?

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Answers

• Ordering inference application
• Write Horn clauses, use Prolog application rules.
• X(a) ? O(a) ?full(a)
• full(A) - X(A).
• full(A) - Y(A).
• Relatively simple backtracking (although lots of
  speedup tricks are used).
• full(3)? If X(3), yes else if Y(3), yes.
• full(P)? If X(P). X(P)? X(0), etc.

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Answers

• Ordering good Rule application
• Order doesnt matter here
• full(A) - X(A).
• full(A) - Y(A).
• But does matter here
• move(A) - empty(A), goodMove(A).
• goodMove(1).
• goodMove(A) - win(A).
• Just put them in the order you wish but this
  may be hard to do.

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Relation to MiniMax?

• In some ways, not much.
• Can Utility measures give us information about
  good rule ordering?

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More on adversarial search

• Alpha-beta pruning good!
• Recognizing repeated states good! (for speed and
  space)
• Recognizing equivalent states good! (for speed
  and space)

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Cant go very deep

• Still, Deep Blue reached 14 plies routinely
• Other heuristics helped

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What to do?

• Evaluate intermediate positions if you cant play
  the end-game out.

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Typical problems

• Non-quiescent positions
• Horizon effect
• Lack of information

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Games of chance

• Expected value of a choice.
• Prob(choice) Value(choice).
• Extend Minimax to use expected value.
• But (averaging over claivorance)
• A B C
• B gtgt A, B gtgt C

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When weak methods dont work

• Use strong ones
• Strong methods might apply across domains
• Might want to do planning.

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Othe games?

• What about, say, non-adversarial games (current
  AI in game research)?
• Might want to do planning.