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More on Adversial Games

More on Adversial Games. Tic-Tac-Toe. Can do minimax And alpha-beta pruning And improved representation Can we use logical inference?   ¬. 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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More on Adversial Games

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  1. More on Adversial Games

  2. Tic-Tac-Toe • Can do minimax • And alpha-beta pruning • And improved representation • Can we use logical inference?   ¬

  3. 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)

  4. 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. • Hmm…permutations bad…

  5. How to move • empty(a)  goodMove(a)  move(a)

  6. What’s a good move? • Winning is good. • X(a)  X(b)  line(a,b,c)  win(c) • win(n)  goodMove(n)

  7. What’s a good move? • Blocking a win is good. • O(a)  O(b)  line(a,b,c)  block:win(c) • block:win(n)  goodMove(n)

  8. What’s 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)

  9. What’s 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)  block:fork/split(c) • block:fork/split(n)  goodMove(n)

  10. What’s a good move? • Just moving somewhere is good (base facts). • goodMove(1), goodMove(2), goodMove(3), etc. • Remember: empty(a)  goodMove(a)  move(a)

  11. Questions… • How to order the inference application? • How to order the goodMove rule application? • How does this differ from minimax?

  12. 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.

  13. Answers • Ordering good Rule application • Order doesn’t 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.

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

  15. More on adversarial search • Alpha-beta pruning: good! • Recognizing repeated states: good! (for speed and space) • Recognizing equivalent states: good! (for speed and space)

  16. Can’t go very deep • Still, Deep Blue reached 14 plies routinely • Other heuristics helped

  17. What to do? • Evaluate intermediate positions if you can’t play the end-game out.

  18. Typical problems • Non-quiescent positions • Horizon effect • Lack of information

  19. 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 >> A, B >> C

  20. When weak methods don’t work • Use strong ones • Strong methods might apply across domains • Might want to do planning.

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

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