Game Playing. Introduction to Artificial Intelligence CS440/ECE448 Lecture 7 FIRST HOMEWORK DUE TODAY. Last lecture. Constraint satisfaction problems Line-drawing interpretation This lecture Games Minimax pruning Reading Chapter 6. Line Drawing Interpretation.
Introduction to Artificial Intelligence
object, label edges as being
Complete junction catalog for line drawings of trihedral-vertex polyhedra
solution is a contingency plan.
3, 2, 2
5, 1, 1
4, 2, 1
3, 3, 1
4, 1, 1, 1
3, 2, 1, 1
2, 2, 2, 1
3, 1, 1, 1, 1
2, 2, 1, 1, 1
2, 1, 1, 1, 1, 1
MAX LosesGrundy’s Game: States
) best achievable payoff against best play by min.
Max want’s largest outcome --- Min want’s smallest.
Yes, if tree is finite.
Yes, against an optimal opponent. Otherwise??
Note: For chess, b 35, m 100 for a “reasonable game.”
Actually only 1040 board positions, not 35100
Suppose we have 100 seconds, explore 104 nodes/second
106 nodes per move.
e.g., depth limit,
(an approximation to the utility used in minimax).
1. Terminal? is replaced by Cutoff?
2. 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
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)
w2 = 5 with
f2(s) = (number of white rooks) - (number of black rooks)
payoff in deterministic games acts as an ordinal utility function.
Bad news may be hidden beyond the cutoff depth.
Black to move..
2 Pruning Example
Prune that branch
Pruning does not affect final result.
Good move ordering improves effectiveness of pruning.
With “perfect ordering”, time complexity = O(bm/2):
A simple example of the value of reasoning about which
computations are relevant (a form of metareasoning).
Checkers: In 1994 Chinookended 40-year-reign of human world champion Marion Tinsley. Used an endgame database defining perfect play for all positions involving 8 or fewer pieces on the board, a total of 443,748,401,247 positions.
Chess:Deep Blue defeated human world champion Gary 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.