AI and computer game playing. Artificial Intelligence. Partial game tree for Tic-Tac-Toe. Game tree search. Minimax Algorithm

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1 AI and computer game playing Artificial Intelligence Introduction to game-playing search imax algorithm, evaluation functions, alpha-beta pruning, advanced techniques Game playing (especially chess and checkers) was the first test application of AI Search is performed to determine the best next move in the game The best move depends on what the opponent might do. As a result, this is sometimes called adversary search Game tree search Partial game tree for Tic-Tac-Toe Initial state: initial board position and player perators: one for each legal move Terminal states: a set of states that mark the end of the game Utility function: assigns numeric value to each terminal state Game tree: represents all possible game scenarios X X X X X X X X X X X X X X X imax Algorithm Invented by Von Neumann & Morgenstern in 19 as part of game theory Used for two player games. ne player (MAX) is trying to maximize the value of the evaluation function. The opponent (MIN) is trying to minimize its value imax Algorithm (cont.) Generate the game tree Apply the utility function to each terminal state to get its value Back-up values through game tree: at each MAX node, take max of successors at each MIN node, take min of successors Select best move for MAX node at root 1

2 Search complexity Space complexity of minimax using depthfirst search is (bd) (where b is branching factor and d is depth of game tree). Why use depth-first search for minimax? Time complexity is (b d ) For chess, b 35, d 100 for reasonable games, making exact solution infeasible Static Evaluation Function Can t search full game tree. Instead search to fixed depth or ply and use estimated values for minimax. (Idea due to Claude Shannon -- the inventor of information theory -- in 1950.) Static evaluation function estimates minimax values of non-terminal states. (Some claim evaluation function should measure P(winning).) This strategy works because backed-up evaluations are better than static evaluations, and improve with the depth of lookahead tree. Evaluating chess states material evaluation: count pieces for each side, giving each a weight (queen=9, rook=5, knight/bishop=3, pawn=1) Most evaluation functions are specified as a weighted sum of features. In chess, some features evaluate piece placement on the board and other features define configurations of several features. Deep Blue has about 6000 features in its evaluation function! The weights for chess features are usually learned using linear regression, hill-climbing, or a similar technique. The performance of a chess-playing program is very dependent on the quality of its evaluation function. Alpha-Beta Pruning In chess, can only search full-width tree to about levels The trick is to prune subtrees Fortunately, the best move is provably insensitive to certain subtrees If there exists a winning move at a node, then its sibling nodes (and their subtrees) need not be examined. 2

3 Alpha-Beta Pruning (cont.) Use bounds on minimax values to prune subtrees Bounds on MAX nodes are called alpha values and are initially -. The alpha value for a MAX node is the value of its best successor and can never decrease. Bounds on MIN nodes are called beta values and are initially. The beta value for a MIN node is the value of its worst successor and can never increase. Alpha cutoff: A MIN node can be pruned if its beta value is the alpha value of its MAX parent Beta cutoff: A MAX node can be pruned if its alpha value the beta value of its MIN parent Effectiveness of Alpha-Beta Alpha-beta pruning does not affect minimax value. It just improves efficiency of search. How much it improves efficiency of search depends on ordering of successors. With perfect ordering, can search twice as deep in given amount of time (i.e., effective branching factor is SQRT(b)). Perfect ordering cannot be achieved, but simple ordering heuristics are very effective. Node ordering Instead of exploring the children of a node in a fixed order (e.g., left to right), base the order of exploration on static evaluation of the nodes Expand children of MAX nodes in decreasing order of static evaluation Expand children of MIN nodes in increasing order of static evaluation This technique significantly improves performance of alpha-beta pruning 3

5 Special-Purpose Hardware and Parallel Processing Evaluation functions and move generators can be encoded in special-purpose chips to make them much faster. For example, 220 special-purpose VLSI chess chips were designed for Deep Blue. Deep Blue also used 32 general processors to conduct parallel search How deep does Deep Blue search? Hard to say when both alpha-beta pruning and secondary search are used: minimum: six moves in typical middle-game positions average: about eight moves maximum: highly variable, but typically in the ten-to-twenty-move range What is the difference between how a human plays chess and how Deep Blue plays chess? An experimenter showed chess positions to experts and non-players. Chess players were able to remember the chess positions accurately; non-players could not. When both groups were shown random chess boards, they did equally badly. This indicates chess experts recognize (and remember) meaningful patterns of the chess board. However, it s difficult to design chess programs with the pattern-recognition abilities of humans (though some have tried). Instead, chess programs rely on what computers are good at lots of calculation and search. Is Deep Blue Intelligent? Saying Deep Blue doesn t really think about chess is like saying an airplane doesn t really fly because it doesn t flap its wings. - Drew McDermott What do you think? Does the success of Deep Blue tell us anything about the nature of intelligence? 5

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