Knuth, D. E., and Moore, R. N., 1975. An analysis of alpha-beta pruning, Artificial Intelligence 6, 293-326.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Subsequent testing on the machine itself con rmed this prediction. Cilk s guarantee of ecient scheduling, together with the performance model it engenders, saved us from a costly design error. 4 Programming alpha beta search This section shows how a parallel version of alpha beta search [28] can be programmed in Cilk. Although Socrates and Cilkchess use di erent parallel variants of minimax search based on Scout search [34] and MTD(f) 35] respectively, all the ideas in these parallel search 13 algorithms can be illustrated in the context of alpha beta search. This section ....

.... child sc; negamax if ( child sc bestscore ) bestscore = child sc; if ( child sc cur.alpha ) cur.alpha = child sc; if ( child sc = cur.beta ) cutoff = TRUE; no need to search further abort; terminate other children The code implements a negamax [28] strategy wherein scores are always viewed from the point of view of the side to move. If the value child sc returned by a child is the best so far, the variable bestscore is updated to record that fact. If the child s score exceeds the current value for alpha, then cur.alpha is updated. Finally, ....

Donald E. Knuth and Ronald W. Moore. An analysis of alpha-beta pruning. Arti cial Intelligence, 6(4):293-326, Winter 1975.

....Fall 1989. 1. Introduction In computer game playing, a b pruning is a commonly used technique for speeding up search processes. Knuth and Moore credit the idea of pruning some nodes from the search process to McCarthy and his group at MIT, and trace back this search reduction method to 1958[1]. However, the first formal treatment of this topic appears to be that of Brudno in 1963[2] In the a b pruning algorithm, two bounds a and b are assigned to an interior node in the search tree that has been visited by a depth first search process, and all the expected merit values of the ....

....nodes in a search tree, it will search the least number of leaf nodes to get the probabilities that describe the root position of the search tree. We show that the probability based a b algorithm can be viewed as a generalization of the a b pruning employed by point value search algorithms [1]. Several variations or applications of the probability based algorithm are also presented. One of them applies a b pruning to range based game tree search. The heuristic information available at interior nodes of a search tree can also be used to improve the a b bounds. We also show how a b ....

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D. E. Knuth and R. W. Moore, An Analysis of Alpha-Beta Pruning, Artificial Intelligence 6, (1975), pp. 293-326.

....was represented by a vector of length 64, where each element stores the contents of a square on the chess board. On each player s turn, the set of legal moves is generated for the current position. To select one of the moves, a minimax tree was generated with alpha beta pruning to a fixed depth [7]. Quiescence was used to selectively extend the search tree to avoid noisy board positions where material exchanges Proceedings of the 2001 IEEE Congress on Evolutionary Computation Seoul, Korea . May 27 30, 2001 0 7803 6657 3 01 10.00 2001 IEEE 995 Computer Champions Human Champions ....

Knuth D. E., Moore R. W., "An analysis of alpha beta pruning", Artificial Intelligence, Vol. 6 (4) (1975), pp. 293-326.

....is the scienti c name for the common fruit y, is popularly used for biological experiments. Similarly, chess is used by many researchers to perform experiments in AI. The techniques pioneered in computer chess have been widely applied to other games, such as Othello and checkers. The search [11, 23] is the most successful algorithm in computer chess. In fact, most of the research in chess has focused on searching chess trees more eciently, a tool which may be applicable directly to other games. The strength of a game playing system is strongly in uenced by the depth of the search tree ....

....reduced because it is not always necessary to visit all the leaf nodes at depth d to determine the minimax value for the root. It can be proved that some nodes cannot a ect the value of the root. Knuth and Moore showed the least number of leaf nodes that must be visited to prove the root value is [23], NBPBest = b 2 e b b 2 c Although NBPBest is still exponential in d, it is much smaller than NBPMinimax . The best case minimax algorithm can search roughly twice as deep as the minimax algorithm. Practical algorithms try to achieve the best case result. The rest of this chapter ....

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D. E. Knuth and R. W. Moore. An Analysis of Alpha-Beta Pruning. Arti cial Intelligence, 6:293-326, 1975.

....a first solution until the search space is completely exhausted. Whenever a better solution is found, the current solution path and its value are updated. Inferior subtrees that are known to be worse than the current solution, are eliminated from the search space ( cut off) The Alpha Beta [15, 26, 28] and SSS [40] game tree search algorithms can also be viewed as DFBB variants operating on trees with alternating levels of AND and OR nodes [19] Best First Search orders the sequence of the node expansions according to a heuristic function describing the relative merit of the successor ....

D.E. Knuth and R.W. Moore. An analysis of alpha-beta pruning. Artif. Intell. 6,4(1975), 293 326.

....modern ones. Games played by computers motivated a sizable amount of research in Artificial Intelligence (AI) with machines being constructed to play specific games and some methodologies developed to evaluate general game trees. Most notable among those is Knuth and Moore s ff fi pruning method [KnM] for efficient evaluation of game trees. This algorithm had many offsprings and ramifications, see e.g. Pearl s book [Pe] Tarsi [Ta] and Saks Wigderson [SW] consider efficient randomized algorithms for evaluating game trees. Having mentioned AI, let us briefly point out some interesting ....

D. E. Knuth and R. N. Moore, An analysis of the alpha-beta pruning, Artificial Intelligence 6(1975) 293 - 322.

....to be included in the model to enable examples of conversational implicature to be produced, as was described in section 1.2.3. The procedure illustrated here is analogous to the alpha beta pruning algorithm used by games playing programmes, the uses of which were reviewed by Knuth and Moore in [78]. The algorithm described here differs from a typical alpha beta pruning in two respects: Firstly, the value of each possible world specification must be recomputed for each agent that considers it, and even for the same agent at different levels of nesting, whereas for two players playing a ....

D. E. Knuth and R. W. Moore. An analysis of alpha-beta pruning. Artificial Intelligence, 6(4):293--326, 1975.

....game states are modeled as vertices and the possible moves as edges. By searching the tree in depth, using the minmax algorithm [4] the best move is computed, according to the value estimation of each vertexposition. In order to accelerate computation, we can use the ( b a, pruning method [1], to search the tree in more depth in the same amount of time. However, it should be clear that in the above case, the programmer s ability in the particular game is a vital component to the success or failure of the experiment, due to the need of correctly modeling several attributes at the same ....

D.E Knuth, R.E Moore. An analysis of alpha beta pruning. Artificial Intelligence 6(4), 293-326, 1975.

....In classical games like Chess, the goal of TS is to find a move among many moves, using an EF as a black box. Numerous publications deal with this problem, and study the well known minimax and Alpha Beta algorithms, as well as their variants [Berliner 79] Stockman 79] Campbell Marsland 83] Knuth 75] Korf 85] A recent trend has favored new approaches, such as the conspiracy numbers approach [McAllester 88] Schaeffer 90] and proof number search [Allis al. 94] Computer Go follows this trend. But in Go, TS is completely different because of the importance of locality. Until now, few ....

, D.E. Knuth, An analysis of alpha-beta pruning, Artificial Intelligence, 6 (4), (1975), pp. 293-326.

....[34] that even if the nal outcomes are restricted to only the two values win or loss, a more ne grained evaluation at interior nodes is bene cial. A big advantage of scalar evaluations is that they can also be used for move ordering. Good move ordering is essential for the speed of alpha beta [19]. In iterative deepening search, using the search result of the previous iteration yields a move ordering that is often close to optimal [32] Finally, scalar evaluation is quite robust against errors in the evaluation function as long as they are systematic. An evaluation function needs only ....

D.E. Knuth and R.W. Moore. An analysis of alpha-beta pruning. Articial Intelligence, 6:293-326, 1975.

....of a strong human player, is a challenging problem that has attracted the attention of many computer scientists over the last fifty years. Two player zero sum games with perfect information, such as chess, Othello 1 and checkers, are programmed using the same basic techniques. The fffi algorithm [9] is used to exhaustively search variations that are d moves deep in a depth first manner to determine the best move and its value. A large hash table, called the transposition table [5] is used to store previously determined best moves and values for positions. The values from this table are ....

....no additional search is needed at this node (a cut off occurs) In effect, this prunes parts of the tree that provably cannot contribute to the minimax value. It has been shown that the fffi algorithm will return the correct minimax value if the root position is searched with ff = Gamma1, fi = 1 [9]. For a depth d tree with b branches at every node, fffi has the potential to search the minimum number of leaf nodes possible to de4 for(i=1; i = numOfSuccessors; i ) move counter score returned by search int sc; AlphaBeta return(alpha) if (alpha = beta) ....

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D. E. Knuth and R. W. Moore. An Analysis of Alpha-Beta Pruning. Artificial Intelligence, 6(3):293--326, 1975. 31

....is given in Figure 2. 3 For a d ply search, it is called by: AlphaBeta( rootnode, 1, 1, d, MAXNODE ) Again assuming a tree of xed branching factor w and search depth d, alphabeta improves the best case of the search tree size to O(w d=2 ) or, to be more precise, w dd=2e w bd=2c 1) [9]. This best case occurs when the move leading to the best minimax score is searched rst at every interior node. 4 If the worst move is search rst at every node, then alpha beta will build an O(w d ) minimax tree. Although the 20 lines of code in Figure 2 look simple, this is misleading. ....

....is often 20 or more pages of code. The reason for this is the exponential di erence in the search e ort between the best and worst alpha beta cases. Considerable e ort has to be invested to ensure a nearly best case result. The consequence is a 3 The Negamax formulation is more concise [9]. 4 At nodes where a cut o occurs, one only needs to search a move that is sucient to immediately cause the cut o . 5 Empirical evidence suggests that this only happens in important tournament games 5 int AlphaBeta( position p, int alpha, int beta, int depth, int type ) Check for a ....

D. Knuth and R. Moore. An analysis of alpha-beta pruning. Articial Intelligence, 6(4):293-326, 1975.

....one move to the child with the maximum or minimum value is made, depending on whose turn it is to move. 6.2 Alpha Beta Pruning One of the most elegant of all AI search algorithms is alpha beta pruning. While it was in use in the late 1950s, a thorough treatment of the algorithm can be found in [29]. The idea, similar to branch and bound, is that the minimax value of the root of a game tree can be determined without examining all the nodes at the search frontier. Figure 5 shows an example of alpha beta pruning. Only the labelled nodes are generated by the algorithm, with the heavy black ....

Knuth, D.E., and R.E. Moore, An analysis of alpha-beta pruning, Artificial Intelligence, Vol. 6, No. 4, 1975, pp. 293-326.

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D.E. Knuth and R.W. Moore, An Analysis of Alpha-beta Pruning, Artificial Intelligence 6, 4 (1975), 293-326.

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D.E. Knuth and R.W. Moore, An Analysis of Alpha-beta Pruning. Artificial Intelligence 6, 4 (1975), 293-326.

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Knuth, D. E., and Moore, R. N., 1975. An analysis of alpha-beta pruning, Artificial Intelligence 6, 293-326.

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D. Knuth and R. Moore. An Analysis of Alpha-Beta Pruning. Arti cial Intelligence, 6(4):293-326, 1975.

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D. E. Knuth and R. W. Moore. An analysis of alpha-beta pruning. Artificial Intelligence, pages 293--326, 1975.

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R. Moore and D. Knuth. An analysis of alpha-beta pruning. Artificial Intelligence, pages 293--326, 1975.

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D. E. Knuth and R. W. Moore. An analysis of alphabeta pruning. Artificial Intelligence, 6(4):293--326, 1975.

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D. Knuth and R. Moore, An analysis of alpha-beta pruning, Artificial Intelligence 6(4),(1975), 293-326.

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D.E. Knuth and R.W. Moore, "An Analysis of Alpha-beta Pruning," Artificial Intelligence 6(4), 293-326 (1975).

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D. E. Knuth and R. W. Moore. An analysis of alpha-beta pruning. Arti#cial Intelligence, 6#4#:293#326, 1975.

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D.E. Knuth, and R.W. Moore. An analysis of alpha-beta pruning. Artif. Intell. 6,4#1975#, 293#326.

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