Overview of Alpha Beta Pruning
Alpha-beta pruning is a type of search algorithm used in the field of artificial intelligence and computer games. This is a common approach used in conjunction with tree search algorithms such as the minimax method described in “Overview of the Minimax Method and Examples of Algorithms and Implementations.
This algorithm is used to efficiently find a solution by reducing unnecessary search when exploring the tree structure of a game. Specifically, it reduces computation time by representing possible combinations of moves in the game as a tree structure and removing unnecessary moves during the search.
Alpha-beta pruning works as follows.
1. both the maximum (Player A’s move) and minimum (Player B’s move) values of alpha and beta for the node being explored. Alpha is the currently confirmed best option, and beta represents the worst possible move the opposing player could take.
2. If, in the course of exploring a node, the alpha value becomes greater than or equal to the beta value (i.e., the best option has been determined), the search for that node is terminated. This is because this move will not be selected in the search at the higher parent node of this node.
3. As the search progresses, the alpha (best option) and beta (worst move of the opposing player) values of the current node are updated. This makes the search more efficient for nodes that may find better moves.
4. repeat this process until all nodes in the tree have been explored or the search is terminated.
Alpha-beta pruning is an effective method that provides significant computational time savings and will be a widely used method in computer games and artificial intelligence.
Algorithms Related to Alpha-Beta Pruning
Alpha-beta pruning is a search algorithm used in combination with the minimax method, which is a common method for exploring the tree structure of a game to find the optimal move. Since using the minimax method as is often requires a very extensive search, alpha-beta pruning is introduced.
The algorithm for alpha-beta pruning is as follows
1. search the tree structure of the game based on the minimax method. Each node represents a player’s turn, and each edge represents the state that results from that move.
2. evaluate the possible moves that each player can take while recursively exploring, and calculate the evaluation value obtained by that move
3. Calculate the evaluation value of the node being explored using alpha-beta values. The alpha value represents the best choice at the moment, and the beta value represents the worst possible move the opposing player could make.
4. If the alpha value is greater than or equal to the beta value, the search for that node is terminated. This is because it indicates that the node is a favorable move for the current player and a deeper search is unnecessary.
5. update the alpha and beta values as the tree is recursively explored
6. repeat this process until all nodes in the tree have been explored or the search is terminated.
Alpha-beta pruning dramatically reduces the computational complexity of the minimax method and improves the efficiency of the search, making it a widely used method in computer games and artificial intelligence.
Application of Alpha Beta Pruning
The following are examples of applications of alpha-beta pruning.
1. board games such as chess and shogi: These games are full information games, where the minimax method is combined with alpha-beta pruning for efficient search to find the best move. Alpha-beta pruning reduces computational cost by efficiently pruning the search tree.
2. board games such as Othello and Reversi: These games, like chess and shogi, also use the minimax method and alpha-beta pruning. Algorithms are used to explore possible combinations of moves in the game and find the optimal move.
3. Go: Go is a very complex game and it is difficult to apply the minimax method and alpha-beta pruning directly, but they are used in combination with an algorithm called Monte Carlo Tree Search (MCTS) described in “Overview of Monte Carlo Tree Search and Examples of Algorithms and Implementations“. MCTS is a method of game search that combines random simulation and tree search, and alpha-beta pruning is to be applied as part of the tree search.
4. computer game AI: Alpha-beta pruning is widely used in computer game AI. For example, it is used in chess and shogi AI, or in various games, such as real-time strategy games, to help the AI calculate optimal moves.
Examples of Alpha Beta Pruning Implementations
An example implementation of alpha-beta pruning is shown. The following example shows Python code for exploring a chess game tree and finding the optimal move using a simple minimax method and alpha-beta pruning.
# Example implementation of chess game tree search using minimax method and alpha-beta pruning
# Functions to search for chess game trees using minimax method and alpha-beta pruning
def alpha_beta_search(board, depth, alpha, beta, maximizing_player):
if depth == 0 or board.game_over():
return evaluate(board)
if maximizing_player:
max_eval = float('-inf')
for move in board.get_legal_moves():
board.make_move(move)
eval = alpha_beta_search(board, depth - 1, alpha, beta, False)
board.undo_move()
max_eval = max(max_eval, eval)
alpha = max(alpha, eval)
if beta <= alpha:
break
return max_eval
else:
min_eval = float('inf')
for move in board.get_legal_moves():
board.make_move(move)
eval = alpha_beta_search(board, depth - 1, alpha, beta, True)
board.undo_move()
min_eval = min(min_eval, eval)
beta = min(beta, eval)
if beta <= alpha: break
return min_eval
# A board evaluation function (as a simple one, returning the sum of the piece values)
def evaluate(board):
piece_values = {'P': 1, 'N': 3, 'B': 3, 'R': 5, 'Q': 9, 'K': 0} # Value of pawns, knights, bishops, rooks, queens, and kings
score = 0
for row in board:
for piece in row:
if piece != '.':
if piece.islower(): # black piece
score -= piece_values[piece.upper()]
else: # white piece
score += piece_values[piece]
return score
# Main execution part
if __name__ == "__main__":
# A list representing a temporary board (e.g., initial chess placement)
board = [
['r', 'n', 'b', 'q', 'k', 'b', 'n', 'r'],
['p', 'p', 'p', 'p', 'p', 'p', 'p', 'p'],
['.', '.', '.', '.', '.', '.', '.', '.'],
['.', '.', '.', '.', '.', '.', '.', '.'],
['.', '.', '.', '.', '.', '.', '.', '.'],
['.', '.', '.', '.', '.', '.', '.', '.'],
['P', 'P', 'P', 'P', 'P', 'P', 'P', 'P'],
['R', 'N', 'B', 'Q', 'K', 'B', 'N', 'R'] ]
# Create a game object for the board (this is a simplified example and should be replaced with a class representing the actual chess board)
class ChessBoard:
def game_over(self):
return False # Game end conditions are omitted here.
def get_legal_moves(self):
return [(0, 0)] # Returns a list of tentative legal moves
def make_move(self, move):
pass # Applying the Temporary Hand
def undo_move(self):
pass # Undo the temporary hand.
# Instantiation of a chess board object
chess_board = ChessBoard()
# Calculate optimal moves using the minimax method and alpha-beta pruning
best_move = None
alpha = float('-inf')
beta = float('inf')
for move in chess_board.get_legal_moves():
chess_board.make_move(move)
eval = alpha_beta_search(chess_board, depth=3, alpha=alpha, beta=beta, maximizing_player=False)
chess_board.undo_move()
if eval > alpha:
best_move = move
alpha = eval
print("Best move:", best_move)This code uses an initial chess arrangement as a temporary board. Also, the ChessBoard class is a simplification and should be replaced with a class representing the actual chess board.
In this example, the alpha_beta_search function implements the alpha-beta pruning algorithm, and the evaluate function evaluates the board. The main function then computes the optimal move using the minimax method and alpha-beta pruning.
Alpha Beta Pruning Challenges and Measures to Address Them
Although alpha-beta pruning is a powerful technique for reducing computational cost by efficiently reducing the search space, several challenges exist. These issues and their solutions are described below.
1. inaccuracy of local evaluation:
Challenge: Alpha-beta pruning relies on a function (heuristic function) to evaluate a station when it reaches the end of the search tree. If this function is not accurate enough, it is difficult to find the optimal move.
Solution: Improve the game evaluation function or introduce more advanced methods. For example, learning the game evaluation function using machine learning techniques or introducing Monte Carlo Tree Search (MCTS).
2. search depth limitation:
Challenge: Alpha-beta pruning searches the search tree to a certain depth, so if there is a depth limitation, the evaluation of the phase may be inaccurate.
Solution: Introduce more advanced pruning methods or algorithms that dynamically adjust the search depth. In addition, the depth of search can be increased by using parallelization or decentralization, where multiple processes or threads perform the search simultaneously.
3. application to non-complete information games:
Challenge: Alpha-beta pruning is mainly used for complete information games, but is difficult to apply to non-complete information games (games in which the opponent’s hand or some information is unknown).
Solution: Consider other search methods such as Monte Carlo Tree Search (MCTS), which can be applied to non-perfect information games.
4. dealing with large search spaces:
Challenge: Large search spaces limit the effectiveness of alpha-beta pruning.
Solution: Use more advanced pruning methods, parallelization, and decentralization to cope with large search spaces. In addition, partial or local search may enable efficient search for solutions.
Reference Information and Reference Books
For general machine learning algorithms including search algorithms, see “Algorithms and Data Structures” or “General Machine Learning and Data Analysis.
“Algorithms” and other reference books are also available.
コメント