我正在尝试在 Haskell 中实现Negamax算法。
为此,我代表游戏在玫瑰树中的未来可能性(Data.Tree.Forest (depth, move, position)
)。但是,通常可以通过两种不同的移动顺序到达某些位置。重新评估重复位置(的子树)是一种浪费(并且很快变得非常慢)。
这是我到目前为止所尝试的:
实现打结的变体以共享常见的子结果。但是,我只能找到为(可能是无限的)列表打结的解释,而没有找到关于重用子树的解释。
我考虑过的另一种方法是在
State
monad 内构建一棵树,其中要保留的状态将是Map (depth, position) (Forest (depth, move, position))
执行显式记忆,但到目前为止我也无法正确设置它。
我认为这两种方法都可能存在只能以核心递归方式构建博弈树的问题:我们不是从叶子到根构建树,而是从根向下懒惰地构建一个(可能是无限的)树。
编辑:给你一个我目前正在使用的代码的例子(太慢了):
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE FlexibleContexts #-}
module ZeroSumGame where
import qualified Control.Arrow
import Data.Tree
import Numeric.Natural (Natural)
(|>) :: a -> (a -> b) -> b
x |> f = f x
infixl 0 |>
{-# INLINE (|>) #-}
class Ord s => Game s where
data Move s
initial :: s -- | Beginning of the game
applyMove :: Natural -> s -> Move s -> s -- | Moving from one game state to the next
possibleMoves :: Natural -> s -> [Move s] -- | Lists moves the current player is able to do.
isGameOver :: s -> Bool -- | True if the game has ended. TODO: Maybe write default implementation using `possibleMoves state == []`?
scorePosition :: Natural -> Move s -> s -> Int -- | Turns a position in an integer, for the Negamax algorithm to decide which position is the best.
type Trimove state = (Natural, Move state, state) -- | Depth since start of game, move to next position, new position
gameforest :: Game s => Natural -> s -> Forest (Trimove s)
gameforest start_depth start_state = unfoldForest buildNode (nextpositions start_depth start_state)
where
buildNode (depth, move, current_state) =
if
isGameOver current_state
then
((depth, move, current_state), [])
else
((depth, move, current_state), nextpositions depth current_state)
nextpositions depth current_state =
current_state
|> possibleMoves depth
|> fmap (\move -> (succ depth, move, applyMove depth current_state move))
scoreTree :: Game s => Ord (Move s) => Natural -> Tree (Trimove s) -> (Move s, Int)
scoreTree depth node =
case (depth, subForest node) of
(0, _) ->
node |> rootLabel |> uncurry3dropFirst scorePosition
(_, []) ->
node |> rootLabel |> uncurry3dropFirst scorePosition
(_, children) ->
children
|> scoreForest (pred depth)
|> map (Control.Arrow.second negate)
|> maximum
uncurry3dropFirst :: (a -> b -> c -> d) -> (a, b, c) -> (b, d)
uncurry3dropFirst fun (a, b, c) = (b, fun a b c)
scoreForest :: Game s => Ord (Move s) => Natural -> Forest (Trimove s) -> [(Move s, Int)]
scoreForest depth forest =
forest
|> fmap (scoreTree depth)
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE FlexibleContexts #-}
module TicTacToe where
import qualified Control.Monad.State
import Control.Monad.State (State)
import qualified Data.Map
import Data.Map (Map)
import qualified Control.Arrow
import Data.Tree
import Data.Array (Array)
import qualified Data.Array
import qualified Data.Maybe
import qualified Data.Foldable
import Numeric.Natural (Natural)
import ZeroSumGame
data CurrentPlayer = First | Second
deriving (Eq, Ord, Show)
instance Enum CurrentPlayer where
fromEnum First = 1
fromEnum Second = -1
toEnum 1 = First
toEnum (-1) = Second
toEnum _ = error "Improper player"
newtype TicTacToe = TicTacToe (Array (Int, Int) (Maybe CurrentPlayer))
deriving (Eq, Ord)
instance Game TicTacToe where
data Move TicTacToe = TicTacToeMove (Int, Int)
deriving (Eq, Ord, Show, Bounded)
initial = TicTacToe initialTicTacToeBoard
possibleMoves _depth = possibleTicTacToeMoves
applyMove depth (TicTacToe board) (TicTacToeMove (x, y)) =
TicTacToe newboard
where
newboard = board Data.Array.// [((x, y), Just player)]
player = case depth `mod` 2 of
0 -> First
_ -> Second
isGameOver state = Data.Maybe.isJust (findFilledLines state)
scorePosition _ _ state =
state
|> findFilledLines
|> fmap fromEnum
|> Data.Maybe.fromMaybe 0
|> (* (-10000))
findFilledLines :: TicTacToe -> Maybe CurrentPlayer
findFilledLines (TicTacToe board) =
(rows ++ columns ++ diagonals)
|> map winner
|> Data.Foldable.asum
where
rows = vals rows_indexes
columns = vals columns_indexes
diagonals = vals diagonals_indexes
rows_indexes = [[(i, j) | i <- [0..2]]| j <- [0..2]]
columns_indexes = [[(i, j) | j <- [0..2]]| i <- [0..2]]
diagonals_indexes = [[(i, i) ]| i <- [0..2]] ++ [[(i, 2 - i) ]| i <- [0..2]]
vals = map (map (\index -> board Data.Array.! index))
winner :: Eq a => [Maybe a] -> Maybe a
winner [x,y,z] =
if x == y && x == z then x else Nothing
winner _ = Nothing
initialTicTacToeBoard :: (Array (Int, Int) (Maybe CurrentPlayer))
initialTicTacToeBoard =
Data.Array.array ((0, 0), (2, 2)) [((i, j), Nothing) | i <- [0..2], j <- [0..2]]
possibleTicTacToeMoves :: TicTacToe -> [Move TicTacToe]
possibleTicTacToeMoves (TicTacToe board) = foldr checkSquareForMove [] (Data.Array.assocs board)
where
checkSquareForMove (index, val) acc = case val of
Nothing -> TicTacToeMove index : acc
Just _ -> acc
printBoard :: TicTacToe -> String
printBoard (TicTacToe board) =
unlines [unwords [showTile (board Data.Array.! (y, x)) | x <- [0..2]] | y <- [0..2]]
where
showTile loc =
case loc of
Nothing -> " "
Just Second -> "X"
Just First -> "O"
(TypeFamilies 用于允许每个Game
实现都有自己的 a 概念,Move
然后需要 FlexibleContexts 来强制Move s
实现Ord
.