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%%HTML
<style>
.container { width:100% }
</style>

Alpha-Beta Pruning with proper Memoization

The global variable Cache is used as a cache for the function valuedefined later.



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Cache = {}

In order to have some variation in our game, we use random numbers to choose between optimal moves.



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import random
random.seed(0)

Given a player p, the function other(p) computes the opponent of p. This assumes that there are only two players and the set of all players is stored in the global variable Players.



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def other(p):
    return [o for o in Players if o != p][0]

The function value takes four arguments:

State is the current state of the game,
player is a player,
alpha is the worst result that can happen to player,
beta is the best result that can happen to player.

The function value returns the value that the given State has for player if both players play optimal game. This values is an element from the set $\{-1, 0, 1\}$.

If player can force a win, the return value is 1.
If player can at best force a draw, the return value is 0.
If player might loose even when playing optimal, the return value is -1.

For reasons of efficiency, this function is memoized.



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def value(State, player, alpha=-1, beta=1):
    global Cache
    if State in Cache:
        val, a, b = Cache[State]
        if a <= alpha and beta <= b:
            return val
        else:
            alpha = min(alpha, a)
            beta  = max(beta , b)
            val   = alphaBeta(State, player, alpha, beta)
            Cache[State] = val, alpha, beta
            return val
    else:
        val = alphaBeta(State, player, alpha, beta)
        Cache[State] = val, alpha, beta
        return val



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def alphaBeta(State, player, alpha, beta):
    if finished(State):
        return utility(State, player)
    val = alpha
    for ns in next_states(State, player):
        val = max(val, -value(ns, other(player), -beta, -alpha))
        if val >= beta:
            return val
        alpha = max(val, alpha)
    return val

The function best_move takes two arguments:

State is the current state of the game,
player is a player.

The function best_move returns a pair of the form $(v, s)$ where $s$ is a state and $v$ is the value of this state. The state $s$ is a state that is reached from State if player makes one of her optimal moves. In order to have some variation in the game, the function randomly chooses any of the optimal moves.



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def best_move(State, player):
    NS        = next_states(State, player)
    bestVal   = value(State, player)
    BestState = random.choice([s for s in NS if -value(s, other(player)) == bestVal])
    return bestVal, BestState

The next line is needed because we need the function IPython.display.clear_output to clear the output in a cell.



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import IPython.display

The function play_game plays on the given canvas. The game played is specified indirectly by specifying the following:

Start is a global variable defining the start state of the game.
next_states is a function such that $\texttt{next_states}(s, p)$ computes the set of all possible states that can be reached from state $s$ if player $p$ is next to move.
finished is a function such that $\texttt{finished}(s)$ is true for a state $s$ if the game is over in state $s$.
utility is a function such that $\texttt{utility}(s, p)$ returns either -1, 0, or 1 in the terminal state $s$. We have that
- $\texttt{utility}(s, p)= -1$ iff the game is lost for player $p$ in state $s$,
- $\texttt{utility}(s, p)= 0$ iff the game is drawn, and
- $\texttt{utility}(s, p)= 1$ iff the game is won for player $p$ in state $s$.



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def play_game(canvas):
    State = Start
    while (True):
        firstPlayer = Players[0]
        val, State  = best_move(State, firstPlayer);
        draw(State, canvas, f'For me, the game has the value {val}.')
        if finished(State):
            final_msg(State)
            break
        IPython.display.clear_output(wait=True)
        State = get_move(State)
        if finished(State):
            draw(State, canvas, '')
            IPython.display.clear_output(wait=True)
            final_msg(State)
            break



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%run Tic-Tac-Toe.ipynb

With memoization, computing the value of the start state takes 95 ms. Without memoization, it takes 5 seconds.



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%%time
val = value(Start, 'X')

We check how many different states are stored in the Cache. Without alpha-beta pruning, we had to inspect 5478 different states.



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len(Cache)

Let's draw the board.



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canvas = create_canvas(Start)
draw(Start, canvas, f'Current value of game for "X": {val}')

Now its time to play. In the input window that will pop up later, enter your move in the format "row,col" with no space between row and column.



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play_game(canvas)



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