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Definition 6.3.1 (Deep α-β pruning) A game G will be said to allow α-β pruning for the minimizer if for any x, y, T, z1, . . , z2i E V  with x ≤ y,

$x\wedge(z_{1}\vee(z_{2}\wedge\cdots\vee(z_{2i}\wedge(y\lor T)))\cdots)=$  $x\wedge(z_{1}\vee(z_{2}\wedge\cdots\lor z_{2i})\cdots)$.  
The game will be said to allow α -β pruning for the maximizer if

$x$$\forall\left(z_{1}\wedge\left(z_{2}\vee\cdots\wedge\left(z_{2i}\vee\left(y\wedge T\right)\right)\right)\cdots\right)=$  $x$$\forall\left(z_{1}\wedge\left(z_{2}\vee\cdots\wedge z_{2i}\right)\cdots\right)$.  
$\angle e,\:\sin i l,\:e$
We will say that G allows pruning if it allows α -β pruning for both players.

As before, the prune allows us to remove the dominated node ( y in Figure 9) and all of its siblings.

The fact that a game allows shallow α β pruning does not mean that it allows pruning in general, as is shown by the following counterexample. The example involves a game with one card that is known to both players; only the suit of the card matters. The game tree appears in Figure 10.

In this tree, a node labelled with a suit symbol is terminal and means that the maximizer wins if and only if the suit of the card matches the given symbol.  So at the root of the given tree, the maximizer (whose turn it is to play) can choose to "turn over" the card, winning if and only if it's a club, or can defer to the minimizer. The minimizer can choose to turn the card ( losing just in case it's a diamond - the suit symbols refer to the maximizer's result), or hand the situation back to the maximizer. If the maximizer defers yet again, the minimizer can either turn over the card, losing if it's a club, or simply declare victory
(presumably his choice).

There is one other wrinkle in this game. At any point in the game, the maximizer can change the card from either a diamond or a spade to a club.

Now let's consider the game itself. At ply 4, the minimizer will obviously choose to win the game. Thus at ply 3, the maximizer will need to choose V, winning just in case the card is a heart. But this means that at ply 2, the minimizer will win the game, since if the card is not a diamond he will move to the left (and win at once) while if the card is not a heart he can win by moving to the right. (Remember that the minimizer knows the suit