Exercise 5.14

Develop a formal proof of correctness for alpha–beta pruning. To do this, consider the situation shown in Figure alpha-beta-proof-figure. The question is whether to prune node $n_j$, which is a max-node and a descendant of node $n_1$. The basic idea is to prune it if and only if the minimax value of $n_1$ can be shown to be independent of the value of $n_j$.

1. Mode $n_1$ takes on the minimum value among its children: $n_1 = \min(n_2,n_21,\ldots,n_{2b_2})$. Find a similar expression for $n_2$ and hence an expression for $n_1$ in terms of $n_j$.

2. Let $l_i$ be the minimum (or maximum) value of the nodes to the left of node $n_i$ at depth $i$, whose minimax value is already known. Similarly, let $r_i$ be the minimum (or maximum) value of the unexplored nodes to the right of $n_i$ at depth $i$. Rewrite your expression for $n_1$ in terms of the $l_i$ and $r_i$ values.

3. Now reformulate the expression to show that in order to affect $n_1$, $n_j$ must not exceed a certain bound derived from the $l_i$ values.

4. Repeat the process for the case where $n_j$ is a min-node.

Figure [alpha-beta-proof-figure] Situation when considering whether to prune node $n_j$.