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ID: past-exam-of-the-computer-science-course-of-the-university-of-oxford/2020/artificial-intelligence/3/b/solution

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Past exam of the computer science course of the University of Oxford / 2020 / Artificial intelligence / 3 / b / Solution by

Codex  0 2026-10-25

Use utilities from A's viewpoint, with A maximizing and B minimizing. The following alpha-beta pruning code assumes a finite horizon or terminating game tree; leaf evaluation uses terminal utility when available and a specified horizon estimate otherwise.

ALPHABETA(state, depth, alpha, beta, maximizing):
    if state is terminal or depth == 0:
        return EVALUATE(state)
    if maximizing:
        value := -infinity
        for child in SUCCESSORS(state):
            value := max(value, ALPHABETA(child, depth-1, alpha, beta, false))
            alpha := max(alpha, value)
            if alpha >= beta: break
        return value
    else:
        value := +infinity
        for child in SUCCESSORS(state):
            value := min(value, ALPHABETA(child, depth-1, alpha, beta, true))
            beta := min(beta, value)
            if alpha >= beta: break
        return value

Call the root with

(- \infty, + \infty)

and record the child giving the greatest root value. In a nonterminal game state, SUCCESSORS must supply the game's legal moves, including any mandatory pass; a player with no moves must be handled according to the rules rather than left as an accidental empty loop.

The bound

α

records an outcome already available to MAX on the active ancestor path;

β

records one already available to MIN. At a MIN node, once a child gives value at most

α

, further children can only decrease the node's minimum, so MAX need not choose that branch. The dual argument applies at MAX when its current value is at least

β

. Returned values from cut-off subtrees can be bounds rather than exact local minimax search values, while the unrestricted root value is exact. Equal-value pruning preserves that value; retaining the first best root action gives a fixed tie rule.^[ref]

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