 Solution

Minimax search applies because this is a finite, deterministic two-player zero-sum game with perfect information. Each move assigns a previously unassigned Boolean variable, so play terminates after exactly $N$ moves. Both players observe the whole partial Boolean valuation, and there are no chance moves. Give terminal positions payoff $+1$ when the propositional formula is true and $-1$ when it is false, measured for $A$. At an $A$-node take the maximum child value and at a $B$-node the minimum. Backward induction then computes the winner and a corresponding winning strategy.