Past exam of the computer science course of the University of Oxford / 2021 / Artificial intelligence / 1 / h / Solution

Take ${\phi }_1=p$ for an $A$ win and ${\phi }_2=p\wedge \neg p$ for a $B$ win. In the first case $A$ assigns $p$ true and play terminates immediately. In the second, either value of $p$ completes an assignment making the contradiction false, so $B$ wins regardless of $A$'s move. These examples force termination and do not rely on a stalling convention.

For the subsequent parts, adopt the explicit convention that an infinite play is a draw, with payoff $0$, between an $A$ win at $+1$ and a $B$ win at $-1$. The source does not assign a value to nontermination. Different conventions can change values of cyclic positions; the forced finite wins above remain valid.