Past exam of the computer science course of the University of Oxford / 2019 / Artificial intelligence / 1 / g

For an action sequence $\pi$, let $T_{\pi }(B)$ be the set of all physical states obtainable by applying it to the initial belief state $B$. Under a deterministic model this is an image; for nondeterministic outcomes it is the union of all permitted images. For $B^{\prime }\subseteq B$,If each action is applicable to every current possibility from $B$, it is also applicable to every possibility from $B^{\prime }$. If the final belief state for $B$ is contained in the physical goal set $G$, then $T_{\pi }(B^{\prime })\subseteq G$. Thus the same plan solves every subset of $B$.

Nothing comparable follows for an arbitrary superset. For example, let an action take $s_1$ to a goal and leave $s_2$ at a non-goal. It solves $\{s_1\}$ but not $\{s_1,s_2\}$. A larger belief state is solved by this particular sequence exactly when all added states also permit it and end in $G$. It may succeed on some supersets and fail on others; it cannot be guaranteed merely from success on $B$.