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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 $B$.

Sensorless planning chooses one action sequence that succeeds from every possible initial physical state without receiving distinguishing observations during execution. A belief state is a set of possibilities. For a deterministic search transition model $T$, an applicable action maps $B$ to $\{T(s,a):s\in B\}$; for nondeterministic outcomes, take the union of all possible successors. Applicability must hold for every member of $B$, and success requires the resulting belief state to be a subset of the physical goal set.

Every fixed action sequence preserves inclusion of the initial belief states. Consequently, a successful plan for $B$ also succeeds for every subset of $B$. Success on a superset is an additional requirement, so cannot be inferred from success on $B$ alone. This property supports incremental belief-state search.

A set $B$ is a superset of $A$ when $A$ is a subset of $B$, meaning that every element of $A$ belongs to $B$. Equality is permitted; a proper superset additionally differs from $A$.