Partially observable search

A search problem in which the available observations need not identify the current physical state. The controller must choose the same action for physical states that its observed history cannot distinguish. A belief state summarizes those possibilities; contingent planning selects future actions according to subsequent observations.

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.

An incremental belief-state search initially plans for a small subset $B_0$ of the possible initial states, then tests the candidate plan on the full initial belief state $B$. If a state in $B$ makes an action inapplicable or fails the final goal test, add that counterexample to $B_0$ and solve the enlarged problem. The new search may need to revise the entire plan, rather than append actions to the previous candidate.

With finitely many initial states and a complete finite-model solver for each restricted problem, at most $|B|$ state additions are required. Failure to find a plan even for a restricted subset proves that none can work for the whole belief state. A candidate that passes full validation is a genuine plan for sensorless planning. The advantage is selective tracking: a small number of representative initial states may force a universally successful plan without constructing the full power set search space.

Without a prior probability distribution, a belief state is the set of physical states still compatible with an observed history. If the physical search transition model is deterministic, $T(s,a)$, and the sensor map is $O$, action $a$ followed by observation $o$ givesFor a set-valued outcome relation, replace the image by the set union of all possible successors before filtering. A guaranteed-success goal test requires every member of the surviving belief state to satisfy the physical goal. A probabilistic belief is instead a posterior probability distribution, requiring a specified prior and observation law.