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Automated planning with several possible successors for an action and no specified probabilities for choosing between them. A plan must say how to act after observed outcomes. A strong plan guarantees eventual success for every possible outcome history; a favorable outcome on just one branch does not establish that guarantee.

This is a finite, discrete, partially observable, single-agent, sequential search problem with an unknown initial state. The cave is static while the agent deliberates. Given a complete world and an action, the described physical effects and ideal sensor readings are deterministic; ignorance of the world does not itself supply a probability distribution. The fixed monster and gold are environmental objects, not competing decision makers. A known action/sensor law with unknown object locations is different from having to learn an unknown search transition model.

Use partially observable search over belief states. A complete physical state can be writtenwhere $p$ is the agent’s room, $m$ the fixed monster room, $P$ the pit set, $g$ the fixed gold room, $\ell$ records whether the monster lives, $r$ whether the arrow remains, $k$ whether gold has been collected and $e$ whether the agent has exited or failed. The known room dimensions and exit are fixed data. Include death or entrapment as failure states. Movement, shooting toward an adjacent room, collection and departure have their physical meanings. In particular, a shot consumes the arrow, and kills the monster exactly when it targets the live monster’s room.

Let $T(s,a)$ be this physical search transition model. For a surviving agent, an ideal observation is the tuple of three truth valuesrecording sound, breeze and glitter. Here $\sim$ denotes geometric adjacency and $1$ an indicator function. Initially $B_0$ contains all worlds compatible with the known facts and any initial observation. After action $a$ and observation $o$, update the belief state byKnown success or failure observations are included as needed. No prior probabilities are required. A guaranteed-success goal test accepts $B$ only when every $s\in B$ has exited carrying gold. A policy must use the same action in all worlds still indistinguishable to the agent.

An AND-OR search has OR choices for the agent’s next action and AND branches for every possible resulting observation with nonempty $B_{a,o}^{\prime }$. Its output is a contingent plan, not merely a route. To demand safe success, reject any action whose possible outcomes include irreversible death or entrapment, and require a successful continuation on every observation branch. In an online execution the agent updates its belief state, follows the observation branch and can plan again. A strong plan need not exist: some admissible worlds have no safe route, and even solvable worlds can be observationally indistinguishable until a dangerous choice. Without an added prior and numerical rewards, expected-risk optimization is not determined by the description.

Use AND-OR search for a strong plan. At an OR node choose a joint action; at its AND node require successful continuations for every possible outcome. A repeated state on the current branch cannot by itself establish progress. On a finite state space, the equivalent strong planning winning-set iteration starts with $W_0=\{(p,p):p\in C\}$ and repeatedly adds states with an action whose nonempty outcome set is contained in the current winning set:A newly admitted state has a plan with at most $t+1$ remaining steps in the worst case. This algorithm terminates on the finite state space and is complete for finite guaranteed-success plans. In a general nondeterministic planning problem it cannot guarantee that a strong plan exists: an uncontrollable self-loop can prevent inevitable progress.

For this particular unobstructed grid, a strong plan does exist. Command Mr Robot to wait throughout and move Mrs Robot along a shortest route to his initial room. Her movement is deterministic and his waiting is unaffected by the horizontal uncertainty. This meets in $D(p_0,q_0)$ steps, regardless of every uncertain outcome, because no uncertain command is used. Thus a complete finite AND-OR search will find a solution here. With an unspecified disconnected maze, existence would instead require both robots to lie in the same reachable graph component.

For a finite nondeterministic planning problem with goal set $G$, initialize $W_0=G$ and putA state in $W_t$ has a strong plan reaching $G$ within $t$ steps. The proof is mathematical induction: an action witnessing admission to $W_{t+1}$ has only successors with plans of depth at most $t$. Conversely, the first action of any guaranteed plan of depth $t+1$ has every successor in $W_t$, again by mathematical induction. Since $W_t$ only grows inside a finite state space, the iteration stabilizes after at most the number of states many strict expansions. Recording a witnessing action at first admission gives a policy whose rank decreases at every nonterminal step.