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A computational problem specifies the required output for every permitted input.

A condition deciding whether a state successfully completes a search problem. Its timing must agree with the search transition model: arriving at a destination before an adversary's compulsory response need not be a successful terminal state.

In a finite reachable search problem with strictly positive step costs and a reachable goal, greedy best-first search using a perfect heuristic selects one parent–child chain until it reaches a goal state. At a selected nonterminal state $s$, an optimal first transition has $h^{\ast }(s^{\prime })=h^{\ast }(s)-c(s,s^{\prime })<h^{\ast }(s)$. Every older search frontier entry has estimate at least $h^{\ast }(s)$, so the next selected entry must be a newly generated child with smaller estimate. Repeating proves the chain property. In an infinite finitely branching problem, a positive uniform lower bound on costs also bounds the chain length by the initial remaining cost divided by that lower bound. Zero-cost ties can break the chain property; the lemma does not establish least accumulated path cost.

A function estimating the remaining cost from a state $s$ to a goal in a search problem. The estimate is an admissible heuristic when it never exceeds the true minimum remaining cost. A consistent heuristic satisfies an additional local inequality along each permitted transition.

Suppose $0\le h(n)\le h^{\ast }(n)\le u(n)$ in a search problem, with $h^{\ast }=\infty$ at dead ends. Every finite $g(n)+u(n)$ bounds some reachable solution cost from above, so their running minimum $B$ is at least the global optimum $C^{\ast }$. Every node of an optimal path has $g+h\le C^{\ast }\le B$, and therefore survives the displayed strict pruning rule. Ordinary A-star search can continue to a selected goal without receiving a path witnessing $u$.

Equality pruning is unsafe without a stored incumbent solution: when $B=C^{\ast }$, it could delete every optimal continuation. A numerical upper bound does not by itself provide a plan to return. This distinguishes an upper-bound certificate from a constructed solution in branch and bound.

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.

Let the input words be $s,t$, let $L$ be their common length, let $\Sigma$ be the finite alphabet, and let $D_L$ contain the dictionary words of length $L$. Unequal input lengths immediately make the puzzle impossible because a mutation preserves length. A word ladder search problem can use the state space $S=D_L\cup \{s\}$, with initial state $s$; including $s$ handles an input word not itself in the dictionary.

At word $w$, an action is a position–letter pair $(i,\ell )$ with $1\le i\le L$, $\ell \in \Sigma$ and $\ell \ne w_i$, permitted precisely when replacing $w_i$ by $\ell$ gives $w^{\prime }\in D_L$. The search transition model returns that $w^{\prime }$, and the goal test is $w=t$. If $s=t$, the empty sequence is accepted; if $t\notin D_L$ and $s\ne t$, no transition can reach $t$.

Assign each permitted transition cost $1$. A search node at search depth $k$ hasThus minimizing path cost is exactly minimizing the number of mutations. Uniform-cost search, with repeated states handled by graph search and the goal test on selection, finds a shortest word ladder if one exists. Because all costs are equal, breadth-first search gives the same optimal-length 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.

The rule giving the successor state of each permitted state-action pair in a search problem. A compound action may include deterministic environmental responses. If a game state is sampled at full-turn boundaries, its search transition model must still detect terminal events occurring inside the turn. Otherwise an intermediate failure can be incorrectly hidden by the final position. All information needed for future successors and the goal test must be encoded in the state or supplied as fixed problem data.

A step cost is the cost of one action or transition in a search problem. The accumulated path cost is the sum of the step costs along its action history. A uniform positive lower bound on step costs makes every bounded-cost action history have bounded search depth; positivity of each individual term alone does not establish such a bound.

A search problem whose states are words of fixed length, with a transition changing exactly one letter and requiring the resulting word to belong to a specified dictionary. Unit path costs make a shortest graph path a minimum-mutation solution. Hamming distance to the target is a consistent heuristic: one transition changes this distance by at most one. Dictionary restrictions may force a longer solution or make the target unreachable.