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Constructing a sequence of actions whose successive applications transform a specified initial state into one satisfying a goal. The formal model must specify action preconditions, state changes and the meaning of facts absent from a state. STRIPS represents changes by add and delete lists; planning state invariants explain which physical properties the action definitions preserve.

The blocks world is a classical planning model of movable blocks arranged in stacks on an unlimited table. Each block has one support, each block supports at most one other block, and the support relation has no directed cycle. A clear block can move onto another clear block or onto the table. In a direct-transfer STRIPS representation, an action includes the entire pickup-and-placement operation; a separate holding or empty-hand planning fluent is unnecessary unless the action model explicitly splits those operations.

The table has unlimited capacity, so its availability must not be deleted when a block is placed there. Planning state invariants connect the predicate describing a clear block to the absence of any block directly above it. Goal interaction in planning makes this small model useful for testing automated planning methods.

Automated planning with a completely specified initial state, deterministic actions and a goal condition. A finite STRIPS model is a common representation. A plan is an action sequence applicable from the initial state, ending in a goal state. Unknown facts about the initial world require an epistemic representation such as a belief state, rather than silently interpreting every unlisted fact as false.

The convention that ground facts not present in the current representation are false. In a basic STRIPS state, listing the true facts therefore determines a complete interpretation for its ground facts. This convention differs from a knowledge base in which an absent assertion is simply unknown.

Expand a concrete state by applying every action whose preconditions hold, and search the resulting state graph for a goal. For STRIPS, the successor is $(S\setminus \mathrm{Del}(a))\cup \mathrm{Add}(a)$. A valid expansion checks the complete precondition set, including resource markers such as an empty hand, rather than only an action's geometric destination.

A condition specifying when a fact remains unchanged by an action. In STRIPS, facts outside the add and delete lists persist. A successor-state axiom combines these persistence conditions with all action effects that can change the fact.

Use direct-transfer STRIPS actions: one action includes picking up a block and putting it down again. There is no separate holding state. Ground $b,y$ as distinct blocks and $x$ as a support, either a block or the table, with $x\ne b$ and $x\ne y$.

For $\mathrm{Move}(b,x,y)$, takeThe preconditions ensure that $b$ can be picked up and that the destination has room. Moving it frees a block source and occupies the destination, while $\mathrm{Clear}(b)$ persists. When $x=\tau$, table availability remains true; its redundant addition is simply omitted. Since $x$ is fixed in each ground action, this case distinction constructs ordinary add lists rather than introducing conditional action effects. The search transition model is $(S\setminus \mathrm{Del})\cup \mathrm{Add}$.

Use a sequential SATPlan encoding with times $t=0,\dots ,T$. The fluent universe isThere are nine support Boolean variables and three clearance Boolean variables per time layer. Static block types, parameter inequalities and permanent table availability are handled during action grounding. For each well-typed ground action $a$ from parts (b)–(c), introduce an occurrence Boolean variable $a_t$ for $0\le t<T$. Allow at most one action per layer, including no action, so that a horizon of $T$ encodes at most $T$ physical actions.

The complete initial-state formula isEvery well-typed fluent not initially true is explicitly false. The final goal formula is simply $G_T=\mathrm{On}(A,B{)}_T\wedge \mathrm{On}(B,C{)}_T$. Leaving unrelated final facts unconstrained is appropriate; leaving unrelated initial facts unconstrained would introduce imaginary starting states.

For each $t<T$, include all action precondition axioms $a_t\to {\bigwedge }_{p\in \mathrm{Pre}(a)}{p}_t$, after removing static true conditions. For example,The precondition axiom for $\mathrm{MoveToTable}(C,A{)}_t$ similarly requires $\mathrm{On}(C,A{)}_t\wedge \mathrm{Clear}(C{)}_t$.

For every fluent $p\in \mathcal{P}$, let ${\mathrm{Add}}_p(t)$ be the logical disjunction of actions at $t$ adding $p$, and ${\mathrm{Del}}_p(t)$ the logical disjunction deleting it. Include the successor-state axiomIt combines action effects with the frame axiom, preventing unsupported changes. A fully grounded example isApply the same construction to every support and clearance fluent, using the exact add/delete lists. If no action occurs, both effect logical disjunctions are false, so all fluents persist.

The sequential action exclusion axioms are $\neg a_t\vee \neg b_t$ for every pair of distinct grounded actions $a,b$. For example,forbids selecting those two occurrences together. Do not add a requirement that some action occur in every layer: that would remove idle padding and change “at most $T$” into an exact-length requirement. One may additionally include true planning state invariants, such as a block having at most one support and a block support carrying at most one block; the unlimited table must be exempt from the latter restriction.

Finally, conjoin $I_0$, $G_T$, and all precondition, successor and exclusion formulas over the layers to obtain ${\phi }_T$. It is a propositional formula; if a solver requires conjunctive normal form, use a satisfiability-preserving Tseitin transformation with auxiliary Boolean variables. The equivalence between this encoding and real action sequences follows from sequential SATPlan correctness: a satisfying assignment induces exactly the stated STRIPS trajectory, and every sufficiently short valid plan supplies such an assignment.

The difficulty is representing dictionary membership and the one-letter relationship in positive conjunctive STRIPS preconditions; a finite dictionary nevertheless admits an exact encoding. Pure STRIPS does not provide string indexing, a “differs in one position” operator, arbitrary disjunctive tests or a procedure that calls a dictionary lookup. Those relationships must be represented by ground facts or compiled into ground actions.

One exact whole-word encoding uses a planning fluent $\mathrm{At}(w)$ and a static predicate $\mathrm{Next}(w,w^{\prime })$ containing exactly the permitted dictionary mutations. For the action schema $\mathrm{Change}(w,w^{\prime })$, useInitialize $\mathrm{At}(s)$ and the static transition facts, and use goal $\mathrm{At}(t)$. Only one $\mathrm{At}$ fact is true initially. Each applicable ground action deletes it and adds exactly its permitted successor, so mathematical induction proves that exactly one current-word fact remains true and that every plan is a legal word ladder. Conversely, every legal word ladder directly supplies a plan. An isolated non-dictionary start is handled as in (c). A shortest plan has the fewest mutations.

This encoding moves the hard lexical relationship into preprocessing. For $N$ dictionary words, checking all ordered pairs takes $O(N^{2}L)$ character comparisons, although indexing can do better; the number of stored transitions is at most $NL(|\Sigma |-1)$ when all states are dictionary words. The STRIPS representation is finite, but its size and preprocessing cost matter.

A position-based representation instead uses planning fluents $\mathrm{Letter}(i,\ell )$. Changing position $i$ must delete its old letter and add its new one; all other positions persist through the STRIPS update rule. Merely requiring each position to contain a letter admits non-dictionary strings. A disjunction over all dictionary words is not a positive conjunctive STRIPS precondition. It can be compiled into separate ground actions, each requiring every letter of a valid source word and changing one position to reach a valid destination. This again enumerates lexical cases and can be large. Arbitrary letter changes followed by an unconstrained “validate” action are unsound: several illegal intermediate strings could otherwise lead to a legal goal. Legality must hold after every mutation, not only at the endpoint.

Use location, holding and hand-capacity planning fluents, together with static object types. Write $\mathrm{In}(b,k)$, $\mathrm{Holding}(b)$, $\mathrm{At}(k)$ and $\mathrm{Empty}$ for a ball's bucket, the held ball, the arm's location and an empty hand. Static predicates $\mathrm{Ball}(b)$, $\mathrm{Bucket}(k)$ and $\mathrm{Different}(k,\ell )$ identify types and distinct buckets. The latter can equivalently be a action grounding restriction.

For typed parameters, the STRIPS actions areUntyped notation adds the appropriate $\mathrm{Ball}$ and $\mathrm{Bucket}$ facts to every precondition set. Static type and inequality facts have no add or delete effects. A state is a set of true ground facts under the closed-world assumption, and an applicable action produces $(S\setminus \mathrm{Del})\cup \mathrm{Add}$.

Start from a physically valid state: each ball is in exactly one bucket or held; at most one ball is held; the arm is at exactly one bucket; and $\mathrm{Empty}$ is true exactly when no ball is held. The operators preserve these planning state invariants. In particular, picking requires an empty hand, dropping releases the held ball, and moving deletes the old arm location. Moving while holding a ball is permitted, so a separate carried-ball movement action is unnecessary.

There are $G=2mn+n(n-1)$ distinct ground actions, or $tG$ time-indexed action variables over $t$ transition steps. Each of STRIPS's pick and drop action schemas has $mn$ type-correct instantiations. A move chooses an ordered pair of distinct buckets, giving $n(n-1)$. Every physical action is a ground action before it receives a time index in SATPlan.These counts exclude static type checks after action grounding and exclude optional persistence/no-op action symbols. Using successor-state axioms supplies persistence without separate no-op action variables. If one action occurrence is allowed per layer, it is selected from the same ground-action set; a parallel encoding also begins with these physical ground actions, but has a different exclusion policy.

Use backward deletion of achieved goal requirements; this decides precondition-free ground STRIPS planning in polynomial time. Let $I$ be the initial set of true facts, $G$ the positive conjunctive goal set and $\mathcal{A}$ the ground actions. Assume the usual disjoint add and delete lists; if both lists contain a fact, first normalize the net effect according to the action semantics.

Maintain residual goal set $R=G$ and an initially empty list $L$. While $R⊈I$, look for an action $a$ satisfyingIf no such action exists, reject. Otherwise record $a$ in $L$ and replace $R$ by $R\setminus \mathrm{Add}(a)$. When $R\subseteq I$, return the actions in reverse order of recording. All actions are applicable because their precondition sets are empty.

For soundness, the selected action can be put last in a plan for the current residual goals: it supplies the removed goals and deletes none of the goals still being required. Reversing all recorded selections makes the final state satisfy the original $G$.

For completeness, suppose $R$ is not already satisfied initially but a plan achieves it. Consider the last action in that plan that changes any fact of $R$. Its effective changes cannot delete any member of $R$, because no later action repairs it; it must add at least one member. Thus a candidate exists. Moreover any candidate chosen by the algorithm is safe: removing some requirements gives a subset of $R$, so a plan for $R$, if one exists, also establishes the smaller goal set. Appending the chosen last action then restores the removed requirements without harming the rest. Mathematical induction therefore preserves solvability during all choices.

Each successful iteration removes at least one residual goal, so there are at most $|G|$ iterations and the returned plan has at most $|G|$ actions. Scanning all actions and their effect lists per iteration is polynomial in the input size, for example $O(|G||\mathcal{A}||P|)$ with simple fact-set representations over $P$. Merely taking the union of all add effects would be unsound, since actions can delete each other's achievements. This algorithm is the last-achiever algorithm for precondition-free planning.

Use $a,w,g,r$ for the agent, monster, gold and arrow, and abbreviate $s{q}_{ij}$ to $s_{ij}$. A physically safe STRIPS plan isEvery movement follows a listed adjacency predicate. The shot is taken from $s_{22}$ while the arrow is held, and kills the monster before entry into $s_{23}$. The pit at $s_{12}$ is never entered. The pickup establishes possession of the gold, and the last four movements restore the exit location while preserving that possession. Thus ten actions achieve the formal goal and the intended safe return; arriving at the exit is the source model’s representation of leaving.

Three different omissions produce concrete erroneous STRIPS plans. Here a route means the corresponding successive $\mathrm{Move}$ actions with the indicated first argument.

No lethal-entry condition. Move the agent along $s_{11},s_{21},s_{22},s_{23},s_{13}$, take the gold at $s_{13}$, and reverse the route. This nine-action STRIPS plan satisfies every supplied precondition and the final goal test, without firing. Physically it fails on entering the live monster’s room: the supplied search transition model neither forbids the entry nor records death. The omitted pit cell happens to be disconnected in the supplied adjacency relation, so this particular instance already excludes pit entry; that accidental protection does not encode the monster hazard.

Supposedly stationary objects can move themselves. The five-action STRIPS planleaves the agent at the exit and brings it the gold. The formal preconditions hold because $x$ in $\mathrm{Move}(x,y,z)$ has no actor restriction. Gold cannot actually propel itself. The same flaw permits moving the supposedly stationary monster.

Unrestricted pickup permits impossible inventory. Follow the first example’s outward route, but insert $\mathrm{Take}(a,w,s_{23})$ immediately after reaching $s_{23}$. Continue to $s_{13}$, take the gold and return. This ten-action STRIPS plan ends with both $\mathrm{have}(a,w)$ and $\mathrm{have}(a,g)$ and satisfies the supplied goal. Its entrance also exposes the first flaw, but the additional, distinct defect is that pickup treats the live monster as a collectible object and deletes its location. A movement safety check alone is not a specification of which objects are collectible. The shot schema likewise lacks restrictions to the designated shooter, projectile and victim; fixing the action roles together prevents these type errors.

A corrected STRIPS model can fix object roles in the action names, leaving only room parameters to instantiate. Use the same $a,w,g,r$ and introduce static $\mathrm{pitfree}(z)$ and $\mathrm{exit}(z)$ facts, plus a planning fluent $\mathrm{clear}(z)$ meaning “no live monster occupies $z$”. Unlike the supplied adjacency relation, let $\mathrm{adj}$ describe all geometrically adjacent rooms; safety is then explicit rather than hidden by missing edges.

Initially the agent, monster and gold have their specified locations, $\mathrm{have}(a,r)$, $\mathrm{alive}(w)$ and $\mathrm{active}$ hold, $\mathrm{exit}(s_{11})$ holds, $\mathrm{pitfree}$ holds at every room except $s_{12}$, and $\mathrm{clear}$ holds at every room except $s_{23}$. All unlisted facts are false by the closed-world assumption. The following positive preconditions, add lists and delete lists are a complete action specification:Require $\mathrm{exited}$ as the goal. If reaching the exit suffices, omit $\mathrm{Leave}$ and use the original two goal facts. Instantiating these schemata over the finite room set gives ordinary ground STRIPS actions.

Only $a$ moves, only $g$ is collectible and only $r$ can be fired at $w$. No action changes the monster’s location; its corpse can stay there, while $\mathrm{clear}$ records that it is harmless. The gold stays at its original room until collection. The arrow is consumed by its only use. By mathematical induction on plan length, every applicable action preserves the planning state invariant that the active agent occupies one pit-free room without a live monster. Consequently every solution is a safe execution. The plan in part a, followed by $\mathrm{Leave}(s_{11})$, is one such execution.

Ordinary classical planning with STRIPS starts from one completely specified state. Here the missing locations express ignorance: not listing a pit or a monster at a room does not establish its absence. Applying the closed-world assumption to those omissions would manufacture knowledge of safety. Moreover, future sensor readings can determine which action is appropriate, so a fixed action sequence need not suffice.

The suitable object is a belief state and a contingent plan that branches on observations. A finite belief state model could in principle be compiled into a larger classical planning instance, but the ordinary physical-state STRIPS representation does not directly represent the required uncertainty and sensing.

In STRIPS, a state records the true ground facts; under the closed-world assumption, an absent fact is false. An action with positive preconditions $P$, add effects $A$ and delete effects $D$ applies when $P\subseteq S$, producing $(S\setminus D)\cup A$.

The given movement action adds the destination fact without deleting the old boat location. After its first crossing the boat can therefore be recorded at both banks, allowing physically impossible loading. Its source-bank symbol is also not a declared action parameter. Repair both issues by using the fully parameterized STRIPS actionHere $\mathrm{notEqual}(s,t)$ excludes a vacuous same-bank crossing. The two static inequality facts already supplied suffice.

The given loading action has no capacity check. Consecutive loads can put arbitrarily many items aboard. Introduce a nullary planning fluent $\mathrm{Empty}$, true initially, and repair loading toUnloading must restore $\mathrm{Empty}$, as in part (c). Including the bank $s$ explicitly as an action parameter removes the other unbound action variable. Equivalently, instantiate separate east-bank and west-bank actions.

These repairs establish the planning state invariants that the boat is at exactly one bank, each item is either at one bank or aboard, and at most one item is aboard. The first two hold initially and are preserved by deleting the former location when adding the new one; the capacity marker permits loading only when no item is aboard. Grounding implicit location variables alone would not repair the missing delete effect or the missing capacity check.

Use the STRIPS actionThe onboard precondition prevents inventing an item at a bank, and deleting $\mathrm{onBoat}(x)$ preserves the item's unique location. Restoring $\mathrm{Empty}$ preserves the one-item planning state invariant.

Replace the one-place marker $\mathrm{Empty}$ by three nullary planning fluents $q_0,q_1,q_2$, meaning that zero, one or two items are aboard. Initially only $q_0$ is true. Exactly one of these facts is maintained. No additional object constants are needed.

For compactness, define the common loading preconditionsand common unloading preconditionsThe four STRIPS actions areHere the first unloading removes one of two items, and the last unloading removes the sole remaining item; items do not require fixed seat identities. The movement action remains as repaired in part (a) and changes none of the occupancy facts.

The planning state invariant is $q_j$ true exactly when $j$ items are aboard. Every action changes the marker by the same amount as the actual onboard count, so it follows by mathematical induction from the initial empty boat. In particular, a third load is impossible.

Keep the corrected one-place STRIPS model, replace the item constants and their initial typing facts by $\mathrm{FOX},\mathrm{RABBIT},\mathrm{CARROT}$, and require all three west in the goal. The farmer travels with the boat, so the boat's bank also records where supervision is available. The new requirement concerns the bank being left: incompatible items may coexist at the farmer's current bank, but must not be left there together.

Here is a compilation using only ordinary positive STRIPS preconditions. Replace $\mathrm{onBoat}(x)$ everywhere by $\mathrm{at}(x,\mathrm{BOAT})$, extending the item's possible locations toThe boat itself still has only bank locations, enforced by $\mathrm{isBank}$. Initialize a static relation $\mathrm{SafePair}(p,q,s)$ for every $p,q\in \mathcal{L}$ and bank $s$ except $p=q=s$. It states that two incompatible items at locations $p,q$ are not both on the bank about to be left.

Introduce planning safety certificates $\mathrm{SafeToSail}(\mathrm{EAST})$ and $\mathrm{SafeToSail}(\mathrm{WEST})$, neither initially true. The following checking action grants a certificate:Add $\mathrm{SafeToSail}(s)$ to the preconditions of $\mathrm{Move}(s,t)$. Every load, unload and move deletes both certificate facts. The other movement and capacity effects remain unchanged.

Deleting certificates on every location change prevents a stale safety check from authorizing a later unsafe departure. A certificate is granted only for the actual current locations and only when both incompatible pairs are separated on the departure bank. During a crossing the remaining bank is therefore safe; on arrival the farmer supervises the destination bank. This proves the required safety property by mathematical induction over crossings. The static $\mathrm{SafePair}$ facts are never changed.