= Solution
\b[Use location, holding and hand-capacity <planning fluents>, together with static object types.] Write $\operatorname{In}(b,k)$, $\operatorname{Holding}(b)$, $\operatorname{At}(k)$ and $\operatorname{Empty}$ for a ball's bucket, the held ball, the arm's location and an empty hand. Static predicates $\operatorname{Ball}(b)$, $\operatorname{Bucket}(k)$ and $\operatorname{Different}(k,\ell)$ identify types and distinct buckets. The latter can equivalently be a <action grounding> restriction.
For typed parameters, the <STRIPS> actions are
$$
\begin{array}{c|l|l|l}
\text{Action}&\text{Preconditions}&\text{Add}&\text{Delete}\\\hline
\operatorname{Pick}(b,k)&\operatorname{In}(b,k),\operatorname{At}(k),\operatorname{Empty}&\operatorname{Holding}(b)&\operatorname{In}(b,k),\operatorname{Empty}\\
\operatorname{Drop}(b,k)&\operatorname{Holding}(b),\operatorname{At}(k)&\operatorname{In}(b,k),\operatorname{Empty}&\operatorname{Holding}(b)\\
\operatorname{Move}(k,\ell)&\operatorname{At}(k),\operatorname{Different}(k,\ell)&\operatorname{At}(\ell)&\operatorname{At}(k)
\end{array}
$$
Untyped notation adds the appropriate $\operatorname{Ball}$ and $\operatorname{Bucket}$ 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 $(S\setminus\operatorname{Del})\cup\operatorname{Add}$.
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 $\operatorname{Empty}$ 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.
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