Some special cases of a problem are not a choice of one value per field: they bound a quantity that reads several fields at once. Zero slack, $d_j = r_j + p_j$, reads the release date (kernel), the processing time (kernel) and the due date (objective), so no single field can express it. Such a special case is a bounded problem: the instances of a problem on which finitely many parameters are at most given numbers. A parameter is any function of the instance into the naturals, so "slack at most $\sigma$", "window at most $k$" and "$p_{\max} \le k$" are all bounds.

A bounded problem is a restriction of its problem, so it is a special case of it by the inclusion of instances, and a reading of the problem is a reading of the bounded problem on the same words.

Theorem. A bounded problem is a special case of its problem.

Theorem. A sound embedding of $A$ in $B$ that maps every instance within the bounds of $A$ to an instance within the bounds of $B$ makes the bounded problem of $A$ a special case of the bounded problem of $B$. If it maps every instance of $A$ within the bounds of $B$, then $A$ itself is a special case of the bounded problem of $B$.

On one machine with release dates and due dates, the due-date slack of a job is $d_j - r_j - p_j$, truncated at $0$, and the parameter is its largest value over the jobs. Bounding it by $0$ gives the problem in which no job has room to move: $d_j \le r_j + p_j$ for every job, so a job is on time exactly when it starts at its release date and completes at its due date $d_j = r_j + p_j$, or it cannot be on time at all.

Theorem. $1 \mid\mid \sum w_j Z_j$ is a special case of $1 \mid r_j \mid \sum w_j U_j$ with due-date slack $0$: the encoding with release dates $r_j = d_j - p_j$ produces only instances of slack $0$.