A parameterized problem on words consists of the words that encode an instance, the yes-instances among them, and a parameter read off the word. Fixed-parameter tractability, fpt-reductions and hardness for a class of such problems are those of the word RAM.

A coding of a family of scheduling problems says, for every specification, which words encode which instance, such that a word encodes at most one instance of a specification and a word that encodes an instance also encodes it when the instance is read with a field-wise more general specification.

Given a coding, a specification $s$ and a parameter $\kappa$ of its problem, the decision version of the problem of $s$ becomes a parameterized problem on words. An instance $x$ with the integral threshold $D$ is encoded by a word of $x$ followed by two entries, $1$ or $0$ according to whether $D$ is negative, and $|D|$. The admissible words are these, the yes-instances are those for which some feasible schedule has cost at most $D$, and the parameter of the word is $\kappa(x)$.

Theorem. Let the fields of the family be sound, let $s$ be field by field a special case of $t$, and let $\kappa_s$ and $\kappa_t$ be parameters with $\kappa_t(x') \le g(\kappa_s(x))$ for a computable $g$ and every instance $x$ of $s$ read as an instance $x'$ of $t$. Then the parameterized problem of $s$ and $\kappa_s$ fpt-reduces to that of $t$ and $\kappa_t$; the reduction is the identity on words. In particular this holds, with the same parameter on both sides, for every parameter of the kernel instance and for every invariant parameter of a constraint field. Consequently fixed-parameter tractability passes from the problem of $t$ to that of $s$, and hardness for a class from the problem of $s$ to that of $t$.