A parameter of a problem assigns a natural number to every instance. An embedding of $A$ into $B$ bounds a parameter $\kappa_B$ of $B$ by a function $g$ of a parameter $\kappa_A$ of $A$ if $\kappa_B(f(x)) \le g(\kappa_A(x))$ for every instance $x$ of $A$, where $f$ is the instance map.

For problems assembled from fields there are two sources of parameters. A kernel parameter depends on the kernel instance only, such as the number of machines. A field parameter depends on the data of one constraint module, such as the number of distinct release dates; it is invariant if reading data with a more general value does not change it.

Theorem. The embedding induced by a field-wise comparison of specifications preserves every kernel parameter and every invariant field parameter exactly.

For reductions of decision versions, let $\kappa_A$ and $\kappa_B$ assign to the instances of $A$ and $B$ values of any kind, for instance pairs of numbers, and let $R$ relate such values. Problem $A$ reduces to $B$ with $R$ if some reduction maps every instance $x$ to an instance $y$ with $R(\kappa_A(x), \kappa_B(y))$. A reduction between two problems of one family is in place if it does not change the kernel instance, and hence no kernel parameter, and changes no invariant field parameter.

Theorem. A sound embedding that relates the parameters gives a reduction with the same relation, and reductions with relations compose.

Reductions are arbitrary maps here: their running time is not modelled.