A count of a kernel assigns a natural number to every instance, such as its number of machines or its number of jobs. The bound field of a count has the values

• at most $k$, for every number $k$: the count is at most $k$, and at least a least number given with the count ($1$ for machines, $0$ for jobs);
• constant: no restriction on the instances; the value records that the count is regarded as a constant, as in the notation $n = k$;
• unlimited: the count is at least the number of jobs, as in the notation $P\infty$ for a machine for every job;
• at most one more than the jobs: the count is at most the number of jobs plus one, as in the notation $n \ge m - 1$ for the number $m$ of machines;
• any: no restriction.

At most $k$ is a special case of at most $l$ for $k \le l$, of constant and of any; the other values are special cases of any.

In the three-field notation, $P2$ denotes two machines. Here it is read as at most two machines and at least one, so that the problems with fewer machines are special cases: an instance with one machine is an instance with at most two. An instance without machines belongs to no problem with a bounded number of machines.

Theorem. Over every kernel, for every count and every least number, the bound field is sound.