An algorithm stated with its exact running time — one program deciding the decision version within $O(T(x))$ instructions on the word of every instance $x$ ($Results.DecidesWithin$) — gives the coarser results by bounding $T$ alone, without looking at the program again. Let $|u|$ be the bit size of the word $u$ of an instance $x$ with a threshold, and $M$ its largest number.

Theorem.

1. If $T(x) \le c\,(|u|+1)^d$, the problem is in P.
2. If $T(x) \le c\,(M+1)^e\,(|u|+1)^d$, it is solvable in pseudo-polynomial time.
3. If $T(x) \le c\,(|u|+1)^{g(\kappa)}$ for a function $g$ of the parameter $\kappa$, it is in P for every fixed value of $\kappa$.
4. If $T(x) \le f(\kappa)\,(|u|+1)^d$ for a computable $f$ and the parameter is computable in polynomial time on the words, it is fixed-parameter tractable in $\kappa$.

Theorem. A program deciding within exactly $T(x)$ instructions decides within $O(T(x))$ ($Results.DecidesExact$).

Theorem. A program deciding within $O(T(x))$ decides within $O(T(x))$ with wide words ($Results.DecidesWithinWide$): the wide words are among the short ones.