Let's take the following definition of a FPT Turing reduction from Flum & Grohe's book.
Let $F$ and $G$ be parameterised problems. For any instance $x$ of $F$, write $k(x)$ for the parameter of $F$ and $|x|$ for the size of $x$. For any instance $y$ of $G$, write $l(y)$ for the parameter of $y$. An FPT Turing reduction from $F$ to $G$ is an algorithm with an oracle for $G$ that, for some computable functions $f, g : \mathbb{N} → \mathbb{N}$ and for some constant $c \in \mathbb{N}$, solves any instance $x$ of $F$ in time at most $f (k(x)) · |x|^c$ in such a way that for all oracle queries the instances $y$ of $G$ satisfy $l(y) ≤ g(k(x))$.
If we go through the definition in high-level terms I understand:
- We have an algorithm for $F$ that is allowed to call an oracle for the problem $G$ (this is done in constant time).
- The Algorithm must run in a time that is a computable function of the size of the parameter of our instance of $F$ multiplied by a polynomial function of the size of the instance of $F$.
I understand things thus far. However the final restriction that instances of the problem $y$ for which we make an oracle query must satisfy $l(y) ≤ g(k(x))$ for a computable function $g$ I do not understand. What does this say in high-level terms and why is it important? Thanks.