Let $$\mathsf C$$ be some class. Let $$a : \mathbb N \to \mathbb N$$ be some function describing the bit length of advice. Let $$C/a := \{L | \exists L' \in \mathsf C \text{ and } \exists (w_n)_{n\in \mathbb N} : |w_n| \leq a(n) \wedge x \in L \iff (x,w_{|x|}) \in L'\}$$ be the class $$C$$ with $$a$$ bits of advice (attributed to @EmilJeřábek in a comment here).
Consider $$\mathsf{DTIME}(2^n)/n$$. Subjectively, I feel this should be the class of languages that on input $$x \in \{0,1\}^n$$ can be decided by some DTM in time $$2^n$$ given $$n$$ bits of advice (let's say on an "advice" tape). However, the (non-uniform) DTM that decides a language $$L \in \mathsf{DTIME}(2^n)/n$$ on inputs of length $$n$$ may actually take $$2^{2n}$$ steps. This is because the input to the DTM for $$L'$$ is actually $$x||w \in \{0,1\}^{2n}$$ (I guess technically, the input and the advice have to be encoded prefix-free, or there needs to be some delimiter between them. I'll ignore that fact.)
Is there some notion/notation for the class that I described above, i.e., where the advice is not included in the input? For classes like $$\mathsf{DTIME}$$ one can quite easily defined such a class (by adjusting the runtime bound in the definition); however is there a standard notation? For general classes $$\mathsf C$$ I'm not sure how to formalize the intuition above, as there is not a priori a notion of runtime.
• I'm not sure that, in the standard definition of advice for classes defined by bounding the running time, then the running time of the machine is measured as a function of the input including the advice. In the book by Arora and Barak, "Computational Complexity: A Modern Approach" (mysmth.net/…) in Definition 6.16 the running time of the machine is bounded by considering the length $n$ of the original input, not that of the input with advice. I don't know if this is consistently done, however.