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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.

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  • $\begingroup$ 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. $\endgroup$
    – a3nm
    Mar 15, 2023 at 9:02
  • $\begingroup$ @a3nm Thanks for the reference. I guess there is just no "standard" definition of a class with non-uniform advice and hence the notation is overloaded. So one should be careful with the exact definition used. $\endgroup$ Mar 16, 2023 at 9:02
  • $\begingroup$ Btw: AroraBarak also ignores the issue of the prefix-free encoding of the advice. $\endgroup$ Mar 16, 2023 at 9:06

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