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(For the below, I referred extensively to this github repo as well as private communication with @aozgaa) Languages can be represented as infinite-length bitstrings (ILB). We will use the two interchangeably. We will also represent strings meant to be inputs to TMs as integers, where a 1 bit in position $w$ in an ILB $A$ means that the $w$th string in $\...


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Unfortunately, I don't possess a copy of Sipser, so I will just define all my notation. Let $T_0,T_1,\ldots$ an enumeration of all oracle Turing machines whose input is a word over some alphabet $\Sigma$. I will denote by $T_i^O(x)$ the output of the execution of $T_i$ on input $x$ with oracle $O$, or $\bot$ if the machine doesn't halt. We say that $T_i$ (...


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