Church's thesis for oracle computations Church's thesis states that any reasonable definition of computability (over subsets or functions of $\mathbb{N}$) coincide. My question is, is there an analogous result/thesis for oracle computations? For instance, suppose I have an oracle Turing machine for which I can give an oracle and an input. Does this model of computation coincide with other reasonable definitions of oracle computations?
I would also view the oracle as a type of input, hence the physical Turing machine. Is there a way of viewing this type of computation (where the oracle can change) outside the Turing machine framework?
 A: Yes- there is an analogue of Church’s thesis for oracle machines just like Turing machines. One way to see this is by a trick called “relativizing,” - since an oracle machine is exactly like a Turing machine with the empty set as oracle, any standard theorem in computability which holds about the empty set should morally hold for the oracle, in the oracle machine setting.
Take for example the equivalence of the Turing machine-computable sets with the quantifier-free definable sets. By relativizing you can see that there is an equivalence between the oracle machine-computable sets and the quantifier-free definable sets in the expanded language with an added predicate representing the oracle.
Modern computability theorists make use of Church’s thesis for oracle computation all the time.
As for your second question, the answer is also yes. The concept you are referring to is represented by a mathematical object called a “Turing Functional” and they are well studied. There are even equivalent formulations for Turing functionals as continuous maps between certain topological spaces, and as a machine model which simply takes oracle inputs.
