Suppose I have an oracle X. Then let Y be an oracle which will answer any computable question about X. In other words, Y takes as input a Turing program which can in turn make calls to X. Y then returns the output of that program (assuming the output is one bit, otherwise, or if the program doesn't halt, then the definition is open to interpretation).
Has this concept been studied? Is there a name for such oracles?
This relates to communication complexity. In that case X would be Alice's input, X' would be Bob's input, and Y and Y' would be sufficient to efficiently compute their strategies. The Y and Y' oracles are a way to express the fact that in communication complexity one usually doesn't care how much time the parties spend on computation.