I would like to understand the rationale behind the oracle machine notation with brackets $\{e\}^A$ which is equivalent to $\phi_e^A$ where $A$ denotes the oracle set and $e$ denotes the index of the partial $A$-computable function.
Does the notation indicate the generalization of the oracle computation? In particular, if $I$ is a set of indices does $I^A$ extend the notation in any originally indented way?
Robert Soare replied to me: "No. I just use the two notations ($\{e\}^A, \phi^A_e$) in your first sentence. I do not use the $I^A$ form you mentioned and have no information on it."
I asked my supervisor Barry Cooper and he could not think of any specific reason for using that notation, however it seems that Sacks was the first to use it in 60'.
So it seems that just as existential quantifiers were denoted $E$ in older papers, perhaps for similar reasons the bracket notation was used and maybe it is just a notation with no obvious justification.