While it doesn't exactly answer your question, I recommend Kihara's paper “Rethinking the notion of oracle”, which considers all sorts of kinds of reductions (oracles you can only use once vs, oracles you can use repeatedly; functions vs. partial functions vs. multifunctions vs. “extended predicates”; on $\mathbb{N}$ vs. on $\mathbb{N}^{\mathbb{N}}$; and so on), and give different points of view on each. I think it's very enlightening.
↧