Relative to P promise - BPP

We show that for determinictic polynomial time computation, oracle access to APP, the class of real functions approximable by probabilistic Turing machines, is the same as having oracle access to promise-BPP. First we construct a mapping that maps every function in APP to a promise problem in prBPP, and that maps complete functions to complete promise problems. Then we show an analogue result in the opposite direction, by constructing a mapping from prBPP into APP, that maps every promise problem to a function in APP, and mapping complete promise problems to complete functions. Second we prove that P APP = P prBPP. Finally we use our results to simplify proofs of important results on APP, such as the APP-completeness of the function f CAPP that approximates the acceptance probability of a Boolean circuit, or the possibility (similarily to the case of BPP) to reduce the error probability for APP functions, or the conditionnal derandomization result APP = AP ii prBPP is easy.