Self-Specifying Machines

We study the computational power of machines that specify their own acceptance types, and show that they accept exactly the languages that ≤ m -reduce to NP sets. A natural variant accepts exactly the languages that ≤ m -reduce to P sets. We show that these two classes coincide if and only if P = P, where the latter class denotes the sets acceptable via at most one question to #P followed by at most a constant number of questions to NP.