Solomonoff induction (e.g. next-bit conditional probabilities of the universal distribution) is a function of discrete domain and real codomain. We sometimes want to compose SI with real domain operations (e.g. in order to calculate a value function we take products and sums). But usually the real domain operations live lower in the hierarchy, so in some sense you’re right that studying real domain hypercomputability is overkill. Most of the results in the paper hold for both discrete and real domain, sometimes discrete domain is nicer.
Okay. I’m just trying to get a sense of what kind of questions all of this is meant to answer. It sounds like we start with “SI is uncomputable, but is computable relative to some oracle”. Now we want to say “this machine I have built out of SI is still computable relative to that same oracle”. That kind of thing? And you’re saying here that while the machine built out of SI is not a function from reals to reals, we can prove these statements by classifying functions from reals to reals involved in its construction.
Solomonoff induction (e.g. next-bit conditional probabilities of the universal distribution) is a function of discrete domain and real codomain. We sometimes want to compose SI with real domain operations (e.g. in order to calculate a value function we take products and sums). But usually the real domain operations live lower in the hierarchy, so in some sense you’re right that studying real domain hypercomputability is overkill. Most of the results in the paper hold for both discrete and real domain, sometimes discrete domain is nicer.
Okay. I’m just trying to get a sense of what kind of questions all of this is meant to answer. It sounds like we start with “SI is uncomputable, but is computable relative to some oracle”. Now we want to say “this machine I have built out of SI is still computable relative to that same oracle”. That kind of thing? And you’re saying here that while the machine built out of SI is not a function from reals to reals, we can prove these statements by classifying functions from reals to reals involved in its construction.