Anomaly hierarchies of mechanized inductive inference

We are interes£ed in the theoretical limitations of agents M which attempt to arrive at explanations for classes of phenomena in the case where M is a machine or robot. If we take a mechanistic philosophical stance, our results can also be construed as theorems in philosophy of science. To these ends we define an ind~cJt~u£ inference ma~ne ~ssentially introduced in [6]) to be an algorithmic device with no a p~o~ bounds on how much time and memory resource it shall use, which takes as its input the graph of a function: N +N an ordered pair at a time, and which, from time to time, as it's receiving its inputs, outputs computer programs. An inductive inference machine M ide~fi66 a function f ~ M fed f (in any order) outputs over time but finitely many computer programs the last of which computes (or explains) f. No restriction is made that we should be able to algorithmically determine when (if ever) M on f has output its last computer program. We say that M ide~fi~6 a ~s S of functions (or phenomena) .~ > M identifies each f in S. For example, the following proposition generalizes a remark in [2]. The notation is from [ii]: ~e is the partial function computed by program e and K in some set representing the halting problem.