Anomaly hierarchies of mechanized inductive inference

Suppose a real world phenomenon F is being investigated scientifically by an agent M. M performs ~c/t~te experiments x on F and receives back corresponding experimental results f(x). By suitable Godel numbering we may treat the f associated with F as a function from N = ~0,i,2 .... }, the set of natural numbers, i~0 N. A ~omplet~ exp~on for F is just a computer program for computing f. Such a program for f gives us predictive power about the results of all possible experiments on F. In what follows we will, in effect, identify F with f. 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. Proposition i. Suppose S is a class of recursive functions r.e. in the halting problem, i.e., suppose S is a class of recursive functions such that (Sg recursive in K) [ S = {~g(i) li EN}]. Then (SM) [ M identifies S]. proof. Suppose the hypothesis. Then there is a recursive function f such that ~x) [ g(x) = lim f(x,y)]. Apply a modification of the enumera-tion technique [2] to the class {~f(x,y) IX,y £ N}. We shall refer …