Inductive inference: A recursion theoretic approach

There are several situations that we are trying more or less to model. One arises from the standard IQ test in which a person is given a finite sequence of integers and asked to produce the next integer in the sequence. Another is provided by the following grossly simplified view of one aspect of physics: Consider a physicist who is trying to find a law to explain a growing body of experimental data. This data is presented as a set of pairs (x,y). Here, x is a description of a particular experiment, e.g., a high energy physics experiment; y is a description of the results obtained, e.g., the particles produced and their respective properties. The law describing these phenomena is essentially an algorithm for computing the function f(x) = y. What an inductive inference machine does is to request and obtain data in the form of pairs (x,y) and then use this to look for an algorithm i that computes (an extension of) f. Other examples arise in grammatical inference, pattern recognition, etc. This paper develops inductive inference along the lines of Solomonoff [9], Gold [4], and Feldman [3]. What is new and distinguishes our work from theirs is our attempt to characterize the set of functions that can be identified by an inductive inference machine. In the process, we have discovered (Theorem 4, part 1) that inductive inference machines can be considerably more powerful than we previously thought possible.