# Some Classical Results in Inductive Inference of Recursive Functions

August 31, 2020

### Abstract

This entry formalizes some classical concepts and results from inductive inference of recursive functions. In the basic setting a partial recursive function ("strategy") must identify ("learn") all functions from a set ("class") of recursive functions. To that end the strategy receives more and more values $f(0), f(1), f(2), \ldots$ of some function $f$ from the given class and in turn outputs descriptions of partial recursive functions, for example, Gödel numbers. The strategy is considered successful if the sequence of outputs ("hypotheses") converges to a description of $f$. A class of functions learnable in this sense is called "learnable in the limit". The set of all these classes is denoted by LIM.

Other types of inference considered are finite learning (FIN), behaviorally correct learning in the limit (BC), and some variants of LIM with restrictions on the hypotheses: total learning (TOTAL), consistent learning (CONS), and class-preserving learning (CP). The main results formalized are the proper inclusions $\mathrm{FIN} \subset \mathrm{CP} \subset \mathrm{TOTAL} \subset \mathrm{CONS} \subset \mathrm{LIM} \subset \mathrm{BC} \subset 2^{\mathcal{R}}$, where $\mathcal{R}$ is the set of all total recursive functions. Further results show that for all these inference types except CONS, strategies can be assumed to be total recursive functions; that all inference types but CP are closed under the subset relation between classes; and that no inference type is closed under the union of classes.

The above is based on a formalization of recursive functions heavily inspired by the Universal Turing Machine entry by Xu et al., but different in that it models partial functions with codomain nat option. The formalization contains a construction of a universal partial recursive function, without resorting to Turing machines, introduces decidability and recursive enumerability, and proves some standard results: existence of a Kleene normal form, the s-m-n theorem, Rice's theorem, and assorted fixed-point theorems (recursion theorems) by Kleene, Rogers, and Smullyan.