Inductive inference of recursive functions: Qualitative theory

Abstract

This survey contains both old and very recent results in non-quantitative aspects of inductive inference of total recursive functions. The survey is not complete. The paper was written to stress some of the main results in selected directions of research performed at the University of Latvia rather than to exhaust all of the obtained results. We concentrated on the more explored areas such as the inference of indices in non-Goedel computable numberings, the inference of minimal Goedel numbers, and the specifics of inference of minimal indices in Kolmogorov numberings.

Goedel numberings have many specific properties which influence the inference process very much. On the other hand, when discussing the desirability inductive inference we usually do not mention these properties. Hence the motivation is valid for inference of indices in non-Goedel computable numberings as well. Section 2 contains several results showing that the inference of indices in computable numberings can differ very much. For instance, there are computable numberings which are difficult for the inference, and only finite classes of total recursive functions can be identified. This shows that computable numberings can be very much removed from Goedel numberings.

We get rather similar results and even very similar methods of proofs when we consider the identification of minimal indices in Goedel numberings. It is difficult to express this similarity explicitly but many proofs can be expressed in parallel. Criteria for the identifiability of the minimal numbers, lattice — theoretical properties of the partial ordering of Goedel numberings with respect to identifiability of the minimal numbers, and identifiable classes with extremal characteristics are considered. Kolmogorov numberings which have special status in defining Kolmogorov complexity turn out to have special properties in inference of minimal numbers as well.

Section 9 presents results in abstract theory of identification types. It turns out that there are "typical" or "complete" classes of functions in many identification types

such that to deside whether or not

it suffices to check whether or not the "

-complete" class

is in

. Unfortunately, the proposed technique for constructing these classes does not work for those types with "identification of minimal indices in specific Goedel numberings".