AN INFINITE CLASS OF FUNCTIONS IDENTIFIABLE USING MINIMAL PROGRAMS IN ALL KOLMOGOROV NUMBERINGS

Abstract

Identification of programs for computable functions from their graphs by algorithmic devices is a well studied problem in learning theory. Freivalds and Chen consider identification of ‘minimal’ and ‘nearly minimal’ programs for functions from their graphs. Freivalds showed that there exists a Gödel numbering in which only finite classes of functions can be identified using minimal programs. To address such problems, Freivalds later considered minimal identification in Kolmogorov Numberings. Kolmogorov numberings are in some sense optimal numberings and have some nice properties. Freivalds showed that for every Kolmogorov numbering there exists an infinite class of functions which can be identified using minimal programs. Note that these infinite classes of functions may depend on the Kolmogorov numbering. It was left open as to whether there exists an infinite class of functions, C, such that C can be identified using minimal programs in every Kolmogorov numbering. We show the existence of such a class.