Co-learnability and FIN-identifiability of enumerable classes of total recursive functions

Abstract

Co-learnability is an inference process where instead of producing the final result, the strategy produces all the natural numbers but one, and the omitted number is an encoding of the correct result. It has been proved in [1] that co-learnability of Goedel numbers is equivalent to EX-identifiability. We consider co-learnability of indices in recursively enumerable (r.e.) numberings. The power of co-learnability depends on the numberings used. Every r.e. class of total recursive functions is co-learnable in some r.e. numbering. FIN-identifiable classes are co-learnable in all r.e. numberings, and classes containing a function being accumulation point are not co-learnable in some r.e. numberings. Hence it was conjectured in [1] that only FIN-identifiable classes are co-learnable in all r.e. numberings. The conjecture is disproved in this paper using a sophisticated construction by V.L. Selivanov.

The research of the two first authors was supported by the grant No. 93.599 from Latvian Science Council. The fourth author is supported, in part, by National Science Foundation Grant 9301339

Research supported in part by the DFG Grant KA 673/4-1, by the ESPRIT BR Grants 7097 and ECUS030