On a question about learning nearly minimal programs

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. The present paper solves the following question left open by Chen: Is it the case that for any collection of computable functions, C, such that some machine can finitely learn a nearly minimal (n + 1)-error program for every function in C, there exists another machine that can learn in the limit an n-error program (which need not be nearly minimal) for every function in C? We answer this question negatively