Learning classes of approximations to non-recursive functions

Abstract

Blum and Blum (Inform. and Control 28 (1975) 125–155) showed that a class B of suitable recursive approximations to the halting problem K is reliably EX-learnable but left it open whether or not B is in NUM. By showing B to be not in NUM we resolve this old problem. Moreover, variants of this problem obtained by approximating any given recursively enumerable set A instead of the halting problem K are studied. All corresponding function classes U(A) are still EX-inferable but may fail to be reliably EX-learnable, for example if A is non-high and hypersimple. Blum and Blum (1975) considered only approximations to K defined by monotone complexity functions. We prove this condition to be necessary for making learnability independent of the underlying complexity measure. The class B ̃ of all recursive approximations to K generated by all total complexity functions is shown to be not even behaviorally correct learnable for a class of natural complexity measures. On the other hand, there are complexity measures such that B ̃ is EX-learnable. A similar result is obtained for all classes U ̃ (A) . For natural complexity measures, B is shown to be not robustly learnable, but again there are complexity measures such that B and, more generally, every class U(A) is robustly EX-learnable. This result extends the criticism of Jain et al. (J. Comput. System Sci. 62(1) (2001) 178–212), since the classes defined by artificial complexity measures turn out to be robustly learnable while those defined by natural complexity measures are not robustly learnable.