In 1986, Osherson, Stob and Weinstein asked whether two variants of anomalous vacillatory learning, TxtFex^*_* and TxtFext^*_*, could be distinguished. These learning criteria place bounds neither on the number of hypotheses between which the learner is allowed to vacillate nor on the number of errors permitted, merely that both are finite. The criteria differ in that the more restrictive one, TxtFext^*_*-learning, requires that all hypotheses output infinitely often must describe the same finite variant of the correct set, while TxtFex^*_* permits the learner to vacillate between finitely many different finite variants of the correct set. In this paper we show that TxtFex^*_* ≠ TxtFext^*_*, thereby answering the question posed by Osherson, et al. We prove this in a strong way by exhibiting a family in TxtFex^*₂\TxtFext^*_*.
 

We consider the arithmetic complexity of index sets of uniformly computably enumerable families learnable under different learning criteria. We determine the exact complexity for the standard notions of finite learning, learning in the limit, behaviourally correct learning and anomalous learning in the limit. In proving the Σ⁰₅-completeness result for behaviourally correct learning we prove a result of independent interest; if a u.c.e. family is not learnable, for any computable learner there is a Δ⁰₂ enumeration witnessing failure.