Anomalous Vacillatory Learning

In 1986, Osherson, Stob and Weinstein asked whether two variants of anomalous vacillatory learning, TxtFex^_ and TxtFext^_, could be distinguished. In both, a machine is permitted to vacillate between a finite number of hypotheses and to make a finite number of errors. TxtFext^_-learning requires that hypotheses output infinitely often must describe the same finite variant of the correct set, while TxtFex^_-learning permits the learner to vacillate between finitely many different finite variants of the correct set. In this paper we show that TxtFex^_ \neq TxtFext^_, thereby answering the question posed by Osherson, \textit{et al}. We prove this in a strong way by exhibiting a family in TxtFex^_2 \setminus {TxtFext}^_*.