Anomalous Vacillatory Learning
This is an incremental theoretical result in computational learning theory, addressing a specific problem for researchers in that field.
The paper resolves a long-standing open question from 1986 by proving that two variants of anomalous vacillatory learning, TxtFex*_* and TxtFext*_*, are distinct, specifically showing TxtFex*_2 is not contained in TxtFext*_*.
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}^*_*.