On a Syntactic Characterization of Classification with a Mind Change Bound

Most learning paradigms impose a particular syntax on the class of concepts to be learned; the chosen syntax can dramatically affect whether the class is learnable or not. For classification paradigms, where the task is to determine whether the underlying world does or does not have a particular property, how that property is represented has no implication on the power of a classifier that just outputs 1’s or 0’s. But is it possible to give a canonical syntactic representation of the class of concepts that are classifiable according to the particular criteria of a given paradigm? We provide a positive answer to this question for classification in the limit paradigms in a logical setting, with ordinal mind change bounds as a measure of complexity. The syntactic characterization that emerges enables to derive that if a possibly noncomputable classifier can perform the task assigned to it by the paradigm, then a computable classifier can also perform the same task. The syntactic characterization is strongly related to the difference hierarchy over the class of open sets of some topological space; this space is naturally defined from the class of possible worlds and possible data of the learning paradigm.