Strongly Non-U-Shaped Learning Results by General Techniques

In learning, a semantic or behavioral U-shape occurs when a learner rst learns, then unlearns, and, nally, relearns, some target concept (on the way to success). Within the framework of Inductive Inference, previous results have shown, for example, that such Ushapes are unnecessary for explanatory learning, but are necessary for behaviorally correct and non-trivial vacillatory learning. Herein we focus more on syntactic U-shapes. This paper introduces two general techniques and applies them especially to syntactic Ushapes in learning: one technique to show when they are necessary and one to show when they are unnecessary. The technique for the former is very general and applicable to a much wider range of learning criteria. It employs so-called self-learning classes of languages which are shown to characterize completely one criterion learning more than another. We apply these techniques to show that, for set-driven and partially set-driven learning, any kind of U-shapes are unnecessary. Furthermore, we show that U-shapes are not unnecessary in a strong way for iterative learning, contrasting an earlier result by Case and Moelius that semantic U-shapes are unnecessary for iterative learning.