Normal Forms for (Semantically) Witness-Based Learners in Inductive Inference

Abstract: We study learners (computable devices) inferring formal languages, a setting referred to as language learning in the limit or inductive inference. In particular, we require the learners we investigate to be witness-based, that is, to justify each of their mind changes. Besides being a natural requirement for a learning task, this restriction deserves special attention as it is a specialization of various important learning paradigms. In particular, with the help of witness-based learning, explanatory learners are shown to be equally powerful under these seemingly incomparable paradigms. Nonetheless, until now, witness-based learners have only been studied sparsely. In this work, we conduct a thorough study of these learners both when requiring syntactic and semantic convergence and obtain normal forms thereof. In the former setting, we extend known results such that they include witness-based learning and generalize these to hold for a variety of learners. Transitioning to behaviourally correct learning, we also provide normal forms for semantically witness-based learners. Most notably, we show that set-driven globally semantically witness-based learners are equally powerful as their Gold-style semantically conservative counterpart. Such results are key to understanding the, yet undiscovered, mutual relation between various important learning paradigms when learning behaviourally correctly.