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

TL;DR

This paper investigates witness-based learners in inductive inference, establishing normal forms and equivalences among different paradigms, thereby deepening understanding of their computational power and relationships.

Contribution

It provides the first comprehensive normal forms for witness-based learners under syntactic and semantic convergence, and shows equivalences among various learning paradigms.

Findings

Witness-based learners are as powerful as non-witness-based in certain settings.

Normal forms for syntactic and semantic convergence are established.

Set-driven and Gold-style learners are equally powerful in semantic learning.

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.