On the Teaching Complexity of Linear Sets

Abstract

Linear sets are the building blocks of semilinear sets, which are in turn closely connected to automata theory and formal languages. Prior work has investigated the learnability of linear sets and semilinear sets in three models – Valiant’s PAC-learning model, Gold’s learning in the limit model, and Angluin’s query learning model. This paper considers a teacher-learner model of learning families of linear sets, whereby the learner is assumed to know all the smallest sets $T_1,T_2,\ldots$ of labelled examples that are consistent with exactly one language in the class $\mathcal {L}$ to be learnt, and is always presented with a sample S of labelled examples such that S is contained in at least one of $T_1,T_2,\ldots$; the learner then interprets S according to some fixed protocol. In particular, we will apply a generalisation of a recently introduced model – the recursive teaching model of teaching and learning – to several infinite classes of linear sets, and show that the maximum sample complexity of teaching these classes can be drastically reduced if each of them is taught according to a carefully chosen sequence. A major focus of the paper will be on determining two relevant teaching parameters, the teaching dimension and recursive teaching dimension, for various families of linear sets.