# The teaching complexity of erasing pattern languages with bounded variable frequency

## TL;DR

This paper investigates how bounding variable frequency in pattern languages affects the complexity of learning and teaching these patterns, focusing on the minimum number of examples needed for unique identification in different models.

## Contribution

It introduces the study of variable frequency bounds in pattern languages and analyzes their impact on teaching complexity in cooperative learning models.

## Key findings

- Bounding variable frequency influences the teaching dimension of pattern classes.
- The paper provides bounds on the number of examples needed for pattern identification.
- It compares teaching complexity across different models with variable frequency restrictions.

## Abstract

Patterns provide a concise, syntactic way of describing a set of strings, but their expressive power comes at a price: a number of fundamental decision problems concerning (erasing) pattern languages, such as the membership problem and inclusion problem, are known to be NP-complete or even undecidable, while the decidability of the equivalence problem is still open; in learning theory, the class of pattern languages is unlearnable in models such as the distribution-free (PAC) framework (if $\mathcal{P}/poly \neq \mathcal{NP}/poly$). Much work on the algorithmic learning of pattern languages has thus focussed on interesting subclasses of patterns for which positive learnability results may be achieved. A natural restriction on a pattern is a bound on its variable frequency -- the maximum number $m$ such that some variable occurs exactly $m$ times in the pattern. This paper examines the effect of limiting the variable frequency of all patterns belonging to a class $\Pi$ on the worst-case minimum number of labelled examples needed to uniquely identify any pattern of $\Pi$ in cooperative teaching-learning models. Two such models, the teaching dimension model as well as the preference-based teaching model, will be considered.