# Regularity of languages generated by non context-free grammars over a singleton terminal alphabet

## TL;DR

This paper proves that any language over a singleton alphabet satisfying the Pumping Lemma is regular, extending known results about the regularity of certain non-context-free languages.

## Contribution

It demonstrates that all languages in the Pumping Lemma superclass over a singleton alphabet are regular, without using Parikh's Theorem, broadening previous understanding.

## Key findings

- Languages satisfying the Pumping Lemma over a singleton alphabet are regular
- The proof is based on a transformational approach, not Parikh's Theorem
- Extends known results to languages not necessarily context-free or satisfying Parikh's Theorem

## Abstract

It is well-known that: (i) every context-free language over a singleton terminal alphabet is regular, and (ii) the class of languages that satisfy the Pumping Lemma is a proper super-class of the context-free languages. We show that any language in this superclass over a singleton terminal alphabet is regular. Our proof is based on a transformational approach and does not rely on Parikh's Theorem. Our result extends previously known results because there are languages that are not context-free, do satisfy the Pumping Lemma, and do not satisfy the hypotheses of Parikh's Theorem.