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

Abstract

It is well-known that: (i) every context-free language over a singleton terminal alphabet is regular [4], and (ii) the class of languages that satisfy the Pumping Lemma (for context-free languages) is a proper super-class of the context-free languages. We show that any language in this super-class over a singleton terminal alphabet is regular. Our proof is based on an elementary transformational approach and does not rely on Parikh's Theorem [6]. 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 [7].