Comparisons of Parikh's condition to other conditions for context-free languages

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

“…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].…”

“…That proof is based on Tarski's fixpoint theorem and it is not based on the Pumping Lemma (contrary to what has been stated in a paper by Andrei et al [1]). Our extension is due to the facts that: (i) our proof does not rely on Parikh's Theorem [6], like the proof in Harrison's book [4], and (ii) there are non context-free languages that do satisfy the Pumping Lemma (see Definition 1) and do not satisfy Parikh's Condition (see Definition 2) (and thus Parikh's Theorem cannot be applied) [7]. Our proof is very much related to one presented in a book by Shallit [8], but we believe that ours is more elementary.…”