On Jaffe’s Pumping Lemma, Revisited

Abstract

We consider Jaffe’s pumping lemma [J. Jaffe. A necessary and sufficient pumping lemma for regular languages. SIGACT News, Summer, 1978] from a descriptional complexity perspective. Jaffe’s pumping lemma is a necessary and sufficient condition for a language for being regular. In this way we improve a result of [A. Yehudai. A note on the pumping lemma for regular languages. Inform. Proc. Lett., 9(3):135–136, 1979] by showing that there is a regular language over the alphabet $$\varSigma $$ of size at least two with deterministic state complexity between p, the minimal pumping constant for Jaffe’s pumping lemma, and $$\sum _{i=0}^{p-1}|\varSigma |^i$$ . This is in line with recent research on minimal pumping constants for various pumping lemma conducted in [J. Dassow and I. Jecker. Operational complexity and pumping lemmas. Acta Inform., 59:337–355, 2022]. Moreover, we also compare the minimal pumping constant of Jaffe’s pumping lemma with those of other well-known pumping lemmata from the literature.