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.
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Notes
- 1.
A language \(L\subseteq \varSigma ^*\) is suffix closed if \(L=\{\,x\mid yx\in L,\,{ forsome}y\in \varSigma ^*\,\}\), i.e., the word x is a member of L whenever yx is in L, for some \(y\in \varSigma ^*\).
- 2.
Let P be a Boolean predicate. Then \(\langle P\rangle :=1\), if P is true; otherwise \(\langle P\rangle :=0\).
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Holzer, M., Rauch, C. (2023). On Jaffe’s Pumping Lemma, Revisited. In: Bordihn, H., Tran, N., Vaszil, G. (eds) Descriptional Complexity of Formal Systems. DCFS 2023. Lecture Notes in Computer Science, vol 13918. Springer, Cham. https://doi.org/10.1007/978-3-031-34326-1_5
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