On Minimal Pumping Constants for Regular Languages

Abstract: The study of the operational complexity of minimal pumping constants started in [J. DASSOW and I. JECKER. Operational complexity and pumping lemmas. Acta Inform., 59:337-355, 2022], where an almost complete picture of the operational complexity of minimal pumping constants for two different variants of pumping lemmata from the literature was given. We continue this research by considering a pumping lemma for regular languages that allows pumping of sub-words at any position of the considered word, if the sub-word is long enough [S. J. SAVITCH. Abstract Machines and Grammars. 1982]. First we improve on the simultaneous regulation of minimal pumping constants induced by different pumping lemmata including Savitch's pumping lemma. In this way we are able to simultaneously regulate four different minimal pumping constants. This is a novel result in the field of descriptional complexity. Moreover, for Savitch's pumping lemma we are able to completely classify the range of the minimal pumping constant for the operations Kleene star, reversal, complement, prefix- and suffix-closure, union, set-subtraction, concatenation, intersection, and symmetric difference. In this way, we also solve some of the open problems from the paper that initiated the study of the operational complexity of minimal pumping constants mentioned above.