Pumping Constants for Infinite Alphabets

It is well known that the pumping lemma for regular languages over finite alphabets does not extend to infinite alphabets. Instead, the generalized version of the pumping lemma for infinite alphabets states that the pumped patterns need not be identical, but only equivalent up to a finite-order permutation of the alphabet. This generalized variant of the pumping lemma involves two parameters: the length of the pumped pattern and an upper bound on the order of the permutation. We improve the known permutation-order bounds from factorial to tight bounds governed by Landau's function. We also show that, if the pumping length is sufficiently large, then permutations of order at most two always suffice. Our main structural result concerns the one-register case: every one-register automaton satisfies the classical pumping lemma, with no alphabet permutation. We prove a quadratic upper bound on the pumping length and give a matching quadratic lower bound up to constant factors. The same classical pumping phenomenon is shown for hierarchical register automata. Finally, we prove that, in general, deciding whether a quasi-regular language is pumpable for given constants is undecidable, while the minimal classical pumping length is computable for non-guessing one-register automata.