发表机构
Technion（以色列理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究无限字母表上泵引理的常数，改进置换阶数上界至兰道函数紧界，证明单寄存器自动机满足经典泵引理并给出二次紧界，同时证明一般情形下可泵性判定不可判定。
AI 中文摘要
众所周知，有限字母表上的正则语言的泵引理并不能推广到无限字母表。相反，针对无限字母表的泵引理的广义版本指出，被泵出的模式不必完全相同，而只需在字母表的有限阶置换下等价。这种广义的泵引理变体涉及两个参数：被泵出模式（pumped pattern）的长度，以及置换阶数的上界。我们将已知的置换阶数上界从阶乘界改进为受兰道函数（Landau's function）控制的紧界。我们还证明，如果泵长度足够大，那么阶数至多为二的置换总是足够的。我们的主要结构结果涉及单寄存器情形：每个单寄存器自动机（one-register automaton）都满足经典泵引理，无需字母表置换。我们证明了泵长度的二次上界，并给出了在常数因子意义下匹配的二次下界。同样的经典泵现象也被证明适用于分层寄存器自动机（hierarchical register automata）。最后，我们证明，一般而言，判定一个准正则语言（quasi-regular language）在给定常数下是否可泵是不可判定的，而对于非猜测型单寄存器自动机（non-guessing one-register automata），最小经典泵长度是可计算的。
英文摘要
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.
CommentsIn Proceedings AFL 2026, arXiv:2608.23071
Journal refEPTCS 451, 2026, pp. 104-122
