一元图片语言中的 Guillotine 与平铺共有限性
Guillotine and Tiling Cofiniteness in Unary Picture Languages
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中文总结 AI 辅助
该研究针对一元图片语言，定义渐近共有限性并给出其算术特征，解决了二维共有限性限制过强的问题，为Guillotine闭包与全平铺闭包提供了统一判定准则。
中文摘要 AI 辅助
受图片语言上闭包运算研究的启发，我们提出问题：一组一元瓦片何时能通过重复水平与垂直连接，或通过平铺运算生成所有足够大的图片。在一维中，一元语言连接闭包内单词的长度构成正自然数的加法子半群，因此是有限生成的；共有限性的特征是这些生成元的最大公约数为1。由于二维普通共有限性限制过强，本质上简化为条带上的退化一维条件，我们引入并研究渐近共有限性：所有足够大的矩形图片均可被生成。直接扩展一维准则（要求瓦片高度的最大公约数与瓦片宽度的最大公约数均为1）并不充分，因为二维中可能持续存在局部同余障碍。我们对任意（可能无限的）一元矩形瓦片集，给出渐近共有限性的精确算术特征；该特征对通过水平与垂直连接得到的Guillotine闭包及全平铺闭包均相同。对于有限瓦片集，证明结合半群论证与用于充分性的显式最小公倍数堆叠构造，必要性则通过适用于不可切片平铺的单位根论证得到；向无限瓦片集的扩展遵循Klarner系统的有限基定理。
英文摘要
Motivated by the study of closure operations on picture languages, we ask when a set of unary tiles can generate all sufficiently large pictures either via repeated horizontal and vertical concatenations or via tiling operations. In one dimension, the lengths of words in the concatenation closure of a unary language form an additive subsemigroup of positive natural numbers; hence, they are finitely generated. Cofiniteness is characterized by the gcd of these generators being one. Since ordinary cofiniteness in two dimensions is too restrictive and essentially reduces to degenerate one-dimensional conditions on strips, we introduce and study asymptotic cofiniteness: all sufficiently large rectangular pictures are generated. A direct extension of the one-dimensional criterion, requiring the gcd of the tile heights and of the tile widths to be one, is not sufficient, because local congruence obstructions may persist in two dimensions. We give an exact arithmetic characterization of asymptotic cofiniteness for arbitrary, possibly infinite, sets of unary rectangular tiles. The characterization is the same for the guillotine closure, obtained by horizontal and vertical concatenation, and for the full tiling closure. For finite tile sets, the proof combines semigroup arguments and an explicit least-common-multiple stacking construction for sufficiency, while necessity is obtained by a roots-of-unity argument applying also to nonsliceable tilings. The extension to infinite tile sets follows from the finite-basis theorem for Klarner systems.
