One way to depict a crystallographic structure is by a periodic (di)graph, i.e., a graph whose group of automorphisms has a translational subgroup of finite index acting freely on the structure. We establish a relationship between periodic graphs representing crystallographic structures and an infinite hierarchy of intersection languages \(\mathcal {DCL}_d,\,d=0,1,2,\ldots \), within the intersection classes of deterministic context-free languages. We introduce a class of counter machines that accept these languages, where the machines with d counters recognize the class \(\mathcal {DCL}_d\). An intersection of d languages in \(\mathcal {DCL}_1\) defines \(\mathcal {DCL}_d\). We prove that there is a one-to-one correspondence between sets of walks starting and ending in the same unit of a d-dimensional periodic (di)graph and the class of languages in \(\mathcal {DCL}_d\). The proof uses the following result: given a digraph \(\Delta \) and a group G, there is a unique digraph \(\Gamma \) such that \(G\le \mathrm{Aut}\,\Gamma ,\,G\) acts freely on the structure, and \(\Gamma /G \cong \Delta \).

中文翻译：

---

计数机和晶体结构。

描绘晶体结构的一种方法是通过周期图（di），即，图的自同构群具有自由作用于该结构的有限折射率的平移子组。我们在确定性上下文无关的交集类中建立表示晶体结构的周期图与交集语言\（\ mathcal {DCL} _d，\，d = 0,1,2，\ ldots \）的无限层次之间的关系语言。我们介绍了一种接受这些语言的计数器机器，其中带有d计数器的机器可以识别类\（\ mathcal {DCL} _d \）。\（\ mathcal {DCL} _1 \）中d语言的交集定义\（\ mathcal {DCL} _d \）。我们证明，在d维周期（di）图的同一单元中开始和结束的行走集合与\（\ mathcal {DCL} _d \）中的语言类别之间存在一对一的对应关系。该证明使用以下结果：给定有向图\（\ Delta \）和G组，存在唯一的有向图\（\ Gamma \）使得\（G \ le \ mathrm {Aut} \，\ Gamma，\ ，G \）在结构上自由作用，并且\（\ Gamma / G \ cong \ Delta \）起作用。