Counter machines and crystallographic structures

Abstract

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 $\Gamma /G \cong \Delta$.