Counter machines and crystallographic structures

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 𝒟𝒞ℒ , d = 0, 1, 2, …, 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 𝒟𝒞ℒ An intersection of d languages in 𝒟𝒞ℒ₁ defines 𝒟𝒞ℒ . 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 𝒟𝒞ℒ . The proof uses the following result: given a digraph Δ and a group G, there is a unique digraph Γ such that G ≤ Aut Γ, G acts freely on the structure, and Γ/G ≅ Δ.