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 ๐๐โd, d = 0, 1, 2, โฆ, 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 ๐๐โd An intersection of d languages in ๐๐โ1 defines ๐๐โd. 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 ๐๐โd. The proof uses the following result: given a digraph ฮ and a group G, there is a unique digraph ฮ such that G โค Aut ฮ, G acts freely on the structure, and ฮ/G โ ฮ.
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