Abstract
We establish a relationship between periodic graphs representing crystallographic structures and an infinite hierarchy of intersection languages \(\mathcal{DCL}_d\), 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 \(\mathcal{DCL}_d\). Each language in \(\mathcal{DCL}_d\) is an intersection of d languages in \(\mathcal{DCL}_1\). 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 \(\mathcal{DCL}_d\).
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