Languages associated with crystallographic symmetry

Abstract

We establish a relationship between periodic graphs representing crystallographic structures and an infinite hierarchy of intersection languages, DCLd 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 d. Each language in DCLd is an intersection of d languages in DCL1. 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 DCLd. © 2014 Springer International Publishing Switzerland.