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Counter machines and crystallographic structures.

N Jonoska1, M Krajcevski1, G McColm1

  • 1Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA.

Natural Computing
|September 13, 2016
PubMed
Summary
This summary is machine-generated.

This study links periodic graphs of crystallographic structures to deterministic context-free languages (DCFLs). A new class of counter machines recognizes these languages, establishing a correspondence with graph walks.

Keywords:
Context-free languagesCounter machinesCrystallographic structuresPeriodic digraphs

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Area of Science:

  • Graph theory
  • Formal languages
  • Computational complexity

Background:

  • Crystallographic structures can be represented by periodic graphs.
  • Automorphism groups with translational subgroups are key to these representations.
  • Deterministic context-free languages (DCFLs) are a class of formal languages.

Purpose of the Study:

  • To establish a relationship between periodic graphs and a hierarchy of intersection languages.
  • To introduce and analyze a class of counter machines for recognizing these languages.
  • To demonstrate a correspondence between graph walks and formal language classes.

Main Methods:

  • Defining an infinite hierarchy of intersection languages (𝒟𝒞ℒd) within DCFLs.
  • Introducing d-counter machines to recognize 𝒟𝒞ℒd.
  • Proving a one-to-one correspondence between walks in d-dimensional periodic graphs and 𝒟𝒞ℒd.

Main Results:

  • The hierarchy of languages 𝒟𝒞ℒd is defined, where 𝒟𝒞ℒd is the intersection of d languages from 𝒟𝒞ℒ1.
  • A class of d-counter machines recognizes 𝒟𝒞ℒd.
  • A bijection is proven between sets of walks in d-dimensional periodic graphs and the language class 𝒟𝒞ℒd.

Conclusions:

  • Periodic graphs representing crystallographic structures are formally linked to intersection classes of deterministic context-free languages.
  • The introduced d-counter machines provide a computational model for these language classes.
  • This work establishes a fundamental connection between graph theory, formal languages, and computational models in the context of crystallography.