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Languages Associated with Crystallographic Symmetry

  • Nataša Jonoska
  • Mile Krajcevski
  • Gregory McColm
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8553)

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 \(\mathcal{DCL}_d\).

Keywords

Abelian Group Regular Language Terminal Vertex Quotient Graph Counter Machine 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Autebert, J.-M., Berstel, J., Boasson, L.: Context-free languages and pushdown automata. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages. Word, Language, Grammar, vol. 1, pp. 111–174. Springer (1997)Google Scholar
  2. 2.
    Beukemann, A., Klee, W.E.: Minimal nets. Z. tür Kristallographie 201(1-2), 37–51 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Chiniforooshan, E., Daley, M., Ibarra, O.H., Kari, L., Seki, S.: One-reversal counter machines and multihead automata: Revisited. Theoretical Computer Science 454, 81–87 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Chung, S.J., Hahn, T., Klee, W.E.: Nomenclature and Generation of Three-Periodic Nets: the Vector Method. Acta Crys. A40, 42–50 (1984)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Cohen, E., Megiddo, N.: Recognizing properties of periodic graphs. J. Applied Geometry and Discrete Mathematics 4, 135–146 (1991)MathSciNetGoogle Scholar
  6. 6.
    Delgado-Friedrichs, O.: Equilibrium placement of periodic graphs and convexity of plane tilings. Discrete Comput. Geom. 33, 67–81 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Delgado-Friedrichs, O., O’Keeffe, M., Yaghi, O.M.: Taxonomy of periodic nets and the design of materials. Phys. Chem. Chem. Phys. 9, 1035–1043 (2007)CrossRefGoogle Scholar
  8. 8.
    Eon, J.-G.: Graph-theoretical characterization of periodicity in crystallographic nets and other infinite graphs. Acta Crys. A 61, 501–511 (2005)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Glusker, J.P.: Brief history of chemical crystallography. ii: Organic compounds. In: Lima-De-Faria, J. (ed.) Historical Atlas of Crystallography, pp. 91–107. Kluwer (1990)Google Scholar
  10. 10.
    Hopcroft, J.E., Ullman, J.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley (1979)Google Scholar
  11. 11.
    Ibarra, O.: Reversal-bounded multicounter machines and their decision problems. J. Assoc. Comput. Mach. 25, 116–133 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Ibarra, O.H., Yen, H.-C.: On two-way transducers. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 300–311. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  13. 13.
    Jonoska, N., Krajcevski, M., McColm, G.: Counter machines and crystalographic structures (in preparation)Google Scholar
  14. 14.
    Jonoska, N., McColm, G.L.: Flexible versus rigid tile assembly. In: Calude, C.S., Dinneen, M.J., Păun, G., Rozenberg, G., Stepney, S. (eds.) UC 2006. LNCS, vol. 4135, pp. 139–151. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Jonoska, N., McColm, G.: Complexity classes for self-assembling flexible tiles. Theoretical Computer Science 410(4-5), 332–346 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Kintala, C.M.: Refining nondeterminism in context-free languages. Mathematical Systems Theory 12(1), 1–8 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Klee, W.E.: Crystallographic nets and their quotient graphs. Cryst. Res. Technol. 39(11), 959–968 (2004)CrossRefGoogle Scholar
  18. 18.
    Liu, L.Y., Weiner, P.: An infinite hierarchy of intersections of context-free languages. Mathematical Systems Theory 7(2), 185–192 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    McColm, G., Clark, W.E., Eddaoudi, M., Wojtas, L., Zaworotko, M.: Crystal engineering using a “turtlebug” algorithm: A de novo approach to the design of binodal metal organic frameworks. Crystal Growth & Design 19(9), 3686–3693 (2011)CrossRefGoogle Scholar
  20. 20.
    Meier, J.: Groups, Graphs and Trees: An Introduction to the Geometry of Infinite Groups. Cambridge U. Pr. (2008)Google Scholar
  21. 21.
    Moore, P.B.: Brief history of chemical crystallography. i: Inorganic compounds. In: Lima-De-Faria, J. (ed.) Historical Atlas of Crystallography, pp. 77–90. Kluwer (1990)Google Scholar
  22. 22.
    O’Keeffe, M., Hyde, B.G.: Crystal Structures I. Patterns and Symmetry. Mineralogical Society of America (1996)Google Scholar
  23. 23.
    Seki, S.: N-dimensional crystallography. Private Communication (2013)Google Scholar
  24. 24.
    Wang, C., Liu, D., Lin, W.: Metal-organic frameworks as a tunable platform for designing functional molecular materials. J. American Chemical Society 135(36), 13222–13234 (2013)CrossRefGoogle Scholar
  25. 25.
    Wells, A.F.: Three-dimensional Nets and Polyhedra. Wiley (1977)Google Scholar
  26. 26.
    Zheng, J., Birktoft, J., Chen, Y., Wang, T., Sha, R., Constantinou, P., Ginell, S., Mao, C., Seeman, N.: From molecular to macroscopic via the rational design of a self-assembled 3D DNA crystal. Nature 461(7260), 74–77 (2009)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Nataša Jonoska
    • 1
  • Mile Krajcevski
    • 1
  • Gregory McColm
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA

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