Advertisement

Generalized Gandy-Păun-Rozenberg Machines for Tile Systems and Cellular Automata

  • Adam Obtułowicz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7184)

Abstract

A concept of a generalized Gandy-Păun-Rozenberg machine for modelling various systems of multidimensional tile-like compartments with common parts (tile faces) of compartment boundaries by graph rewriting is introduced, where some massive parallelism of computations or evolution processes generated by these systems is respected. The representation of Gandy–Păun–Rozenberg machines by Gandy machines in [16] is extended to the case of generalized Gandy–Păn–Rozenberg machines, where the machines represented by Gandy machines are equivalent to Turing machines.

Keywords

Cellular Automaton Turing Machine Label Graph Membrane Computing Massive Parallelism 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Becker, F.: Pictures worth a thousand tiles, a geometrical programming language for self-assembly. Theoret. Comput. Sci. 410, 1495–1515 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brun, Y.: Solving satisfiability in tile assembly model with constant-size tilset. J. Algorithms 63, 151–166 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bruni, R., Meseguer, J., Montanari, U.: Symmetric monoidal and cartesian double categories as a semantic framework for tile logic. Math. Structures Comput. Sci. 12, 53–90 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chemero, A., Turvey, M.T.: Autonomy and hypersets. Biosystems 91, 320–330 (2008)CrossRefGoogle Scholar
  5. 5.
    Cherubini, A., Reghizzi, S.C., Pardella, M., San Pietro, P.: Picture languages: Tiling systems versus tile rewriting grammars. Theoret. Comput. Sci. 356, 90–103 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gandy, R.: Church’s thesis and principles for mechanisms. In: Barwise, J., et al. (eds.) The Kleene Symposium, pp. 123–148. North-Holland, Amsterdam (1980)CrossRefGoogle Scholar
  7. 7.
    Gardner, M.: Mathematical games. Scientific American 223, 120–123 (1970)CrossRefGoogle Scholar
  8. 8.
    Giavitto, J.-L., Michel, O.: The topological structures of membrane computing. Fund. Inform. 49, 123–145 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Giavitto, J.-L., Spicher, A.: Topological rewriting and the geometrization of programming. Physica D 237, 1302–1314 (2008)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Haeseler, F.V., Peitgen, H.-O., Skordev, G.: Cellular automata, matrix substitutions and fractals. Annals of Mathematics and Artificial Intelligence 8, 345–362 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hartmann, P.: Parallel replacement systems on geometric hypergraphs: a mathematical tool for handling dynamic geometric sceneries. In: Parcella 1994, pp. 81–90. Akademie Verlag, Potsdam (1994)Google Scholar
  12. 12.
    Klavins, E.: Directed self-assembly using graph grammars. In: Foundations of Nanoscience: Self Assembled Architectures and Devices, Snowbird UT (2004)Google Scholar
  13. 13.
    Lanthrop, J.I., Lutz, J.H., Summers, S.M.: Strict self-assembly of discrete Sierpiński Triangles, arXiv: 0903.1818v1 [cs.DM] (March 10, 2009)Google Scholar
  14. 14.
    Lienhardt, P.: Topological models for boundary representation: A comparison with n-dimensional generalized maps. Comput. Aided Geom. Design 23, 59–82 (1991)zbMATHGoogle Scholar
  15. 15.
    Margenstern, M.: An Algorithmic Approach to Tilings of Hyperbolic Spaces: 10 Years Later. In: Gheorghe, M., Hinze, T., Păun, G., Rozenberg, G., Salomaa, A. (eds.) CMC 2010. LNCS, vol. 6501, pp. 37–52. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  16. 16.
    Obtułowicz, A.: Gandy–Păun–Rozenberg machines. Romanian J. of Information Science and Technology 13, 181–196 (2010)zbMATHGoogle Scholar
  17. 17.
    Obtułowicz, A.: Randomized Gandy-Păun-Rozenberg Machines. In: Gheorghe, M., Hinze, T., Păun, G., Rozenberg, G., Salomaa, A. (eds.) CMC 2010. LNCS, vol. 6501, pp. 305–324. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  18. 18.
    Păun, G., Rozenberg, G., Salomaa, A.: The Oxford Handbook of Membrane Computing, Oxford (2009)Google Scholar
  19. 19.
    Rosen, R.: Life itself. Columbia Univ. Press, New York (1991)Google Scholar
  20. 20.
    Rothemund, P.W.K., Papadakis, N., Winfree, E.: Algorithmic self-assembly of DNA Sierpiński triangles. PLoS Biology 2, 2041–2053 (2004)CrossRefGoogle Scholar
  21. 21.
    Sieg, W., Byrnes, J.: An abstract model for parallel computations: Gandy’s Thesis. The Monist 82(1), 150–164 (1999)CrossRefGoogle Scholar
  22. 22.
    Stewart, J., Mossio, M.: Is “Life” computable? On the simulation of closure under efficient causation. In: Proceedings of 4th Int. Conf. on Enactive Interfaces, Grenoble, France, November 19-22 (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Adam Obtułowicz
    • 1
  1. 1.Institute of MathematicsPolish Academy of SciencesWarsawPoland

Personalised recommendations