Lorentz Lattice Gases and Many-Dimensional Turing Machines

  • Leonid A. Bunimovich
  • Milena A. Khlabystova


We study lattice gas models of parallel, many-tape Turing machines generated by the motion of point objects on a lattice. Each read/write head of the Turing machine is seen as an object that hops from one vertex of the lattice to another according to a rule (symbol) written in the vertex. The symbols written in the lattice vertices represent the scattering rules, or scatterers, of the lattice gas model. Initially, the scatterers are randomly distributed among the vertices of the lattice. The random environment formed by the scatterers may either be fixed or may evolve as a result of collisions with moving objects. The collisions, in fact, simulate the writing of symbols into the lattice vertices. We investigate models of this type with one (many-tape single-head Turing machine) and many (many-tape many-head Turing machine) propagating objects on different types of lattices (different topologies of Turing tapes). We explore the localization and propagation properties of these models. Experiments with lattice gas models of the Turing machine demonstrate that both multiplicity of the Turing heads and non-regularity of the Turing tape may cause a localization of orbits in the corresponding model. The propagation is shown to occur in the one-particle model on the regular triangular lattice, where a moving object always propagates in one direction with random velocity.


Periodic Orbit Cellular Automaton Turing Machine Periodic Point Cellular Automaton 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adamatzky A., Melhuish C. and Holland O. Morphology of patterns in lattice swarm: interval parameterization Mathml. Comput. Modell. 30 (1999) 35–59.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bennett C.H. Universal computation and physical dynamics Physica D 86 (1995) 268–273.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Boon J.P. How fast does Langton’s ant move? J. Stat. Phys. 102 (2001) 355–360.MATHCrossRefGoogle Scholar
  4. 4.
    Bunimovich L.A. Many-dimensional Lorentz cellular automata and Turing machines Int. J. Bif. Chaos 6 (1996) 1127–1136.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bunimovich L.A. On localization of vorticity in Lorentz lattice gases J. Stat. Phys. 87 (1997) 449–457.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bunimovich L.A. and Troubetzkoy S.E. Topological properties of flipping Lorentz lattice gas models J. Stat. Phys. 72 (1993) 297–307.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bunimovich L.A. and Troubetzkoy S.E. Rotators, periodicity and absence of diffusion in cyclic cellular automata J. Stat. Phys. 74 (1994) 1–10.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bunimovich L.A. Motion of particles in random media and many-dimensional Turing machines. Multi. Val. Logic. (2001), in print.Google Scholar
  9. 9.
    Bunimovich L.A. and Troubetzkoy S.E.Recurrence properties of Lorentz.lattice gas cellular automata J. Stat. Phys.67(1992) 289–302.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Bunimovich L.A. and Troubetzkoy S.E. Mechanisms which produce nongaussian Lorentz lattice gas cellular automata In Dynamics of Complex and Irregular Structures, Blanchard Ph. (Editor) (World Scientific: Singapore, 1994) 86–92.Google Scholar
  11. 11.
    Bunimovich L.A. and Khlabystova M.A. Localization and propagation in random lattices J. Stat. Phys. 104 (2001) 1155–1171.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Burgin B.S. Inductive Turing machines Notices of the Academy of Sciences of the USSR 270 (1991) 1289–1293.MathSciNetGoogle Scholar
  13. 13.
    Christ N.H., Friedberg R. and Lee T.D. Random lattice field theory: general formulation Nucl. Phys. B 202 (1982), 89–125.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cohen E.G.D. New types of diffusion in lattice gas cellular automata In Microscopic Simulations of Complex Hydrodynamic Phenomena Mareschal M. and Holian B.L. (Editors) (Plenum: New York, 1992) 137–152.Google Scholar
  15. 15.
    Dewdney A.K. Two-dimensional Turing machines and Turmites make tracks on a plane Scientific American September (1989) 180–183.Google Scholar
  16. 16.
    Ehrenfest P. Collected Scientific Papers (North Holland: Amsterdam, 1959) 229.MATHGoogle Scholar
  17. 17.
    Friedberg R. and Ren H.-C. Field theory on a computationally constructed random lattice Nucl. Phys. B 35[FS11] (1984) 310–320.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Grosfils P., Boon J.P., Cohen E.G.D. and Bunimovich L.A. Propagation and self-organization in lattice random media J. Stat. Phys. 97 (1999) 575–608.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Gunn J.M.F. and Ortuño M. Percolation and motion in a simple random environment J. Phys. A18 (1985) 1095–1099.Google Scholar
  20. 20.
    Hartmanis J. and Stearns R.E. On the computational complexity of algorithms Trans. Amer. Math. Soc. 117 (1965) 285–306.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Hemmerling A. Concentration of multidimensional tape-bounded systems of Turing automata and cellular spaces In Budach L. (Editor) Fundamentals of Computation Theory (Berlin: Akademie-Verlag, 1979) 167–174.Google Scholar
  22. 22.
    Hoperoft J.E. and Ullman J.D. Formal Languages and their Relation to Automata (Addison-Wesley, 1969).Google Scholar
  23. 23.
    Jiang T., Seiferas J.I. and Vitanyi P.M.B. Two heads are better than two tapesJ. ACM 44 (1997) 237–256.MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Koiran P. and Moore C. Closed-form analytic maps in one and two dimensions can simulate universal Turing machines Theor. Comput. Sci. 210 (1999) 217–223.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Kurka P. On topological dynamics of Turing machines Theor. Comput. Sci. 174 (1997) 203–216.MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Langton C.G. Studying artificial life with cellular automata Physica D22 (1986) 120–149.MathSciNetGoogle Scholar
  27. 27.
    Lorentz H.A. The motion of electrons in metallic bodies Proc. Amst. Acad. 7 (1905), 438, 585, 604.Google Scholar
  28. 28.
    Moore C. Unpredictability and undecidability in dynamical systems Phys. Rev. Lett. 64 (1990) 2354–2357.MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Moukarzel C. Laplacian growth on a random lattice PhysicaB190 (1992) 13–23.Google Scholar
  30. 30.
    Moukarzel C. and Herrmann H.J. A vectorizable random lattice J. Stat. Phys 68 (1992) 911–923.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Petersen K. Ergodic Theory (Cambridge Univ. Press: Cambridge, 1983).MATHGoogle Scholar
  32. 32.
    Ruijgrok T.W. and Cohen E.G.D. Deterministic lattice gas models Phys. Lett. A133 (1988) 415–419.MathSciNetGoogle Scholar
  33. 33.
    Siegelmann H.T. The simple dynamics of super Turing theories Theor. Comput. Sci. 168 (1996) 461–472.MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Wang F. and Cohen E.G.D. Diffusion on random lattices, J. Stat. Phys. 84 (1996) 233–261CrossRefGoogle Scholar
  35. 35.
    Worsch T. On parallel Turing machines with multi-head control units Parallel Comput. 23 (1997) 1683–1697.MathSciNetCrossRefGoogle Scholar
  36. 36.
    Worsch T. Parallel Turing machines with one-head control units and cellular automataTheor. Comput. Sci. 217 (1999) 3–30.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2002

Authors and Affiliations

  • Leonid A. Bunimovich
  • Milena A. Khlabystova

There are no affiliations available

Personalised recommendations