Lorentz Lattice Gases and Many-Dimensional Turing Machines

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.

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