Cellular Automata Coupled by Overlap or Common Boundary

Cellular automata (CA) are often treated as isolated systems with simple cyclic or Dirichlet boundary conditions. Realistic systems, in contrast, interact with the environment through a boundary. Boundaries can be as simple as solid body surfaces, as complex as walls of living cells, or even non-geometric boundaries of social systems. A boundary, having an intricate structure and being a coupling link to the environment, can strongly influence the system dynamics. Here two elementary CA colored red and blue interact via a boundary consisting of black shared cells. The configuration of coupling is schematically shown on the image at the lower-left corner of the graphic. Even the single cell boundary can significantly alter the dynamics of CA, as shown on the fourth row of snapshots. To see clearly whether the dynamics was altered, switch a few times between the given and zero values of the control "overlap" for comparison. If the overlap is made significantly large, it can be considered as a 2-color range-2 CA interacting with two elementary CA. The boundary in this case is the line separating CA of different color. The CA exchange information via the bordering cells. A few examples of cases when the dynamics are significantly altered by the CA coupling are shown on the third row of snapshots. The Demonstration also shows how larger neighborhood CA patterns arise from the multiple action of smaller neighborhood CA. Simply compare the red and blue patterns to the black ones, which should be done with a substantial number of black cells (with the control "overlap" much larger than 5). A few illustrative examples are shown in the first and second rows of snapshots. Browse the bookmarks; for more information, see the Details section.



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This Demonstration shows the evolution of two elementary CA sharing several black cells. Unshared cells are red and blue to distinguish between the different CA. The number of shared cells is set by the control "overlap". When the "overlap" is zero, no cells are shared and the red and blue CA evolve independently. In this zero-overlap case evolution takes place on isolated red and blue loops due to cyclic boundary conditions. Increasing the control "overlap" does NOT change the number of cells in the red and blue loops, but makes some of the cells common to both CA. This is schematically shown on the image at the lower-left corner of the graphic. In some way the "overlap" control acts as a "zipper" fusing the red and blue loops together. With equal numbers of red and blue cells it is possible to completely "zip up" the loops. At a single step in the evolution, the "red rule" is applied to the red loop including the black shared cells, and then the "blue rule" is applied to the blue loop including the same black cells once again. Thus, during a single step of the evolution, black shared cells are acted upon twice by two generally different 2-color range-1 CA rules. Such double action is equivalent to a single action of a 2-color range-2 CA rule. All three CA (red, blue, and black) interact through the bordering cells exchanging information.
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