Dynamical behavior and complexity of Langton's ant

O ne of the first models of artificial life, proposed back in the 1980s by Christopher Langton, founder of the field, was the virtual ant [1,2]. This simple cellular automaton is defined on the square grid in the following way: each square (“cell”) of the grid can be in one of two states, white or black, and the ant is represented by a short arrow that stands on one cell and points to the north, the west, the east, or the south. At each time step, it moves to the cell it was pointing to, and it turns 90 degrees to the left if this cell is white or 90 degrees to the right if it is black; in addition, the state of the cell is switched. Figure 1a shows the situation after 5 time steps, starting with a background of only white cells. The interesting part starts when we let the ant go on with its walk. At iterations 96 and 184, rotational symmetry of order 2 is observed; at iteration 368 (Figure 1b), it is almost of order 4. When one could expect further symmetrical patterns, the symmetry breaks down, and after the step 500, the ant seems to walk at random, for more than 9000 iterations (Figure 1c). Again, when one would expect this chaos to go on forever, the ant suddenly starts building a pattern that is periodic but for a drift; it is the so called “highway,” and in the absence of obstacles, the ant will draw it forever (Figure 1d). This brief history involving a break of symmetry, a “chaotic” phase, and sudden order, all generated by a rule that could hardly be simpler, becomes more intriguing when we notice that it repeats with other initial patterns. In fact, the highway has appeared, sooner or later, in all the simulations that have been started with a finite amount of black (or of white) cells (we call them configurations with finite support). Nobody has demonstrated Langton’s ant, with its intriguing behavior, has been elusive to theoretical results, particularly from the point of view of the system’s complexity. We summarize here some recent work of our group, that sheds some new light on the ant.