History [of tag systems]

The tag systems that I consider are generalizations of those first discussed by Emil Post in 1920 as simple idealizations of certain syntactic reduction rules in Alfred Whitehead and Bertrand Russell's *Principia Mathematica* (see page 1149). Post's tag systems differ from mine in that his allow the choice of block that is added at each step to depend only on the very first element in the sequence at that step (see however page 670). (The lag systems studied in 1963 by Hao Wang allow dependence on more than just the first element, but remove only the first element.) It turns out that in order to get complex behavior in such systems, one needs either to allow more than two possible colors for each element, or to remove more than two elements from the beginning of the sequence at each step. Around 1921, Post apparently studied all tag systems of his type that involve removal and addition of no more than two elements at each step, and he concluded that none of them produced complicated behavior. But then he looked at rules that remove three elements at each step, and he discovered the rule {3, {{0, _, _} {0, 0}, {1, _, _} {1, 1, 0, 1}}}. As he noted, the behavior of this rule varies considerably with the initial conditions used. But at least for all the initial conditions up to length 28, the rule eventually just leads to behavior that repeats with a period of 1, 2, 6, 10, 28 or 40. With more than two colors, one finds that rules of Post's type which remove just two elements at each step can yield complex behavior, even starting from an initial condition such as {0, 0}. An example is {2, {{0, _} {2, 1}, {1, _} {0}, {2, _} {0, 2, 1, 2}}}. (See also pages 1113 and 1141.)