Conditions for Open-Ended Evolution in Immigration Games

The Immigration Game (invented by Don Woods in 1971) extends the solitaire Game of Life (invented by John Conway in 1970) to enable two-player competition. The Immigration Game can be used in a model of evolution by natural selection, where fitness is measured with competitions. The rules for the Game of Life belong to the family of semitotalistic rules, a family with 262,144 members. Woods' method for converting the Game of Life into a two-player game generalizes to 8,192 members of the family of semitotalistic rules. In this paper, we call the original Immigration Game the Life Immigration Game and we call the 8,192 generalizations Immigration Games (including the Life Immigration Game). The question we examine here is, what are the conditions for one of the 8,192 Immigration Games to be suitable for modeling open-ended evolution? Our focus here is specifically on conditions for the rules, as opposed to conditions for other aspects of the model of evolution. In previous work, it was conjectured that Turing-completeness of the rules for the Game of Life may have been necessary for the success of evolution using the Life Immigration Game. Here we present evidence that Turing-completeness is a sufficient condition on the rules of Immigration Games, but not a necessary condition. The evidence suggests that a necessary and sufficient condition on the rules of Immigration Games, for open-ended evolution, is that the rules should allow growth.