On the Long String Limit

Description/Abstract

To gain a clearer understanding of the dynamics of evolving populations the limit of infinitely long strings is examined. The limit clarifies some of the behaviour of evolving systems. For example, this limit removes any complications due to transient behaviour. Care is required in taking the limit to ensure that the quantities considered are meaningful. For concreteness a simple Genetic Algorithm acting on the problem of ones-counting is considered. A population evolving under selection and mutation alone is first examined. In this limit a remarkably simple picture of the evolutionary flow emerges. The discussion is widened to include crossover. This gives a new insight into how diversity induced by crossover improves search. Finally the approach to the limit is examined. The limit provides a way of separating the initial transient dynamics from the long time (asymptotic) dynamics.