Exact Schema Theorems for GP with One-Point and Standard Crossover Operating on Linear Structures and Their Application to the Study of the Evolution of Size

In this paper, firstly we specialise the exact GP schema theorem for one-point crossover to the case of linear structures of variable length, for example binary strings or programs with arity-1 primitives only. Secondly, we extend this to an exact schema theorem for GP with standard crossover applicable to the case of linear structures. Then we study, both mathematically and numerically, the schema equations and their fixed points for infinite populations for both a constant and a length-related fitness function. This allows us to characterise the bias induced by standard crossover. This is very peculiar. In the case of a constant fitness function, at the fixed-point, structures of any length are present with non-zero probability. However, shorter structures are sampled exponentially much more frequently than longer ones.