Exact GP Schema Theory for Headless Chicken Crossover and Subtree Mutation

In this paper a new general GP schema theory for headless chicken crossover and subtree mutation is presented. The theory gives an exact formulation for the expected number of instances of a schema at the next generation. The theory includes four main results: microscopic schema theorems for both headless chicken crossover and subtree mutation, and two corresponding macroscopic theorems. The microscopic versions are applicable to headless chicken crossovers and subtree mutation operators. The macroscopic versions are valid for slightly more restricted sets of headless chicken and mutation operators in which the probability of selecting the crossover/mutation point(s) depends only on the size and shape of the parent program(s). In the paper we provide examples which show how the theory can be specialised to specific operators.