General Schema Theory for Genetic Programming with any Subtree-Swapping Crossover

In this paper a new general schema theory for genetic programming is presented. Like other recent GP schema theory results (Poli 2000a, Poli 2000b), the theory gives an exact formulation (rather than a lower bound) for the expected number of instances of a schema at the next generation. The theory is based on a Cartesian node reference system which makes it possible to describe programs as functions over the space and allows to model the process of selection of the crossover points of subtreeswapping crossovers as a probability distribution over . The theory is also based on the notion of variable-arity hyperschema, which generalises almost all previous definitions of schema or hyperschema introduced in GP. The theory includes two main theorems describing the propagation of GP schemata: a microscopic schema theorem and a macroscopic one. The microscopic version is applicable to any crossover operator which swaps a subtree in one parent with a subtree from the other parent to produce the offspring. Therefore, this theorem is equally applicable to standard GP crossover (Koza 1992) with and without uniform selection of the crossover points, as it is to one-point crossover (Poli and Langdon 1997b, Poli and Langdon 1998), size-fair crossover (Langdon 1999, Langdon 2000), strongly-typed GP crossover (Montana 1995), context-preserving crossover (D’haeseleer 1994) and many others. The macroscopic version is applicable to crossover operators in which the probability of selecting any two crossover points in the parents depends only on their size and shape. This is still a very general results which can be used to model most untyped GP systems. In the paper we provide examples which show how the theory can be specialised for any specific crossover operator. We also show how the theory can be used to derive other important general results such as an exact definition of the notion of effective fitness and a general size-evolution equation for GP.