The Troubling Aspects of a Building Block Hypothesis for Genetic Programming

In this paper we rigorously formulate the Schema Theorem for Genetic Programming (GP). This involves defining a schema, schema order, and defining length and accounting for the variable length and the non-homologous nature of GP'S representation. The GP Schema Theorem and the related notion of a GP Building Block are used to construct a testable hypothetical account of how GP searches by hierarchically combining building blocks. Since building blocks need to have consistent above average fitness and compactness, and since the term in the GP Schema Theorem that expresses compactness is a random variable, the proposed account of GP search behavior is based on empirically questionable statistical assumptions. In particular, low variance in schema fitness is questionable because the performance of a schema depends in a highly sensitive manner on the context provided by the programs in which it is found. GP crossover is likely to change this context from one generation to the next which results in high variance in observed schema fitness. Low variance in compactness seems fortuitous rather than assured in GP because schema-containing programs change their sizes essentially at random.