An Analysis of the MAX Problem in Genetic Programming

We present a detailed analysis of the evolution of GP populations using the problem of finding a program which returns the maximum possible value for a given terminal and function set and a depth limit on the program tree (known as the MAX problem). We confirm the basic message of \citeGathercole:1996:aicrtd that crossover together with program size restrictions can be responsible for premature convergence to a sub-optimal solution. We show that this can happen even when the population retains a high level of variety and show that in many cases evolution from the sub-optimal solution to the solution is possible if sufficient time is allowed. In both cases theoretical models are presented and compared with actual runs. Price’s Covariance and Selection Theorem is experimentally tested on GP populations. It is shown to hold only in some cases, in others program size restrictions cause important deviation from its predictions