A theoretical analysis of the HIFF problem

We present a theoretical analysis of Watson's Hierarchical-if-and-only-if (HIFF) problem using a variety of tools. These include schema theory and course graining, the concept of effective fitness, and statistical analysis. We first review the use of Stephen's exact schema equations and schema basis to compute the changes in population distributions over time. We then use the tools described above to solve for the limit distributions of the 2 and 4-bit HIFF problems, and show that these limit distributions are essentially one-dimensional. We also show that a combination of fitness and the number of break points (a rough measure of distance in crossover space) in a string can be used to almost completely explain the limit distribution in the 4-bit HIFF problem.