Search, Binary Representations and Counting Optima

Choosing a good representation is a vital component of solving any search problem. However, choosing a good representation for a problem is as diicult as choosing a good search algorithm for a problem. Wolpert and Macready's No Free Lunch theorem proves that no search algorithm is better than any other over all possible discrete functions. We elaborate on the No Free Lunch theorem by proving that there tend to be a small set of points that occur as local optima under almost all representations. Along with the analytical results, we provide some empirical evaluation of two representations commonly used in genetic algorithms: Binary Reeected Gray coding and standard Binary encoding.