Typology of Boolean Functions Using Walsh Analysis

Abstract

Much previous works deal with the functions that cause a genetic algorithm (GA) to diverge from the global optimum. It is now a fact: Walsh analysis allows the identification of GA-hard problems. But what about genetics-based machine learning? Do we know what makes a problem hard for a classifier system (CS)? In order to try to answer these questions, we describe the relation between CS performance and the structure of a given boolean function when it is expressed as a Walsh polynomial. The analysis of the relative magnitude of Walsh coefficients allows us to set up a typology of boolean functions according to their hardness for a CS. Thus, each function can be placed in a specific class of difficulty. Using the converse process, we can start from well chosen Walsh coefficients in order to build boolean functions hard for a CS to learn.