Properties of Symmetric Fitness Functions

The properties of symmetric fitness functions are investigated. We show that the search spaces obtained from symmetric functions have the zero-correlation structures between fitness and distance. It is also proven that symmetric functions induce a class of the hardest problems in terms of the epistasis variance and its variants. These analyses suggest that the existing quantitative measures cannot discriminate among symmetric functions, which reveals critical limitations of the measures. To take a closer look at the symmetric functions, additional analyses are performed from other viewpoints including additive separability and boundedness. It is shown that additive separability in a symmetric function is closely related to the symmetry of its subfunctions. This elucidates why most of the well-known symmetric fitness functions are additively inseparable. The properties of two-bounded symmetric functions are investigated and they are utilized in designing an efficient algorithm to check additive separability for the two-bounded functions. Throughout this paper, we heavily use the generalized Walsh transform over multary alphabets. Our results have an independent interest as a nontrivial application of the generalized Walsh analysis.