We study the k-epistasis of a fitness function over a search space. This concept is a natural generalization of that of epistasis, previously considered by Davidor, Suys and Verschoren and Van Hove and Verschoren [Y. Davidor, in: Foundations of genetic algorithms, Vol. 1, (1991), pp. 23–25; D. Suys and A. Verschoren, ‘Proc Int. Conf. on Intelligent Technologies in Human-Related Sciences (ITHURS’96), Vol. II (1996), pp. 251–258; H. Van Hove and A. Verschoren, Comput. Artificial Intell.14 (1994), 271–277], for example. We completely characterize fitness functions whose k-epistasis is minimal: these are exactly the functions of order k. We also obtain an upper bound for the k-epistasis of nonnegative fitness functions.
