Construction of high-order deceptive functions using low-order Walsh coefficients

Abstract

This paper constructs deceptive functions over bit strings of any length using low-order Walsh coefficients. Specifically, a partially deceptive construct is achieved with coefficients no higher than second order, and a fully deceptive construct is achieved with coefficients no higher than third order. Previous work on deception tacitly assumed that the very highest order Walsh coefficient, the order-l coefficient in a length-l problem, must be non-zero to achieve full deception, that condition where all schemata of orderl−1 or less containing the complement of the global optimum are superior to their competitors. The constructions of this paper show such assumptions to be false and lead to a more general theory of functions whose value only depends on the number of ones (or zeros) in the function's argument, so-calledfunctions of unitation. These results also help shed some light on Tanese's puzzling results on functions that contained large numbers of Walsh terms of fixed order.