Why complex-valued fuzzy? Why complex values in general? A computational explanation

In the traditional fuzzy logic, as truth values, we take all real numbers from the interval [0; 1]. In some situations, this set is not fully adequate for describing expert uncertainty, so a more general set is needed. From the mathematical viewpoint, a natural extension of real numbers is the set of complex numbers. Complex-valued fuzzy sets have indeed been successfully used in applications of fuzzy techniques. This practical success leaves us with a puzzling question: why complex-valued degree of belief, degrees which do not seem to have a direct intuitive meaning, have been so successful? In this paper, we use latest results from theory of computation to explain this puzzle. Namely, we show that the possibility to extend to complex numbers is a necessary condition for fuzzy-related computations to be feasible. This computational result also explains why complex numbers are so efficiently used beyond fuzzy, in physics, in control, etc.