Separable Boolean functions and generalized Fibonacci sequences

It is well known that the Boolean functions can be represented by two-layer perceptrons, and a part of them, namely separable Boolean functions, can be represented by one-layer perceptrons. How many separable Boolean functions of n variables there are is an open problem. On the other hand, given a n-element set X, how many antichains does P(X) have is also an open problem. This paper established an inequality reflecting the relationship between these two open problems. Second, this paper introduced two classes of Boolean functions which are generalizations of AND-OR functions and OR-AND functions, respectively, and proved that they are all separable and the weights in representing them are exactly terms of corresponding generalized Fibonacci sequences. (C) 2000 Elsevier Science Ltd. All rights reserved.