Blanket Algebra for Multiple-valued Function Decomposition

We consider functions of the form f : are only partially speciied. In general, f assigns to any output y i a nonempty set d 2 D of elements of E. If d = feg for some e 2 E, then y i has the value e. If d = E, then y i is a \complete don't care", i.e., is allowed to take on any value in E. In other cases, y i is a \limited don't care", allowed to take on any value in d. Moreover, the functions are represented in a compact notation, which is a generalization of the cube notation for Boolean functions. In this representation, we call these functions \set functions." We consider the problem of decomposing a set function f(X) in the form h(U; g(V)), where X = fx 1 ; : : :; x n g is the set of input variables, U and V are two subsets of X whose union is X, and g and h are set functions. It has been previously shown that Boolean function decompositions can be obtained with the aid of blankets, which are a generalization of set systems. In this paper we extend blanket algebra methods to the decomposition of multiple-valued set functions.