An E cient Method for Finding an Optimal Bi - Decomposition

This paper presents a new e cient method for nding an \optimal" bi-decomposition form of a logic function. A bi-decomposition form of a logic function is the form: f(X) = (g1(X ); g2(X )). We call a bi-decomposition form optimal when the total number of variables in X and X is the smallest among all bi-decomposition forms of f . This meaning of optimal is adequate especially for the synthesis of LUT (Look-Up Table) networks where the number of function inputs is important for the implementation. In our method, we consider only two bi-decomposition forms; (g1 g2) and (g1 g2). We can easily nd all the other types of bi-decomposition forms from the above two decomposition forms. Our method e ciently nds one of the existing optimal bi-decomposition forms based on a branch-and-bound algorithm. Moreover, our method can also decompose incompletely speci ed functions. Experimental results show that we can construct better networks by using optimal bi-decompositions than by using conventional decompositions. key words: functional decomposition, bi-decomposition, AND, XOR, look-up table