Four-variable AND-EXOR minimum expressions and their properties

Abstract

An AND-EXOR expression (exclusive-or sum-of-products expression: ESOP) is obtained by EXORing arbitrary product terms. First, some properties are shown of minimum ESOPs which are useful for the minimization of ESOPs. Second, a catalog of minimum ESOPs for the representative functions of 4-variable NP-equivalence classes is presented. Minimality is defined as first minimizing the number of the product terms and then the number of the literals in the ESOP. Minimum ESOPs are obtained by an exhaustive method. The average number of products to realize the 4-variable functions by AND-OR sum-of-products expressions is 4.13 and 3.66 by ESOPs. Minimum ESOPs with four variables or less are obtained easily by the table look-up of the catalog. The catalog is useful for the minimization and the complexity analysis of ESOPs with five or more variables.