Generalized Inclusive Forms—New Canonical Reed-Muller Forms Including Minimum ESOPs

Abstract

This paper describes two families of canonical Reed-Muller forms, called inclusive forms (IFs) and their generalization, the generalized inclusive forms (GIFs), which include minimum ESOPs for any Boolean function. We outline the hierarchy of known canonical forms, in particular, pseudo-generalized Kronecker forms (PGKs), which led us to the discovery of the new families. Next, we introduce special binary trees, called the S/D trees, which underlie IFs and permit their enumeration. We show how to generate IFs and GIFs and prove that GIFs include minimum ESOPs. Finally, we present the results of computer experiments, which show that GIFs reduce the search space for minimum ESOP by several orders of magnitude, and this reduction grows exponentially with the number of variables.