Canonical restricted mixed-polarity exclusive-OR sums of products and the efficient algorithm for their minimisation

Abstract

The concept of canonical restricted mixed polarity (CRMP) exclusive-OR sum of products forms is introduced. The CRMP forms include the inconsistent canonical Reed-Muller forms and the fixed-polarity Reed-Muller (FPRM) forms as special cases. The set of CRMP forms is included in the set of exclusive-OR sum-of-product (ESOP) expressions. An attempt to characterise minimal CRMP forms for completely specified Boolean functions is presented as well as an insight into the complexity of computation needed to find such a form. Some fundamental properties unique to CRMPs are proven. It is also proven that the upper bound on the number of terms in the CRMP form is smaller than that in the conventional normal forms and equal to that of the ESOPs. A theorem providing a lower bound on the number of CRMP terms is given. Finally, based on these theoretical results, a heuristic algorithm and its implementation to obtain a quasiminimal CRMP form for a multioutput function are presented.