Simplified Reed–Muller Expressions for Residue Threshold Functions

Abstract

Residue threshold functions are a broad class of symmetric functions that include all the unit-weighted threshold functions. In this paper, we investigate the complexity of the Reed–Muller (RM) expressions for these functions. We prove that an important subclass of the functions has very simple RM expansions and determine the conditions that define such a subclass. Additionally, we show that such an expansion is also the optimal one concerning its polarity. As an interesting practical application, an analysis of the RM expansion of the output functions for parallel counters is performed.