Pivotal decompositions of functions

TL;DR

This paper generalizes Shannon decomposition to broader function classes through pivotal decompositions, revealing new structural insights and characterizing functions by their unary components.

Contribution

It introduces pivotal decompositions extending Shannon's approach, applies them to various function classes, and explores their structural and definitional implications.

Findings

Pivotal decompositions hold for lattice polynomial and multilinear polynomial functions.

New function classes characterized by pivotal decompositions are defined.

Connections between functions characterized by pivotal decompositions and unary members are investigated.

Abstract

We extend the well-known Shannon decomposition of Boolean functions to more general classes of functions. Such decompositions, which we call pivotal decompositions, express the fact that every unary section of a function only depends upon its values at two given elements. Pivotal decompositions appear to hold for various function classes, such as the class of lattice polynomial functions or the class of multilinear polynomial functions. We also define function classes characterized by pivotal decompositions and function classes characterized by their unary members and investigate links between these two concepts.