Pivotal decompositions of functions

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.

💡 Research Summary

The paper introduces “pivotal decomposition,” a broad generalization of the classic Shannon decomposition that has long been a cornerstone in Boolean function theory. While Shannon’s expansion expresses a Boolean function f(x₁,…,xₙ) as f = x_k·f|{x_k=1} + ¬x_k·f|{x_k=0}, the authors replace the fixed Boolean constants 0 and 1 by an arbitrary pair of elements a and b from the function’s codomain and introduce auxiliary unary functions φ_k and ψ_k that depend only on the k‑th variable. A function f admits a pivotal decomposition with respect to variable x_k if it can be written as

  f(x) = φ_k(x_k)·f(x₁,…,x_{k‑1},a,x_{k+1},…,xₙ) + ψ_k(x_k)·f(x₁,…,x_{k‑1},b,x_{k+1},…,xₙ).

When a=1, b=0, φ_k(x_k)=x_k, and ψ_k(x_k)=1−x_k, the definition collapses to the ordinary Shannon expansion. The key idea is that every unary section (the function obtained by fixing all variables except one) depends only on the two “pivot” values a and b, not on the full range of the variable.

The authors demonstrate that this abstract scheme captures several important families of functions. First, lattice polynomial functions—functions built from the meet and join operations of a lattice—naturally satisfy a pivotal decomposition when the pivots are chosen as the lattice’s least and greatest elements. Second, multilinear polynomials over the real interval