On Partial Differential Encodings of Boolean Functions

TL;DR

This paper introduces partial differential encodings and programs for Boolean functions, providing new bounds on polynomial complexity, and demonstrates exponential separation in hypergraph isomorphism problems.

Contribution

It presents a novel framework of partial differential encodings and programs, offering new bounds and insights into polynomial families and hypergraph isomorphism complexity.

Findings

Derived bounds on Chow-Rank of polynomials

Showed partial differential programs describe determinants and permanents

Established exponential separation in hypergraph isomorphism instances

Abstract

We introduce partial differential encodings of Boolean functions as a way of measuring the complexity of Boolean functions. These encodings enable us to derive from group actions non-trivial bounds on the Chow-Rank of polynomials used to specify partial differential encodings of Boolean functions. We also introduce variants of partial differential encodings called partial differential programs. We show that such programs optimally describe important families of polynomials including determinants and permanents. Partial differential programs also enables to quantitively contrast these two families of polynomials. Finally we derive from polynomial constructions inspired by partial differential programs which exhibit an unconditional exponential separation between high order hypergraph isomorhism instances and their sub-isomorphism counterparts.