On Partial Differential Encodings of Boolean Functions

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.