  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.
