We say that a circuit C over a field F functionally computes an
n-variate polynomial P if for every x∈{0,1}n we have that C(x)=P(x). This is in contrast to syntactically computing P, when C≡P as
formal polynomials. In this paper, we study the question of proving lower
bounds for homogeneous depth-3 and depth-4 arithmetic circuits for
functional computation. We prove the following results :
1. Exponential lower bounds homogeneous depth-3 arithmetic circuits for a
polynomial in VNP.
2. Exponential lower bounds for homogeneous depth-4 arithmetic circuits
with bounded individual degree for a polynomial in VNP.
Our main motivation for this line of research comes from our observation that
strong enough functional lower bounds for even very special depth-4
arithmetic circuits for the Permanent imply a separation between #P and
ACC. Thus, improving the second result to get rid of the bounded individual
degree condition could lead to substantial progress in boolean circuit
complexity. Besides, it is known from a recent result of Kumar and Saptharishi
[KS15] that over constant sized finite fields, strong enough average case
functional lower bounds for homogeneous depth-4 circuits imply
superpolynomial lower bounds for homogeneous depth-5 circuits.
Our proofs are based on a family of new complexity measures called shifted
evaluation dimension, and might be of independent interest