  We say that a circuit $C$ over a field $F$ functionally computes an
$n$-variate polynomial $P$ if for every $x \in \{0,1\}^n$ we have that $C(x) =
P(x)$. This is in contrast to syntactically computing $P$, when $C \equiv P$ 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.
