What is the difference between $AC^2$ and $FO (\log \log n)$ complexity classes?

Question

$FOLL$ or $FO (\log \log n)$ is a complexity class of all problems solvable by uniform poly-size circuit families of unbounded fan-in and depth $O(\log \log n)$.

$AC^2$ is a complexity class of all problems recognized by boolean circuits with depth $O(\log ^{2}n)$ and polynomial number of unlimited fan in AND and OR gate.

Question. : What is a difference between $FOLL$ and $AC^2$?

In the definition of $FOLL$ they have used "uniform " poly-size and in the definition of $AC^2$ they are saying boolean circuits.

Answer 1

The functions $\log \log n$ and $\log^2 n = (\log n)^2$ are quite different. The latter grows much faster than the former: $\log \log n = o(\log^2 n)$. This shows that FOLL is a subset of AC² (and even of the smaller class AC¹).

Uniformity is a different aspect in which the two classes differ: FOLL is a class of uniform circuits, whereas the AC hierarchy captures non-uniform circuits. As such, even AC⁰ circuits can compute uncomputable functions, something that FOLL cannot do.