Formula size lower bounds for AC0 functions Question:


  What is the best known formula size lower bound for an explicit function in AC0? Is there an explicit function with an $\Omega(n^2)$ lower bound?


Background:

Like most lower bounds, formula size lower bounds are hard to come by. I am interested in formula size lower bounds over the standard universal gate set {AND, OR, NOT}.

The best known formula size lower bound for an explicit function over this gate set is $\Omega(n^{3-o(1)})$ for a function defined by Andreev. This bound was shown by Håstad, improving Andreev's lower bound of $\Omega(n^{2.5-o(1)})$. 
Another explicit lower bound is Khrapchenko's $\Omega(n^2)$ lower bound for the parity function.

However, these two functions are not in AC0. I'm wondering if we know an explicit function in AC0 with a quadratic (or better) lower bound. The best bound I am aware of is the $\Omega(n^2/\log n)$ lower bound for the Element Distinctness function, as shown by Nechiporuk. Note that the element distinctness function is in AC0, so I'm looking for a lower bound for an explicit AC0 function that is better than $\Omega(n^2/\log n)$, preferably $\Omega(n^2)$.

Further reading:

An excellent resource on the topic is "Boolean Function Complexity: Advances and Frontiers" by Stasys Jukna. A draft of the book is available for free on his website.
A nice question! Khrapchenko can definitely not give quadratic lower bounds for $AC^0$ functions. His lower bound is in fact at least square of average sensitivity. And all functions in $AC^0$ have poly-logarithmic average sensitivity. Subbotovskaya-Andreev can apparently also not give such a function because the argument they use (random restriction results in much smaller formulas) is exactly the reason forcing large $AC^0$ circuit size; Hastad's Switching Lemma (am not quite sure, just an intuition). The only hope is Nechiporuk. But his argument cannot give more than $n^2/\log n$, by information theoretic reasons. So, can it be that everything in $AC^0$ has formulas of quadratic (or even smaller) size? I don't believe in it, but couldn't quickly find a counterexample.  

Actually, the Allender-Koucky phenomenon arises also in other context - in graph complexity. Say that a circuit of $2n$ variables represents a bipartite $n\times n$ graph $G$ on vertices $V=\{1,\ldots,2n\}$ if for every input vector $a$ with exactly two 1s is, say, positions $i$ and $j$ ($i\leq n$, $j>n$) the circuit accepts $a$ iff vertices $i$ and $j$ are adjacent in $G$. Problem: exhibit an explicit graph $G$ requiring at least $n^{\epsilon}$ gates to be represented by a monotone $\Sigma_3$-circuit. Seems like an innocent question (since most graphs require about $n^{1/2}$ gates. But any such graph would give us a boolean function of $2m=2\log n$ variables requiring non-monotone log-depth circuits of superlinear size (by results of Valiant). Thus, even proving $n^{\epsilon}$ lower bounds for depth-3 circuits may be a challenge.