We consider a subclass of $\mathbf{AC}^0[2]$ circuits that simultaneously captures $\mathrm{DNF} \circ \mathrm{Xor}$ and depth-$3$ $\mathbf{AC}^0$ circuits. For this class we show a technique for proving lower bounds inspired by the top-down approach. We give lower bounds for the middle slice function, inner product function, and affine dispersers.

Fixed a bug in the writing of Theorem 3.1

We consider a subclass of $\mathbf{AC}^0[2]$ circuits that simultaneously captures $\mathrm{DNF} \circ \mathrm{Xor}$ and depth-$3$ $\mathbf{AC}^0$ circuits. For this class we show a technique for proving lower bounds inspired by the top-down approach. We give lower bounds for the middle slice function, inner product function, and affine dispersers.