Why sum of two binary numbers cannot be determined in $NC^0$ but in $AC^0$?

Question

Why sum of two binary numbers cannot be determined in $NC^0$ but it can be determined in $AC^0$?

Answer 1

In an $\mathsf{NC^0}$ circuit, every output bit depends on a bounded number of input bits. But the $k$th bit of the output (counting from the LSB) depends on the first $k$th bits of each input.

To see that $\mathsf{AC^0}$ circuits can compute addition, we need to produce such a circuit. Hopefully you have seen such circuits, and otherwise perhaps you can construct them on your own. Give it a shot.

Answer 2

an $NC^0$ can only consider circuits of fan-in 2. If we try to adding with a Full-Adder with Lookahead Gatter to calculate the carry, needs every Full-Adder 3 Input signals. But in $NC^0$ are only 2 inputs allowed. If we try to replace Full-Adder with other logic gatters, we hurt the depth of the circuit.

image-source: https://upload.wikimedia.org/wikipedia/commons/thumb/0/04/4-bit_carry_lookahead_adder.svg/1000px-4-bit_carry_lookahead_adder.svg.png