Double inversion is the gate-level version of the involution law, = A. Flip a bit twice and it's back where it started.
It shows up in three places:
- Inverter chains. Two NOT gates in series pass A through unchanged. In general, an even number of inverters passes A and an odd number gives .
- Bubbles. Two bubbles on the same wire, one at a gate's output and one at the next gate's input, cancel out. This is the key move in bubble pushing.
- Algebra. A bar over a bar on the same thing disappears, as in = .
Because a pair of inversions changes nothing, you can add a pair to any wire whenever it helps. That is exactly how an AND-OR circuit becomes nand nand logic: add two bubbles to each internal wire, then regroup them into the gates.
In hardware, two inverters in a row aren't useless. Every gate outputs a fresh, full-strength level, so back-to-back inverters act as a buffer that restores a weak signal or drives a large fan out.
Only inversions of the same signal cancel. Bars of different lengths are different operations: is not ; by De Morgan it is .