Algebraic normal form (ANF) is a way of describing digital circuits made up of AND and XOR gates.
The below is an example of an ANF expression which evaluates to true if two or more of it's three inputs are true ($\oplus$ being XOR, implicit multiplication being AND)
$x_0x_1 \oplus x_0x_2 \oplus x_1x_2$
When a digital circuit is expressed this way, you can evaluate it as a polynomial, taking 1 or 0 as inputs for the variables, doing a multiplication for an AND gate, addition for an XOR gate, and doing a modulus by 2 on the final result.
$y = x_0x_1 + x_0x_2 + x_1x_2$
you can verify a truth table by plugging in 0s and 1s for the various $x$ parameters and seeing that it comes out to the correct values.
What I'm curious about is what if we don't use whole numbers? I'm sure that's a well studied thing but I haven't been able to find any information about it.
it seems sort of like fuzzy logic, but fuzzy logic is a well defined thing with different operations.
Here's some values plugged in and their output.
\begin{array}{|c|c|c|c|}\hlinex_0 & x_1 & x_2 & output & output \% 2 \\\hline0 & 0 & 0 & 0 & 0 \\1 & 1 & 0 & 1 & 1 \\1 & 1 & 1 & 3 & 1 \\0.5 & 0.5 & 0 & 0.25 & 0.25\\0.9 & 0.9 & 0.9 & 2.43 & 0.43 \\1.0 & 1.0 & 0.5 & 2.0 & 0.0 \\\hline\end{array}
The analog value input gives output that seems especially wrong in the last two rows.
It seems like maybe this just "doesn't work", but it also feels like maybe it does work, or does do something interesting, perhaps with some modifications?
Has anyone come across non digital values used in ANF or similar?
Thanks!
