My question is more from a digital logic background, and what is possible on computers, though I am curious what the answer is for real numbers.
Consider a function that performs a bitwise AND on two inputs, perhaps implemented as logic gates in hardware or a simple function in software:
$$and_2(x,y) = x \space \& \space y$$
This function takes two variables as input, and evaluates two variables to produce the output. Now consider a three variable function:
$$and3(x,y,z) = x \space \& \space y \space \& \space z$$
This function takes three input variables, but it doesn't necessarily evaluate three at a time. It could be rewritten to only evaluate two variables at a time (and iirc this is what most hardware design software does):
$$and_3(x,y,z) = (x \space \& \space y) \space \& \space z$$
which is the same as
$$and_3(x,y,z) = and_2(and_2(x,y), z)$$
If I had to guess, I would say that rewriting a multi-input function as a combination of two-input functions is possible for any associative binary operator (I think that's part of the definition...).
I thought perhaps the ternary operator (aka if/then/else, used in C) might be a three input function, but this is straightforward to re-write as a combination of two-input functions.
$$ternary(a,b,c) = a \space ? \space b : c$$
$$ternary(a,b,c) = (a \space \& \space b) \space | \space (\tilde a \space \& \space c)$$
$$ternary(a,b,c) = or_2(and_2(a,b),and_2(not(a),c))$$
To rephrase my original question: can all multi-input functions be re-written as a combination of two input functions (specifically for digital logic, and if you have the time, for any function in general)?
If I used any terminology wrong or missed some relevant tags, please correct, I'm not too familiar with the theory side of this.
These may be related but didn't answer my question:
Can any function be written as a composition of other functions?
Non-associative, non-commutative binary operation with a identity
