Problem: Express a boolean logic formula using polynomials. I.e., if an input variable $ x$ is set to $ 0$, that is interpreted as false, while $ x=1$ is interpreted as true. The output of the polynomial should be 0 or 1 according to whether the formula is true or false as a whole.
Solution: You can do this using a single polynomial.
Illustrating with an example: the formula is $ \neg[(a \vee b) \wedge (\neg c \vee d)]$ also known as
not((a or b) and (not c or d))
The trick is to use multiplication for “and” and $ 1-x$ for “not.” So $ a \wedge b$ would be $ x_1 x_2$, and $ \neg z$ would be $ 1-z$. Indeed, if you have two binary variables $ x$ and $ y$ then $ xy$ is 1 precisely when both are 1, and zero when either variable is zero. Likewise, $ 1-x = 1$ if $ x$ is zero and zero if $ x$ is one.
Combine this with deMorgan’s rule to get any formula. $ a \vee b = \neg(\neg a \wedge \neg b)$ translates to $ 1 – (1-a)(1-b)$. For our example above,
$ \displaystyle f(x_1, x_2, x_3, x_4) = 1 – (1 – (1-a)(1-b))(1 – c(1-d))$
Which expands to
$ \displaystyle 1 – a – b + ab + (1-d)(ac + bc – abc)$
If you plug in $ a = 1, b = 0, c = 1, d = 0$ you get True in the original formula (because “not c or d” is False), and likewise the polynomial is
$ \displaystyle 1 – 1 – 0 + 0 + (1-0)(1 + 0 – 0) = 1$
You can verify the rest work yourself, using the following table as a guide:
0, 0, 0, 0 -> 1 0, 0, 0, 1 -> 1 0, 0, 1, 0 -> 1 0, 0, 1, 1 -> 1 0, 1, 0, 0 -> 0 0, 1, 0, 1 -> 0 0, 1, 1, 0 -> 1 0, 1, 1, 1 -> 0 1, 0, 0, 0 -> 0 1, 0, 0, 1 -> 0 1, 0, 1, 0 -> 1 1, 0, 1, 1 -> 0 1, 1, 0, 0 -> 0 1, 1, 0, 1 -> 0 1, 1, 1, 0 -> 1 1, 1, 1, 1 -> 0
Discussion: This trick is used all over CS theory to embed boolean logic within polynomials, and it makes the name “boolean algebra” obvious, because it’s just a subset of normal algebra.
Moreover, since boolean satisfiability—the problem of algorithmically determining if a boolean formula has a satisfying assignment (a choice of variables evaluating to true)—is NP-hard, this can be used to show certain problems relating to multivariable polynomials is also hard. For example, finding roots of multivariable polynomials (even if you knew nothing about algebraic geometry) is hard because you’d run into NP-hardness by simply considering the subset of polynomials coming from boolean formulas.
Here’s a more interesting example, related to the kinds of optimization problems that show up in modern machine learning. Say you want to optimize a polynomial $ f(x)$ subject to a set of quadratic equality constraints. This is NP-hard. Here’s why.
Let $ \varphi$ be a boolean formula, and $ f_\varphi$ its corresponding polynomial. First, each variable $ x_i$ used in the polynomial can be restricted to binary values via the constraint $ x_i(x_i – 1) = 0$.
You can even show NP-hardness if the target function to optimize is only quadratic. As an exercise, one can express the subset sum problem as a quadratic programming problem using similar choices for the constraints. According to this writeup you even express subset sum as a quadratic program with linear constraints.
The moral of the story is simply that multivariable polynomials can encode arbitrary boolean logic.
Something related that people might find interesting is “homomorphic encryption over the integers” which also encodes boolean logic into relatively simple math. The purpose there is to be able to give encrypted data to an untrusted machine to do work with, and then give you back the still encrypted data. When you unencrypt it, it’s as if the work was done on the plain text data, but the machine doing the work never had knowledge of the plain text data. Useful for things including utilizing cloud computing to process secret / sensitive data.
A simplified version of how it works:
Pick a random odd number k, and encrypt a 0 or 1 by adding it to k, to get C, the cipher bit.
If you want to AND two bits encrypted with the same key, you multiply them together. If you want to XOR two bits encrypted with the same key, you add them together.
Note: you can do those two same operations with unencrypted plain text bits (imagine the untrusted computing machine wanted to multiply your encrypted values by a constant).
To decrypt an encrypted bit, it’s P = (C % k) % 2
More info on this here if you are curious! https://blog.demofox.org/2015/09/05/super-simple-symmetric-leveled-homomorphic-encryption-implementation/
A fun thing here is that it’s possible to encrypt a bit such that using different keys to decrypt give it different values as output which lets you put a bit or multiple bits into a “super position” between values. You can operate on these encrypted bits as per usual, but then you can choose what key to decrypt them with later, giving you the correct answer as if the bits were the values encrypted from the point of view of that key. In other words, you get a sort of “SIMD” effect… doing an operation once on the encrypted data, but getting results for multiple inputs at once. I haven’t written up the details of that yet, and it is a known thing in the HE circles, so isn’t novel, but it is interesting!