The Three Ways Of XOR

We all know the XOR operation. It’s the cousin of NOT, AND and OR. Like AND and OR, it is a binary boolean function. Unlike them, most programming languages don’t have an explicit “logical operator” for it, but rather show only the bitewise version. Regardless, cryptographers love it and machine learning folks fear it. And for very good reasons. Despite its simplicity, there are quite interesting things to say about it, and this article presents three of them.

To begin with, XOR looks like this, in truth-table form:

The first way of XOR is that it is true when both arguments are different and false when they are the same. So, to paraphrase the description of logical OR, a XOR b is true when either a is true or b is true, but not both. This is the understanding most common when writing program logic. XOR doesn’t appear as an equivalent to NOT, AND and OR, as a logical operator on booleans, but you can use != to much the same effect, but without the coercion to boolean operators found in many languages. This not such a great loss however. XOR doesn’t play well with short-circuiting and can lead to bugs in code, if one expects a XOR to guarantee at least a single execution of a block.

The second way of XOR is that it is a controlled NOT. The first argument controls whether the second argument is negated or not. This is the understanding most common in cryptography, where XOR appears as a basic building block of ciphers. The key is used to control the negation, while the plaintext is what is negated. It also appears as a nifty tool in circuit design. Also notice that XOR is symmetrical. The arguments can swap their roles, depending on what we need.

The third way of XOR is that it is the most “complex” binary boolean function. There are \(2^{2^{2}} = 16\) distinct boolean functions of two operators. Half of them are the negation of another function, so let’s list the \(8\) which are distinct.

We can see XOR looks slightly odd among the set. While there are other functions with two zeros and two ones, they’re all just copies of one of the arguments. From an information theoretic point of view, it has the highest entropy in the set (along side the other functions with \(2\) ones). Furthermore, from a geometric and machine learning point of view, the function is not linearly separable. It’s the only one which is not so from the set. It was in fact the example used against Perceptrons to highlight their limitations. This in turn was one of the seeds for the AI Winter of the late 80s. As commenters @dvhwgumby and @JamesPaulWhite pointed out, the AI winter was a separate event from the falling-out-of-favour of neural nets. There’s some 10-15 years between the two events. While both are part of AI folklore, it’s not correct to use causation here.

Some extra treats: XOR can be considered an imparity function. When the number of inputs is even, it outputs zero, while when it is odd, it outputs one. It is also the sum part of an adder, that is, without the carry part. It’s also a function whose output changes when any of the inputs change, which can’t be said about AND and OR. Also, from a XOR b = c, knowing either two of the variables, we can obtain the third. This is again useful in cryptography, to perform decryption of something encrypted via XOR. There’s also an correspondence between XOR and the outer product of the Clifford algebra[1]

Quite the punch for such a small function.

Edit: loads of comments to the post, especially about XOR being present as != in programming languages. Walked right into that one. I’ve also updated the entropy computations and the AI winter bits. I’ve also added a lot of other ways of XOR, picked up from the comments on Hacker News and Reddit.

[1] I understand each word in the sentence, but not the sentence as a whole, unfortunately. My math-foo is not that great :D.