Notes on negation

The well known formulas

Most readers will be familiar with -x = ~x + 1 = ~(x - 1). These are often just stated without justification, or even an explanation for why they are equivalent. There are some algebraic tricks, but I don't think they explain much, so I'll use the rings from visualizing addition, subtraction and bitwise complement. ~x + 1, in terms of such a ring, means "flip it, then draw a CCW arrow on it with a length of one tick". ~(x - 1) means "draw a CW arrow with a length of one tick, then flip". Picking CCW first is arbitrary, but the point is that the direction is reversed because flipping the ring also flips the arrow if it is drawn first, but not if it is drawn second. Equivalent to drawing an arrow, you may rotate the ring around its center.

So they're equivalent, but why do they negate. The same effect also explains
a - b = ~(~a + b), which when you substitute a = 0 almost directly gives -b = ~(b - 1). Or using the difference between one's complement and proper negation as I pointed out in that visualization post: the axis of flipping is offset by half a tick, so the effect of flipping introduces a difference of 1 which can be removed by rotating by a tick.

Bit-string notation

I first saw this notation in The Art of Computer Programming v4A, but it probably predates it. It provides a more "structural" view of negation: $$-(a10^k) =\; {\sim} (a10^k - 1) =\; {\sim} (a01^k) = ({\sim} a)10^k$$ Here juxtaposition is concatenation, and exponentiation is repetition and is done before concatenation. a is an arbitrary bit string that may be infinitely long. It does not really deal with the negation of zero, since the input is presumed to end in 10^k, but the negation of zero is not very interesting anyway.

What this notation shows is that negation can be thought of as complementing everything to the left of the rightmost set bit, a property that is frequently useful when working with the rightmost bit. A mask of the rightmost set bit and everything to the right of it can be found with
x ^ (x - 1) or, on a modern x86 processor, blsmsk. That leads to negation by XOR: $$-x = x\oplus {\sim}\text{blsmsk}(x)$$ which is sort of cheating since ~blsmsk(x) = x ^ ~(x - 1) = x ^ -x, so this said that
-x = x ^ x ^ -x. It may still be useful occasionally, for example when a value of "known odd-ness" is being negated and then XORed with something, the negation can be merged into the XOR.

Negation by MUX

Using that mask from blsmsk, negation can be written as $$-x = \text{mux}(\text{blsmsk}(x), {\sim} x, x)$$ which combines with bit-level commutativity in some fun ways:

• (~x + 1) + (x - 1) = mux(blsmsk(x), ~x, x) + mux(blsmsk(x), x, ~x) = ~x + x = -1
• (~x + 1) | (x - 1) = ~x | x = -1
• (~x + 1) ^ (x - 1) = ~x ^ x = -1
• (~x + 1) & (x - 1) = ~x & x = 0

All of these have simpler explanations that don't involve bit-level commutativity, by rewriting them back in terms of negation. But I thought it was nice that it was possible this way too, because it makes it seem as though a +1 and a -1 on both sides of an OR, XOR or AND cancel out, which in general they definitely do not.

The formula that I've been using as an example for the proof-finder on haroldbot.nl/how.html,
(a & (a ^ a - 1)) | (~a & ~(a ^ a - 1)) == -a, is actually a negation-by-MUX, written using mux(m, x, y) = y & m | x & ~m.

Bit-level commutativity

By bit-level commutativity I mean that a binary operator has the property that swapping any subset of bits between the left and right operands does not change the result. The subset may be any old thing, so in general I will call an operator o bit-level commutative if it satisfies the following property $$\forall m,a,b: a \circ b = \text{mux}(m, a, b) \circ \text{mux}(m, b, a)$$ For example, by setting m = b we get a ⊗ b = (a & b) ⊗ (a | b), sort of "bitwise sorting" the operands, with zeroes moved to the left operand and ones moved to the right operand (if possible).

Anyway, obviously AND, OR and XOR (and their complemented versions) are all bit-level commutative, indeed any purely bitwise operation (expressible as a vectorized function that takes two booleans as input) that is commutative is necessarily also bit-level commutative, for obvious reasons. Interestingly, addition is also bit-level commutative, which may be less obvious (at least in a recent coding competition, it seemed that people struggled with this). It may help to consider addition on a slightly more digit-by-digit level: $$ a + b = \sum_i 2^i (a_i + b_i)$$ It should be clear from the bit-level "exploded" sum, that the individual bits a_i and b_i can be either swapped or not, independently for any i. This should get more obvious the more you think about what representing a number in a positional numeral system even means in the first place: it was always a sum, so adding two numbers is like taking the sum of two "big" sums, of course it does not matter which of the big sums any particular contribution comes from.

Alternatively, the old a + b = (a ^ b) + 2(a & b) (ie computing bit-level sums and then adding the carries separately) can explain it: both XOR and AND are bit-level commutative, so the whole expression is, too.

Anyway, a consequence is that a + b = (a & b) + (a | b), which I have more commonly seen derived as:

a + b = (a ^ b) + 2(a & b)  // add carries separately
      = (a | b) - (a & b) + 2(a & b) // see below
      = (a | b) + (a & b)

Where (a ^ b) = (a | b) - (a & b) can be explained as XOR being like OR, except that unlike OR it is 0 when both operands are set, so just subtract that case out. I always like having two (or more!) explanations from completely different directions like that.

Multiplication (including carryless multiplication and OR-multiplication) is of course not bit-level commutative. For example if one operand is zero and the other is odd and not 1, then the lowest bit could be swapped to make neither operand zero, and a non-zero result could be produced that way. Operations such as comparison and (by extension) min and max are obviously not bit-level commutative.

There is probably more to this, I may add some stuff later.