[Strilanc]

> The given function is clearly only defined over naturals.

The post doesn't specify. Programming languages do define it over the negatives, however. I just didn't want anyone going into python and asserting that a & b >= min(a,b).

> there's no indication of how to encode negatives

Just do the standard thing: use 2s complement arithmetic [1]. More concretely: set b to infinity and take the limit under the 2-adic metric [2]. For example, -2 & -1 has partial sums 0, 10, 110, 1110, 11110, etc. It's limiting to ...1111110, which is the 2's complement representation of -2.

1: https://en.wikipedia.org/wiki/Two%27s_complement

2: https://en.wikipedia.org/wiki/P-adic_number

[thedufer]

The laws given in the post were explicitly derived from the equation. Please explain how you used that to determine f(0, -1)?

[Strilanc]

Pick any `b` you want, even infinity, and evaluate:

    sum_n^b 2^n (floor(0/2^n) mod 2) (floor(-1/2^n) mod 2)

    sum_n^b 2^n * 0 * 1

    sum_n^b 0

    0

[thedufer]

Excellent. I'm glad we agree. You'll notice that the stated identity f(x,x) = x fails to hold over x < 0. I hope this is sufficient to make it clear that the equation does not consider the negatives?

[Strilanc]

Seems to work fine for -1:

    f(-1, -1)
    = sum_{n=0}^b 2^n (floor(-1/2^n) mod 2) (floor(-1/2^n) mod 2)
    = sum_{n=0}^b 2^n * 1 * 1
    = 1 + 10 + 100 + 1000 + ... [in binary]
    = ...11111 [in the 2-adics]
    = -1

[TTPrograms]

I'm not quite sure they have enough paper for numbers with infinite boolean expansions.