Expressing bitwise operations in terms of other functions

Question

I'm asking in the spirit of these two questions: can bitwise operations (AND, OR, XOR) be expressed in terms of other (more familiar?) functions?

I had been playing around with the bitwise operations in Mathematica a while back, and was at first struck by this identity for bitwise NOT: ~ n == -1 - n (I'll use C-ish notation/syntax for this question, but I'm using Mathematica's definitions, which assume a two's complement representation for negative integers.)

Try as I might, I have not managed to figure out alternative ways of expressing the other bitwise operations. I have also tried Using The Fabulous Search Engine to see if other people have looked into this, but I probably am not using the right keywords.

I have noticed that the implementation of the bitwise operators in Mathematica satisfies de Morgan's theorem:

i | j == -1 - ((-1 - i) & (-1 - j))

i & j == -1 - ((-1 - i) | (-1 - j))

and the following identity for bitwise XOR is satisfied as well:

i ^ j == (i | j) & (-1 - (i & j))

I suppose then that any alternate expression for the bitwise operators would have to satisfy these identities as well.

(I am not intending to replace the bitwise expressions in actual programming, of course; they are already optimized, and there is really no practical reason to replace them with "more mathematical" expressions. I am asking merely for curiosity's sake.)

References and sundry information will be appreciated.

Answer 1

If you are asking for a simple mathematical identity in addition, subtraction, multiplication and division, then I am afraid that I do not know of any other than those already mentioned. Since you mentioned, however, that you were asking "merely for curiosity's sake", I will attempt to satisfy at least some of that curiosity.

Since bitwise operations are, after all, iterative, there exists for all bitwise operations a sequential sum that, for each given input, returns an identical output to that of the bitwise operation. Unfortunately, these are computationally incredibly expensive, but will serve all the same as a purely mathematical solution.

The mathematical equivalents to right- and left-shifts are already widely known, so I will not repeat them here. The simplest of the sequential sums that I could find, following these, was that of a bitwise NOT. While you have already stated that it can be expressed as $-1 - n$, this is more of a hack exploiting the fact that $-1$ (in 32-bit) is equal to 0xFFFFFFFF (hexadecimal, continuing your precedent of c-style notation). NOT, mathematically, can be expressed as follows: $$ \operatorname{not}(a)=\sum_{n=1}^{\left \lfloor \log_{2}(a) \right \rfloor} \left [ n \left ( 1 - \left ( \left \lfloor \frac{a}{2^{n}} \right \rfloor \bmod 2 \right ) \right ) \right ] $$

I will now move fairly quickly through the remaining bitwise operations.

OR: $$ \operatorname{or}(a,b)=\sum_{n=1}^{\left \lfloor \log_{2}(a) \right \rfloor} \left [ n \left \lceil \frac{ \left ( \left \lfloor \frac{a}{2^{n}} \right \rfloor \bmod 2 \right ) + \left ( \left \lfloor \frac{b}{2^{n}} \right \rfloor \bmod 2 \right ) }{2} \right \rceil \right ] + b - b \bmod 2^{\left \lfloor \log_{2}(a) \right \rfloor} $$

AND (using the floor of an addition): $$ \operatorname{and}_1(a,b)=\sum_{n=1}^{\left \lfloor \log_{2}(a) \right \rfloor} \left [ n \left \lfloor \frac{ \left ( \left \lfloor \frac{a}{2^{n}} \right \rfloor \bmod 2 \right ) + \left ( \left \lfloor \frac{b}{2^{n}} \right \rfloor \bmod 2 \right ) }{2} \right \rfloor \right ] $$

AND (using multiplication): $$ \operatorname{and}_2(a,b)=\sum_{n=1}^{\left \lfloor \log_{2}(a) \right \rfloor} \left [ n \left ( \left \lfloor \frac{a}{2^{n}} \right \rfloor \bmod 2 \right ) \left ( \left \lfloor \frac{b}{2^{n}} \right \rfloor \bmod 2 \right ) \right ] $$

XOR: $$ \operatorname{xor}(a,b)=\sum_{n=1}^{\left \lfloor \log_{2}(a) \right \rfloor} \left [ n \left ( \left ( \left ( \left \lfloor \frac{a}{2^{n}} \right \rfloor \bmod 2 \right ) + \left ( \left \lfloor \frac{b}{2^{n}} \right \rfloor \bmod 2 \right ) \right ) \bmod 2 \right ) \right ] + b - b \bmod 2^{\left \lfloor \log_{2}(a) \right \rfloor} $$

I am aware of the fact that this thread has not seen any activity in a while, but I myself spent some time looking (unsuccessfully) on the internet for this before deciding to work it out myself. Since I did work this out by hand, however, there may be errors in the formulas. If anyone finds any, please let me know of them, and I will attempt to fix them.

Note that I could not find any mathematical way of performing these operations without the use of modulus, floors, ceilings and the sequential additions. If anyone finds any alternative methods of solving these, I would be interested in hearing of them (hypothetically, substitution of the equation for triangular numbers into the sequential sums may be able to simplify them).

Also note that, programatically, the floor and ceiling functions can be replaced by integer operations.

Hopefully somebody, sometime, will find a use for these equations.

In case anybody is wondering, the reason why I put these together was to discover if it is possible to apply calculus to bitwise operations. It may or may not be, depending on whether or not sequential sums, floors, ceilings and modulo operations can be used in calculus.

Answer 2

I have no references; I've just puzzled a bit on this nice question. I'm not sure the result resembles what you were looking for, but one simple way is to use induction to define bitwise and.

$ \newcommand{\bnot}{\mathop{\boxed{\lnot}}} \newcommand{\band}{\mathbin{\boxed{\land}}} \newcommand{\bor}{\mathbin{\boxed{\lor}}} \newcommand{\bxor}{\mathbin{\boxed{\not\equiv}}} \newcommand{\beq}{\mathbin{\boxed{\equiv}}} \newcommand{\divtwo}[1]{\text{div2}(#1)} \newcommand{\modtwo}[1]{\text{mod2}(#1)} $In this answer $\;a,b\;$ range over $\;\mathbb Z\;$, and for consistency I will write $\;\bnot a\;$ for bitwise negation, and similarly $\;a \band b\;$, $\;a \bor b\;$, $\;a \bxor b\;$, and $\;a \beq b\;$ for bitwise and, or, xor, and equivalence, all of which are again values in $\;\mathbb Z\;$.

Of course we have the standard 2-complements negation $$ \bnot a \;=\; -a-1 $$

And assuming we have $\;\band\;$, then $\;\bor\;$ can be derived by the nice $$ a + b \;=\; (a \band b) + (a \bor b) $$ (This should be equivalent to DeMorgan's laws, using the above definition of $\;\bnot\;$.)

And for equivalence and xor we have the standard \begin{align} a \beq b & \;=\; (a \band b) \bor \bnot(a \bor b) \\ a \bxor b & \;=\; \bnot (a \beq b) \\ \end{align}

Finally, to define $\;\band\;$ inductively, we note that every $\;a\;$ can uniquely be split into its "leading part" and its "least significant bit": $$ a \;=\; \divtwo{a} \times 2 + \modtwo{a} $$ where $$ \divtwo{a} = \lfloor a/2 \rfloor $$ As far as I can see, $\;\band\;$ is now fully defined by \begin{align} \divtwo{a \band b} & \;=\; \divtwo{a} \band \divtwo{b} \\ \modtwo{a \band b} & \;=\; \modtwo{a} \times \modtwo{b} \\ \end{align}

I haven't done any proofs yet, but I think the above equations fully define the bitwise operators on $\;\mathbb Z\;$. I may be missing some details, though.

So in summary, this means that the bitwise operators can be inductively defined in terms of addition, subtraction, multiplication, and 'floored division' by 2.

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In principle, this is no different than the infinite sums of the first answer

Answer 3

Exclusive OR is addition in $\mathbb{Z}_2^n$. The AND operation corresponds to set intersection, OR to set union, NOT to complementation (and XOR to symmetric difference).

Answer 4

Here are some:

• a+b = a^b + 2 a&b
• a^b = a|b - a&b
• a^b >= |a-b|

Answer 5

I tested Mmarss functions, but "OR" and "XOR" didn't work. I played a bit and managed to get these two improvements for them.

XOR: $$ \operatorname{xor}(a,b)=\sum_{n=1}^{\left \lfloor \log_{2}(max(a,b)) \right \rfloor} \left [ 2^{n} \left ( \left ( \left ( \left \lfloor \frac{a}{2^{n}} \right \rfloor \bmod 2 \right ) + \left ( \left \lfloor \frac{b}{2^{n}} \right \rfloor \bmod 2 \right ) \right ) \bmod 2 \right ) \right ] + (a+b)\bmod 2 $$

OR: $$ \operatorname{or}(a,b)= \operatorname{xor}(a,b) +\operatorname{and}(a,b) $$

Answer 6

Looking at VZAP's response, first it appears to be correct and work, and secondly I believe the equation can be simplified as such:

$$\operatorname{xor}(a,b)=\sum_{n=0}^{⌊log_2(max(a,b))⌋} \left[2^n\left(\left(⌊\frac{a}{2^n}⌋+⌊\frac{b}{2^n}⌋\right) \pmod 2\right)\right].$$

as at $n=0,$ the summation would be $((a+b)\pmod2)$, the last part of the equation. Also, $[(a\mod2)+ (b\mod2)]\mod2 = (a + b)\pmod 2$.

Let me know if there are any issues with this formula!

Answer 7

Here I provide a PARI/GP code for Bitwise AND operator:

AND(A,B) = sum(n=0,floor(log(A)/log(2)), 2^n*((floor(A/2^n)%2)*(floor(B/2^n)%2)))

Notice there are slightly changes, sum starting at $0$, and $n$ factor to $2^n$.