bitwise not - not what I expected

[Elaine Jackson]

Is there a function that takes a number with binary numeral a1...an to the
number with binary numeral b1...bn, where each bi is 1 if ai is 0, and vice
versa? (For example, the function's value at 18 [binary 10010] would be 13
[binary 01101].) I thought this was what the tilde operator (~) did, but when I
went to try it I found out that wasn't the case. I discovered by experiment (and
verified by looking at the documentation) that the tilde operator takes n
to -(n+1). I can't imagine what that has to do with binary numerals. Can anyone
shed some light on that? (In case you're curious, I'm writing a script that will
play Nim, just as a way of familiarizing myself with bitwise operators. Good
thing, too: I thought I understood them, but apparently I don't.)

Muchas gracias for any and all helps and hints.

Peace,
EJ

 

[Graham Fawcett]

Is there a function that takes a number with binary numeral a1...an to the
number with binary numeral b1...bn, where each bi is 1 if ai is 0, and vice
versa? (For example, the function's value at 18 [binary 10010] would be 13
[binary 01101].) I thought this was what the tilde operator (~) did, but when I
went to try it I found out that wasn't the case. I discovered by experiment (and
verified by looking at the documentation) that the tilde operator takes n
to -(n+1). I can't imagine what that has to do with binary numerals.

It has a lot to do with binary! Google for "two's complement".

In the meantime, try this:
13

The '~' operator cannot care about precision -- that is, how many bits
you're operating on, or expecting in your result. In your example, you
represent decimal 18 as '10010', but '000000010010' is also correct,
right?

In two's complement math, both inverses, '01101' and '111111101101'
respectively, are equivalent to decimal -19.

And-ing with a mask that is the of length 'n' will ensure that you only
get the least significant n bits -- and this is what you're looking for.
Since you're operating on five bits in your example, I chose decimal 31,
or '11111'.

-- Graham

 

[Michael Peuser]

....

I'm writing a script that will
play Nim, just as a way of familiarizing myself with bitwise operators. Good
thing, too: I thought I understood them, but apparently I don't.)

Hi Elaine,
You have described a general misconception you seem to be not the only one
to live with.
The enlightening answers having been posted might sufficwe, but a schuld
liek to add some more "enlightenment":
Bit complements have a lot to do with set complents and the aritmetic
negation (sometimes called two's complement for obvious reasons). Consider
the set of of "red" of "blue". Now whats the complement? "green" and
"yellow" is obviously the wrong answer. You in fact cannot give any answer
befor you define the total set you are dealing with. The same applies to
logical bit operations. Generally you take a "processor" word or a part of
it to be defined. Some high level languages are more flexible; and even some
computers ("vector processors") are.

The only rule is, that ~(~x) == x

The same situation with numbers: What is the negation of +5. You have to
think very hard! This is a trick question and you probably will give a
"trick answer": -5. You should be aware that this is just a trick. "-5"
contains no other information as that it is some "complement" of 5. (same
with complex "imaginary" numbers: 5j (in Python) just says it is some fancy
5.)

Now we define a transformation between positive numbers and bit patterns 5 =
LoL. Note that 5 == ...000005 or LoL == ....ooooLoL does not help any
understanding so you generally skip this part.

Now you do some arithmetic "inversion": 5 -> -5 This however can (and
should) stay a secret of the processor! By no means should you be interested
in how the machine represents "-5". If you are courious then know that
there had been times when computers represente -5 as ...LLLoLo. Yes it
worked! And you had two diffrent "zeros" then: +0 and -0 !!!!

Most computers do not distinguish between the representation of negativ
numbers and complemented sets (let alone note a special "total set" the
complemt was referring to). Thus the "secret" of modern two's-complement
computern arithmetic is always disclosed to you.

Note that there is no use in something like "masking" the MSB, i.e. that
bits-complements only work on 31 bits. This will lead to ~5 == ~LoL == ~
LL..LLLLoL == 2,147,483,643 Not much improvement, eh!?

Kindly
Michael P

 

[Irmen de Jong]

While others explained how the ~ operator works, let me suggest
another possibility: the bitwise exclusive or.
.... l = ['0000', '0001', '0010', '0011', '0100', '0101', '0110', '0111',
.... '1000', '1001', '1010', '1011', '1100', '1101', '1110', '1111']
.... s = ''.join(map(lambda x, l=l: l[int(x, 16)], hex(i)[2:]))
.... if s[0] == '1' and i > 0:
.... s = '0000' + s
.... return s
....

You still have to think about the number of bits you want to invert.
x ^ 0x1f inverts the 5 least significant bits of x.
x ^ 0xff inverts the 8 least significant bits of x, and so on.


--Irmen de Jong

 

[Dennis Lee Bieber]

Elaine Jackson fed this fish to the penguins on Saturday 16 August 2003
09:58 pm:

Is there a function that takes a number with binary numeral a1...an to
the number with binary numeral b1...bn, where each bi is 1 if ai is 0,
and vice versa? (For example, the function's value at 18 [binary
10010] would be 13
[binary 01101].) I thought this was what the tilde operator (~) did,
[but when I
went to try it I found out that wasn't the case. I discovered by
experiment (and verified by looking at the documentation) that the
tilde operator takes n to -(n+1). I can't imagine what that has to do
with binary numerals. Can anyone shed some light on that? (In case
you're curious, I'm writing a script that will play Nim, just as a way
of familiarizing myself with bitwise operators. Good thing, too: I
thought I understood them, but apparently I don't.)

Muchas gracias for any and all helps and hints.

You've had lots of answers at the moment though I haven't seen anyone
explain away the "+1" part...

Most computers use twos-complement arithmetic to avoid the problem of
having two valid values for integer 0, which is what appears in ones
complement arithmetic.

For argument, assume an 8-bit integer. The value of "5" would be
represented as 00000101. The one's complement negative would be
11111010. So far there isn't any problem... But consider the value of
0, represented as 00000000. A one's complement negative would become
11111111 -- But mathematically, +0 = -0; in one's complement math, this
does not hold true.

So a little trick is played, to create twos complement... To negate a
number, we take the ones complement, and then add 1 to the result. The
"5" then goes through: 00000101 -> 11111010 + 1 -> 11111011... Looks
strange, doesn't it... But watch what happens to that 8-bit 0: 00000000
-> 11111111 + 1 -> (overflows) 00000000.... Negative 0 is the same as
positive 0.

So when you complemented your number, you first neglected to take into
account that you complement the entire bit width, including all those 0
bits to the left, and then when displaying the result, were confused by
what the computer does to display... Namely, seeing a MSB set to 1, it
interpreted the result as a negative number, put out a "-" sign, then
generated a twos complement to create a positive value for output. The
twos complement has that +1 step, so the ones complement "18" became
"19"


--

 

[Michael Peuser]

Elaine Jackson fed this fish to the penguins on Saturday 16 August 2003
09:58 pm:

Is there a function that takes a number with binary numeral a1...an to
the number with binary numeral b1...bn, where each bi is 1 if ai is 0,
and vice versa? (For example, the function's value at 18 [binary
10010] would be 13
[binary 01101].) I thought this was what the tilde operator (~) did,
[but when I
went to try it I found out that wasn't the case. I discovered by
experiment (and verified by looking at the documentation) that the
tilde operator takes n to -(n+1). I can't imagine what that has to do
with binary numerals.

[..]

You've had lots of answers at the moment though I haven't seen anyone
explain away the "+1" part...

Most computers use twos-complement arithmetic to avoid the problem of
having two valid values for integer 0, which is what appears in ones
complement arithmetic.

For argument, assume an 8-bit integer. The value of "5" would be
represented as 00000101. The one's complement negative would be
11111010. So far there isn't any problem... But consider the value of
0, represented as 00000000. A one's complement negative would become
11111111 -- But mathematically, +0 = -0; in one's complement math, this
does not hold true.

So a little trick is played, to create twos complement... To negate a
number, we take the ones complement, and then add 1 to the result. The
"5" then goes through: 00000101 -> 11111010 + 1 -> 11111011... Looks
strange, doesn't it... But watch what happens to that 8-bit 0: 00000000
-> 11111111 + 1 -> (overflows) 00000000.... Negative 0 is the same as
positive 0.

[..]

I have the impression (may be wrong) that you are working under the
misconception that there can be a "natural" binary represensation of
negative numbers!?
Three conventions have commonly been used so far:
1- Complement
2-Complement
Sign tag plus absolut binary values

All of them have their pros and cons. For a mixture of very technical
reasons (you mentioned the +0/-0 conflict, I might add the use of binary
adders for subtraction) most modern computers use 2-complement, and this now
leads to those funny speculations in this thread. ;-)

Kindly
Michael P

 

[Dennis Lee Bieber]

Michael Peuser fed this fish to the penguins on Sunday 17 August 2003
02:41 pm:

I have the impression (may be wrong) that you are working under the
misconception that there can be a "natural" binary represensation of
negative numbers!?

Apologies if I gave that impression... the +/- 0 technical affair is
the main reason I went into the whole thing...

Three conventions have commonly been used so far:
1- Complement
2-Complement
Sign tag plus absolut binary values

All of them have their pros and cons. For a mixture of very technical
reasons (you mentioned the +0/-0 conflict, I might add the use of
binary adders for subtraction) most modern computers use 2-complement,
and this now leads to those funny speculations in this thread. ;-)

From a human readable standpoint, your third option is probably the
most "natural"; after all, what is -19 in human terms but a "pure" 19
prefaced with a negation tag marker... (I believe my college
mainframe's BCD hardware unit actually put the sign marker in the
nibble representing the decimal point location -- but it has been 25
years since I had to know what a Xerox Sigma did for COBOL packed
decimal <G>).

ie, 00010011 vs -00010011 <G>

1s complement is electrically easy; just "not" each bit.

2s complement is mathematically cleaner as 0 is 0, but requires an
adder to the 1s complement circuit... Though both complement styles
lead to the ambiguity of signed vs unsigned values

--

 

[Elaine Jackson]

| I have the impression (may be wrong) that you are working under the
| misconception that there can be a "natural" binary represensation of
| negative numbers!?
| Three conventions have commonly been used so far:
| 1- Complement
| 2-Complement
| Sign tag plus absolut binary values

The last alternative sounds like what I was assuming. If it is, I would argue
that it's pretty darn natural. Here's a little function to illustrate what I
mean:

def matilda(n): ## "my tilde"
if 0<=n<pow(2,29):
for i in range(1,31):
iOnes=pow(2,i)-1
if n<=iOnes:
return iOnes-n
else:
raise

 

[Grant Edwards]

I have the impression (may be wrong) that you are working under the
misconception that there can be a "natural" binary represensation of
negative numbers!?
Three conventions have commonly been used so far:
1- Complement
2- Complement
Sign tag plus absolut binary values

All of them have their pros and cons. For a mixture of very technical
reasons (you mentioned the +0/-0 conflict, I might add the use of binary
adders for subtraction)

The latter is _far_ more important than the former. Being able
to use a simple binary adder to do operations on either signed
or unsigned values is a huge savings in CPU and ISA design. I
doubt that anybody really cares about the +0 vs. -0 issue very
much (IEEE FP has zeros of both signs, and nobody seems to
care).

 

[Michael Peuser]

Everyone says "two's complement" and then usually starts talking about numbers
that are bigger than two. I'll add another interpretation, which is what I thought
when I first heard of it w.r.t. a cpu that was designed on the basis that all its
"integer" numbers were fixed point fractions up to 0.9999.. to whatever precision
the binary fractional bits provided. There was no units bit. And if you took one
of these fractional values 0.xxxx and subtracted it from 2.0, you would have a
complementary number with respect to two. Well, for addition and subtraction, that turns
out to work just like the "two's complement" integers we are used to. But since the
value of fractional bits were all in negative powers of two, squaring e.g., .5 had
to result in a consistent representation of 0.25 -- i.e. in binary squaring 0.1
resulted in 0.01 -- which is shifted one bit from what you get looking at the numbers
as integers with the lsb at the bottom of the registers and the result.

I.e., a 32-bit positive integer n in the fractional world was n*2**-31. If you square
that for 64 bits, you get n**2, but in the fractional world that looks like (n**2)*2**-63,
where it's supposed to be (n*2**-31)**2 => (n**2)*2**-62 with respect to the binary point.
The fractional model preserved an extra bit of precision in multiplies.

So on that machine we used to count bits from the left instead of the right, and place imaginary
binary points in the representations, so a binary 0.101 could be read as "5 at 3" or "2.5 at 2"
or "10 at 4" etc. And the multiplying rule was x at xbit times y at ybit => x*y at xbit+ybit.

You can do the same counting the bit positions leftwards from lsb at 0, as we usually do now,
of course, to play with fixed point fractions. A 5 at 0 is then 1.25 at 2 ;-)

Anyway, my point is that there was a "two's complement" implementation that really meant
a numeric value complement with respect to the value two ;-)

Regards,
Bengt Richter

A very good point! I might add that this is my no means an exotic feature.
Mathematically speaking there is great charme in computing just inside the
invervall (-1,+1). And if you have no FPU you can do *a lot* of pseudo real
operations. You have get track of the scale of course - it is a little bit
like working with sliding rules if anyone can remember those tools ;-)

Even modern chips have support for this format, e.g. there is the 5$ Atmel
Mega AVR which has two kinds of multiplication instructions: one for the
integer multiplication and one which automatically adds a left shift after
the multiplication! I leave it as an exercise to find out why this is
necessary when multiplying fractional numbers ;-)

Negative numbers are formed according to the same rule for fractionals and
integers:
Take the maximum positive number: 2**32-1 or 0.999999
Extend your scope
Add one bit: 2*32 or 1
Double it: 2*33 or 2
Subtract the number in question
Reduce your scope again

Kindly Michael P

 

[Grant Edwards]

A very good point! I might add that this is my no means an exotic feature.
Mathematically speaking there is great charme in computing just inside the
invervall (-1,+1). And if you have no FPU you can do *a lot* of pseudo real
operations. You have get track of the scale of course - it is a little bit
like working with sliding rules if anyone can remember those tools ;-)

Sure. I've got two sitting at home.

FWIW, it used to be fairly common for process-control systems
to define operations only over the interval (-1,+1). This made
implimentation easy, and the input and output devices
(temp/pressure sensors, valves, whatnot) all had pre-defined
ranges that mapped logically to the (-1,+1) interval.

 

[Michael Peuser]

Sure. I've got two sitting at home.

FWIW, it used to be fairly common for process-control systems
to define operations only over the interval (-1,+1). This made
implimentation easy, and the input and output devices
(temp/pressure sensors, valves, whatnot) all had pre-defined
ranges that mapped logically to the (-1,+1) interval.

--

Yes it simplifies a lot of matters, even when using full floating point
numbers. Take OpenGL e.g. The colour space is a 1x1x1 cube. Very fine! No
magic numbers near 256 ;-)

Kindly
Michael P

 

[Dennis Lee Bieber]

Grant Edwards fed this fish to the penguins on Monday 18 August 2003
06:11 am:

Sure. I've got two sitting at home.

Only two? <G>

I've got five (the most recent, new-in-box, cost me more than I paid
for an HP25 back in 1978 <G>)... Regretably, I missed my chance at a
lovely plastic over bamboo laminate back then... As I recall, a
deci-trig log-log model, being cleared out by my college bookstore at
half price (which put it about $25 -- the HP25 was $100 or so, and also
a clear-out as no one else was smart enough to buy an RPN calculator).

--