Digital Binary Computation with Fingers and Circuits

CS 301 Lecture, Dr. Lawlor

"Never trust a man that can count to 1023 on his fingers."
- Filippo Gioachin's signature

So here's normal, base-1 counting on your fingers. In base 1, you just raise the number of fingers equal to the value you're trying to represent:

To represent two, raise two fingers.
To represent six, raise six fingers.
To represent 67, grow more fingers.

Base-1 computation on a human hand: all fingers count as 1 unit

This is funky base-2 counting on your fingers. Each finger represents a different value now, so you have to start counting with '1' at your pinky, then '2' with just your ring finger, and '3=2+1' is pinky and ring finger together. '4' is a single raised middle finger. Then '5=4+1' is middle finger and pinky, and so on. Just 10 digits actually allows you to count all the way to 1023, but we'll ignore the thumbs and just use 8 fingers, to count up to 255=128+64+32+16 (left hand palm-up, pinky is 16) +8+4+2+1 (right hand palm-down, pinky is 1).

To represent one, raise the 1 finger.
To represent three, raise the 2 and 1 fingers together.
To represent ten, raise the 8 and 2 fingers together.
To represent twenty, raise the 16 (left pinky) and 4 fingers.
To represent 67, raise the 64 (left middle finger), 2, and 1 fingers.

This is actually somewhat useful for counting--try it!

(Note: the numbers four, sixty-four, and especially sixty-eight should not be prominently displayed. Digital binary counting is not recommended in a gang-infested area.)

Base-2 computation on a human hand: finger values are 8, 4, 2, and 1

Counting on your fingers is "digital" computation--it uses your digits!

Bitwise Operations

So most of the usual work you do in C/C++/Java/C# manipulates integers or strings. For example, you'll write a simple line like:
x = y + 4;
which adds 4 to the value of y.

But sometimes you have to understand how this works internally. For example, on a 32-bit machine, this code returns... 0.

long x=1024;
long y=x*x*x*4;
return y;

(Try this in NetRun now!)

Why? The real answer is 4 billion (and change), which requires 33 bits: a 1 followed by 32 zero bits. But on a 32-bit machine, all you get is the zeros; the higher bits "overflow" and (at least in C/C++) are lost! Understanding the bits underneath your familiar integers can help you understand errors like this one. (Plus, by writing assembly code, you can actually recover the high-order bits after a multiplication if you need them.)

So because bits are important, C/C++/Java/C# include "bitwise" operators that manipulate the underlying bits of the integer. It's like the computer counts on its (32 or 64) fingers, does the operation on those bits, and then converts back to an integer. Except, of course, deep down the hardware only knows about bits, not integers!

Bitwise Left Shift: <<

Makes values bigger, by shifting the value's bits into higher places, tacking on zeros in the vacated lower places.

As Ints	As Bits
3<<0 == 3	0011<<0 == 0011
3<<1 == 6	0011<<1 == 0110
3<<2 == 12	0011<<2 == 1100

Interesting facts about left shift:

1<<n pushes a 1 up into bit number n, creating the bit pattern 100... n zeros here ...00.
The value of (k<<n) is actually k*2ⁿ. This means bit shifting can be used as a faster multiply.
k<<0 == k, for any k.
(k<<n) >= k, for any n and k (unless you have "overflow"!).
On a 32-bit machine, (k<<32) == 0, plus a compiler warning, because all the bits of k have overflowed away.
Left shift always shifts in fresh new zero bits.
You can left shift by as many bits as you want.
You can't left shift by a negative number of bits.

In C++, the << operator is also overloaded for iostream output. I think this was a poor choice, in particular because "cout<<3<<0;" just prints 3, then 0! To actually print the value of "3<<0", you need parenthesis, like this: "cout<<(3<<0);". Operator precedence is screwy for bitwise operators, so you really want to use excess parenthesis!

Bitwise Right Shift: >>

Makes values smaller, by shifting them into lower-valued places. Note the bits in the lowest places just "fall off the end" and vanish.

As Ints	As Bits
3>>0 == 3	0011>>0 == 0011
3>>1 == 1	0011>>1 == 0001
3>>2 == 0	0011>>2 == 0000
6>>1 == 3	0110>>1 == 0011

Interesting facts about right shift:

The value of (k>>n) is actually k/2ⁿ. This can be used as a faster divide.
(k<<n)>>n == k, unless overflow has happened.
Right shift can do strange things to negative values (we'll talk about negative values later).
On a 32-bit machine, (k>>32) == 0, plus a compiler warning, because all the bits of k have fallen off the end.

If you're dyslexic, like me, the left shift << and right shift >> can be really tricky to tell apart. I always remember it like this:

k<<n pumps up the value of k (the point of the << is injecting bigness into k)
k>>n drains away the value of k (the point of the >> is draining bigness from k)

Bitwise AND: &

Output bits are 1 only if both corresponding input bits are 1. This is useful to "mask out" bits you don't want, by ANDing them with zero.

As Ints	As Bits
3&5 == 1	0011&0101 == 0001
3&6 == 2	0011&0110 == 0010
3&4 == 0	0011&0100 == 0000

0=A&0 (AND by 0's creates 0's--used for masking)
A=A&~0 (AND by 1's has no effect)
A=A&A (AND by yourself has no effect)

Bitwise AND is a really really useful tool for extracting bits from a number--you often create a "mask" value with 1's marking the bits you want, and AND by the mask. For example, this code figures out if bit 2 of an integer is set:
    int mask=(1<<2); // in binary: 100
    int value=...;           // in binary: xyz
    if (0!=(mask&value)) // in binary: x00
       ...

In C/C++, bitwise AND has the wrong precedence--leaving out the parenthesis in the comparison above gives the wrong answer! Be sure to use extra parenthesis!

Bitwise OR: |

Output bits are 1 if either input bit is 1. E.g., 3|5 == 7; or 011 | 101 == 111.

As Ints	As Bits
3\|0 == 3	0011\|0000 == 0011
3\|3 == 3	0011\|0011 == 0011
1\|4 == 5	0001\|0100 == 0101

A=A|0 (OR by 0's has no effect)
~0=A|~0 (OR by 1's creates 1's)
A=A|A (OR by yourself has no effect)

Bitwise OR is useful for sticking together bit fields you've prepared separately. Overall, you use AND to pick apart an integer's values, XOR and NOT to manipulate them, and finally OR to assemble them back together.

Bitwise XOR: ^

Output bits are 1 if either input bit is 1, but not both. E.g., 3^5 == 6; or 011 ^ 101 == 110. Note how the low bit is 0, because both input bits are 1.

As Ints	As Bits
3^5 == 6	0011&0101 == 0110
3^6 == 5	0011&0110 == 0101
3^4 == 7	0011&0100 == 0111

A=A^0 (XOR by zeros has no effect)
~A = A ^ ~0 (XOR by 1's inverts all the bits)
0=A^A (XOR by yourself creates 0's--used in cryptography)

The second property, that XOR by 1 inverts the value, is useful for flipping a set of bits.

Bitwise NOT: ~

Output bits are 1 if the corresponding input bit is zero. E.g., ~011 == 111....111100. (The number of leading ones depends on the size of the machine's "int".)

As Ints	As Bits
~0 == big value	~...0000 == ...1111

I don't use bitwise NOT very often, but it's handy for making an integer whose bits are all 1: ~0 is all-ones.

Non-bitwise Logical Operators

Note that the logical operators &&, ||, and ! work exactly the same as the bitwise values, but for exactly one bit. Internally, these operators map multi-bit values to a single bit by treating zero as a zero bit, and nonzero values as a one bit. So (2&&4) == 1 (because both 2 and 4 are nonzero), but (2&4) == 1 (because 2==0010 and 4 == 0100 don't have any overlapping one bits).

Use of Bitwise Operators

Say you're Google. You've got to search all the HTML pages on the net for any possible word. One way to do this is for each possible word, store a giant table of every HTML document on the net (maybe 10 billion documents) containing one bit per document: 1 if the word appears in that document, 0 if the word doesn't appear. This table is 10 billion bits, about 1GB uncompressed, or only a few dozen megabytes compressed. Given two search words, you can find all the pages that contain both words by ANDing both tables. The output of the bitwise AND, where both bits are set to 1, is a new table listing the HTML pages that contain both search terms; now sort by pagerank, and you're done! Note that storing the big table by bits saves a lot of space, and doing a bitwise AND instead of a regular logical AND saves a lot of time (over 10x speedup in my testing!):

enum {n=1}; // Number of integers in our tables (== size of internet / 32)
unsigned int funky_table[n]={(1<<24)|(1<<17)|(1<<12)|(1<<4)};
unsigned int aardvark_table[n]={(1<<31)|(1<<24)|(1<<15)|(1<<6)|(1<<4)};

/* Match up the bits of these two tables using bitwise operations */
void both_tables(const unsigned int *a,const unsigned int *b,unsigned int *o) {
	for (int i=0;i<n;i++) o[i]=a[i]&b[i]; /* bitwise AND */
}

/* Match up the bits of these two tables using logical (one-bit) operations */
void both_tables_logical(const unsigned int *a,const unsigned int *b,unsigned int *o) 
{
	for (int i=0;i<n;i++) {
		o[i]=0;
		for (int bit=0;bit<32;bit++)
		{
			unsigned int a_bit=a[i]&(1<<bit);
			unsigned int b_bit=b[i]&(1<<bit);
			if (a_bit && b_bit) /* logical AND */
				o[i]=o[i]|(1<<bit);
		}
	}
}

int foo(void) {
	unsigned int output_table[n];
	both_tables(funky_table,aardvark_table,output_table);
	return output_table[0];
}

(Try this in NetRun now!)

The same bitwise testing idea shows up in the "region codes" of Cohen-Sutherland clipping, used in computer graphics.

As Ints	As Bits
3\|0 == 3	0011\|0000 == 0011
3\|3 == 3	0011\|0011 == 0011
1\|4 == 5	0001\|0100 == 0101