# The "Logic" part of the ALU

The ALU is the arithmetic-logic unit.  We know that the ALU does arithmetic operations such
as ADD and MULT, but what is this "logic" stuff?

The ALU can carry out operations other than just airthmetic.  It can do comparisons, such as
whether X  is greater than Y, and it also lets you play with bits.  Typical ALUs are 32 bits and
64 bits (more for floating-point operations rather than integer).  Two of the logic operations
are the bitwise OR and the bitwise AND.  Here, the 32-bit "numbers" are not actually
numbers at all--they are what are known as "bit strings".  Each bit in a bit string represents
something--not a power of 2 in this case.  To illustrate, let's have a 16-bit string for UT
students (we're numbering the bits right-to-left from 1 to 16--again, these are NOT powers
of 2).

___  ___  ___  ___     ___  ___  ___  ___    ___  ___  ___  ___    ___  ___  ___  ___
16    15    14    13        12    11   10     9        8      7      6      5        4      3      2      1

meanings: 0 = false  1 = true
bits
______________________________________________________________________
1  grad student                                          9  GPA is 2.00+
2  undergrad                                             10 GPA is 2.50+
3  in-state                                                 11  GPA is 2.75+
4  has hope scholarship                           12  GPA is 3.00+
5  freshman                                              13  GPA is 3.5+
6  sophomore                                            14  GPA is 3.75+
7  junior                                                     15  lives on campus
8  senior                                                     16  on probation

Donnie Wahlberg:  0101  1111   0100   1110:  lives on campus, GPA is 3.5+, junior, hope scholar,
Ludwig Snarf:         1000   0000  1000  0110:  on probation, GPA is less than 2.0 (otherwise bit
9 would be a 1), senior, lives off-campus (otherwise bit 15 would be a 1), in-state, undergrad.

These are patterns of bits that contain information about students.  Note that home address,
etc. are not represented here--that info would be in the student database.

Now--bitwise operations.  Bitwise means that we're not adding, etc--no carry-in or carry-out.
We're comparing/matching two bit patterns, column by column:  what occurs in column 9, for
example, does not affect anything in other columns.  You might think of 1 as being "true", 0
as being "false".

BITWISE AND.  If both bits being compared are 1's, the bitwise result is a 1--otherwise the
result is 0.  Think of someone saying  "I'm a UT student and my GPA is 3.9".  Is the student
telling the truth?  Suppose the GPA is indeed 3.9 but the student is at Pellissippi, not UTK.
Or suppose the student is at UTK but the GPA is 2.3.  Both parts must be true for the whole
statement to be true.

BITWISE OR.  If either (or both) bits is a 1, the result is 1.  Only if both bits are 0 is the result
0.  "My GPA is 3.9 or I'm a liar".  If either part is true, the statement is true.  Only if both parts
are false is the statement false (this could be confusing--if the statement is false, the person
making it is lying--so perhaps "My GPA is 3.9 or I always lie" would be better).

1011               1001
AND  0110        OR 1100
---------        --------
0010               1101

A "BIT MASK" is a particular pattern of bits that we're looking for.   Suppose we want to know
whether a student can continue their Hope scholarship (I think this now requires a 2.75 GPA).
To continue the scholarship, bits 2, 3, and 4 must be 1's (undergrad, in-state, already has a Hope
scholarship (??) and bit 11 must also be a 1 (GPA is 2.75 or better).  So for each student we will
do a bitwise AND with the following pattern:

0000  0100  0000  1110

Donnie Wahlberg:    0101   1111  0100   1110               Ludwig Snarf:   1000  0000  1000  0110
bit mask          :    0000   0100  0000   1110                         mask     :    0000  0100  0000  1110
AND result     -------------------------                ---------------------------
0000   0100   0000  1110                                            0000  0000  0000  0110

If we now did a comparison, treating the results as numbers, if we subtract the mask from the
result, in Wahlberg's case the result is 0--meaning all the right bits were 1's, and the result
in Snarf's case is negative--meaning at least one bit which should have been a 1 is a zero--
and hence fails to match the criteria.

--------------------------------------------------------------------------
BIT SHIFTS
The ALU can also take a bit string (or number) and perform a shift operation--meaning that
each bit gets shifted left or right by some number of bits:  a 1-bit left  shift of 001101 would
produce 011010, for example.  This may not seem very useful.  But suppose we have a true-
false exam with 32 questions, and we have a resulting 32-bit pattern showing which questions
the student answered correctly.  For a 6-question TF test, 110100 would mean the student
answered questions 3, 5, and 6 correctly, 1, 2, and 4 were incorrect.  So how could we figure
out how many questions the student answered correctly?  We can use bit shifts and a bit mask,
for example.

110100
AND   000001
-------------
This looks only at the rightmost bit--if the bitwise AND is a 000001, we have a correct answer
and we add 1 to a running total of correct answers.  We then shift the pattern 110100 right 1 bit
and do another bitwise AND.  Each time the bitwise AND is 1, we add 1 to our total--this adds up
all the 1's in the pattern.  To answer your unasked question--"Yes, we could also have shifted the