Computing via
Look-up Tables:
Combinational Logic
Comp 101 - Spring 2008
1/21/08
L04 - Computing via Table Lookup
To COMPUTE, what does it mean?
• Computing = Processing Information
• A computing system transforms information presented on its
inputs to a result, which it generates on its outputs
fåéìíë
Comp 101 - Spring 2008
ô
Computational
System
or Block
1/21/08
õ
lìíéìíë
L04 - Computing via Table Lookup
What is Digital?
• Computing systems can be either “discrete” or “continuous”
• Inputs and outputs of a continuous system can take be any of
an infinite number of values {ex. a + b = c}
a
b
+
c
• “Discrete” implies that all inputs and outputs can take on one
value of a finite number of values
(ex. digits of a decimal number can take on one of 10 discrete values {0,1,2,3,4,5,6,7,8,9})
• “Digital” implies a that inputs and outputs can take on one of
two possible binary values 0 or 1.
Comp 101 - Spring 2008
1/21/08
Why Digital?
L04 - Computing via Table Lookup
Representing Continuous Information
Let’s use old fashion B&W “Analog” Television as an example. Each picture is
transmitted as a series of left-to-right scan lines where each point (x, y) is
represented as a voltage between 0 and 1 volts:
0 volts:
1 volt:
0.37 volts:
etc.
BLACK
WHITE
37% Gray
Frames are transmitted at a rate of 30 frames
per second, and each is divided into 525 scanlines, where roughly 480 are visible. The details
are slightly more complicated.
Comp 101 - Spring 2008
1/21/08
How much information
at each point?
L04 - Computing via Table Lookup
Information Processing = Computation
First, let’s introduce some processing blocks:
Comp 101 - Spring 2008
v
Copy
v
v
INV
1-v
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L04 - Computing via Table Lookup
Let’s build a system!
input
Copy
INV
Copy
INV
Copy
INV
Copy
INV
(In(Reality)
Theory)
?
output
Comp 101 - Spring 2008
1/21/08
L04 - Computing via Table Lookup
Why Did Our System Fail?
Why doesn’t reality match theory?
1. COPY Operator doesn’t work right
2. INVERSION Operator doesn’t work right
3. Theory is imperfect
4. Reality is imperfect
5. Our system architecture stinks
ANSWER: all of the above!
Noise and inaccuracy are inevitable; we can’t reliably reproduce infinite
information-- we must design our system to tolerate some amount of error if
it is to process information reliablyK
Comp 101 - Spring 2008
1/21/08
L04 - Computing via Table Lookup
The Key to System Design
A SYSTEM is a structure that is guaranteed to exhibit a specified behavior,
assuming all of its components obey their specified behaviors.
How is this achieved?
Contracts
Every system component will have clear obligations and
responsibilities. If these are maintained we have every
right to expect the system to behave as planned. If
contracts are violated all bets are off.
Comp 101 - Spring 2008
1/21/08
L04 - Computing via Table Lookup
The Digital Panacea ...
Why DIGITAL?
… because it keeps the contracts SIMPLE!
What is the price we pay for this robustness?
0 or 1
All the information that we transfer
between components is only 1 crummy bit!
But, in exchange, we get a guarantee of a reliable system.
Comp 101 - Spring 2008
1/21/08
L04 - Computing via Table Lookup
The Digital Abstraction
“Ideal”
Digital World
Real World
0/1
Manufacturing
Variations
Bits
Noise
Volts or
Electrons or
Ergs or Gallons
Remember, the world is not digital, we engineer it to behave that way.
We use real physical phenomena to implement digital designs!
Comp 101 - Spring 2008
1/21/08
L04 - Computing via Table Lookup
Computing Discretely
• How does our decision to compute using a discrete,
digital, information representation impact what we
can compute?
• Do *you* have any previous experience computing in a
discrete world?
Comp 101 - Spring 2008
1/21/08
L04 - Computing via Table Lookup
Grade School Arithmetic
• Despite the fact that you know how to
perform arithmetic on any number, you
*learned* to perform arithmetic using
discrete tricks.
• In fact, most of what you know about
math originated from memorizing a few
tables, which, even today, form the
basis for your ability to do arithmetic
• Computers use tables too.
Comp 101 - Spring 2008
1/21/08
L04 - Computing via Table Lookup
1st Grade Review
• The first table that you learned
was a table for adding any digit to
any other digit
• This table can be represented as a
computational system
digitA
digitB
Comp 101 - Spring 2008
+
carry
digitC
1/21/08
+
0
1
2
3
4
5
6
7
8
9
0
0
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
10
2
2
3
4
5
6
7
8
9
10
11
3
3
4
5
6
7
8
9
10
11
12
4
4
5
6
7
8
9
10
11
12
13
5
5
6
7
8
9
10
11
12
13
14
6
6
7
8
9
10
11
12
13
14
15
7
7
8
9
10
11
12
13
14
15
16
8
8
9
10
11
12
13
14
15
16
17
9
9
10
11
12
13
14
15
16
17
18
L04 - Computing via Table Lookup
Composing Tables
• You learned add larger
numbers, with more digits
by reapplying the table
that you had committed
to memory
digitA10
digitB10
digitA1
digitB1
Comp 101 - Spring 2008
+
+
digitC100
+
digitC10
54
+78
12
12
132
digitC1
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L04 - Computing via Table Lookup
Multiplication
• Another table, another box...
digitA
digitB
x
B C
A
X
D
X
X
+
+
+
digitC10
digitC1
+
+
B
X
AB
X CD
DB
DA
CB
CA
PPPP
x
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
9
2
0
2
4
6
8
10
12
14
16
18
3
0
3
6
9
12
15
18
21
24
27
4
0
4
8
12
16
20
24
28
32
36
5
0
5
10
15
20
25
30
35
40
45
6
0
6
12
18
24
30
36
42
48
54
7
0
7
14
21
28
35
42
49
56
63
8
0
8
16
24
32
40
48
56
64
72
9
0
9
18
27
36
45
54
63
72
81
Product bits
Comp 101 - Spring 2008
1/21/08
L04 - Computing via Table Lookup
A Digital Processing Element
• A combinational device is a circuit element that has
Ó one or more digital inputs
Ó one or more digital outputs
A
B
C
Y
Ó a functional specification that details
the value of each output for every possible
combination of valid input values
(A lookup table)
0
0
0
0
0
0
1
0
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
1
1
1
0
1
1
1
1
1
input A
input B
input C
Comp 101 - Spring 2008
Output a “1” if at
least 2 out of 3 of
my inputs are a “1”.
Otherwise, output “0”.
output Y
1/21/08
L04 - Computing via Table Lookup
Digital Processing Elements
Some digital processing elements occur so frequently that we give them special
names and symbols
A
A
B
I will copy and
restore my input
buffer
to my output
I will only output
a ‘1’AND
if all my
inputs are ‘1’
A
B
Comp 101 - Spring 2008
Y
A
A
Y
B
I will only output a
‘1’ if an XOR
odd number
of my inputs are ‘1’
1/21/08
I will output the
complement of
my input
inverter
I will output a
‘1’ if any
OR of my
inputs are ‘1’
Y
Y
Y
L04 - Computing via Table Lookup
Digital Processing Elements
Some digital processing elements occur so frequently that we give them special
names and symbols
A
Y
buffer
A
B
AND
B
inverter
A
B
Y
A
Comp 101 - Spring 2008
A
XOR
1/21/08
OR
Y
Y
Y
In honor of the richest
man in the world we will
henceforth refer to
digital processing
elements as “GATES”
L04 - Computing via Table Lookup
A Slight Diversion
Are we sure we have all the gates we need?
How many two-input gates are there?
OR
NAND
NOR
SURGE
AND
Hum… all of these have 2-inputs (no surprise)
… 2 inputs have 4 permutations, giving 22 output cases
24
How many permutations of 4 outputs are there? ___
2N
Generalizing, there are 2 , N-input gates!
Comp 101 - Spring 2008
1/21/08
L04 - Computing via Table Lookup
There Are Only So Many Digital Gates
There are only 16 possible 2-input gates
… some we know already, others are just silly
What are the
most common
gates, used by
computers?
The answer
might surprise
you.
Nope. After all, we describe them all using AND, OR, and NOT.
Do we need all of these gates?
Comp 101 - Spring 2008
1/21/08
L04 - Computing via Table Lookup
A Binary Adder
Binary addition used by computers is similar to the the
addition you learned in grade school, except the tables
are simpler.
1 1 0 1
Adding two N-bit
numbers produces
an (N+1)-bit result
A: 1101
B:+ 0101
10010
Carries from
previous column
A
CO
B
FA
CI
S
Let’s start by building a block that adds one column:
Then we can cascade them to add two numbers of any size…
A3 B3
A2 B2
A1 B1
A0 B0
A
CO
A
CO
A
CO
A
CO
B
FA CI
S
S4
S3
Comp 101 - Spring 2008
B
FA CI
S
S0
B
FA CI
S
S1
B
FA CI
S
S0
1/21/08
L04 - Computing via Table Lookup
A Binary Adder Implementation
A binary adder’s function,
as given by its truth table,
can be implemented
“Carry”
using the simpler
Logic
functions, AND,
OR, and XOR.
CO
A
CO
AB
CI
S
B
FA
CI
“Sum”
Logic
S
Comp 101 - Spring 2008
1/21/08
L04 - Computing via Table Lookup
Binary Multiplication
The multiplication table
for binary is much simpler,
than the decimal one.
A multiplier for one bit
can be implemented
with a single gate.
“Binary”
Multiplication
Table
X
0
1
0
0
0
1
0
1
A
B
Y
AND
Moreover, every result
generates only one output
bit (digit). Recall that multiplying two decimal digits might result in a two digit
result. For example, 6 x 7 = 42.
Comp 101 - Spring 2008
1/21/08
L04 - Computing via Table Lookup
Adding Up Product Terms
In binary multiplication, multiplying the bits is the easy part.
The more difficult part
Binary multiplication is implemented using the same basic
is adding up all of the
longhand algorithm that you learned in grade school.
terms resulting from
A
A
A
A
B
B
B
x B
the digit-by-digit
A B A B AB A B
A B is a “partial product”
multiplications.
A B A B AB A B
3
2
1
0
3
2
1
0
2 0
1 0
0 0
2 1
1 1
0 1
A3B2 A2B2 A1B2
A2B3 A1B3 A0B3
A0B2
3 0
j i
3 1
+ A 3B 3
Multiplying N-digit number by M-digit number gives (N+M)-digit result
Easy part: forming partial products (just an AND gate since BI is either 0 or 1)
Hard part: adding M, N-bit partial products
Comp 101 - Spring 2008
1/21/08
L04 - Computing via Table Lookup
Implementing Tables
The BIG lesson: A large fraction of “computation”, in both
humans and computers, is carried out using tables.
How are tables implemented?
Gates
Memory
In fact, there is a
very general purpose
gate, called a
memory.
Comp 101 - Spring 2008
1/21/08
L04 - Computing via Table Lookup
A General Purpose Gate
Every possible 2-input gate can be implemented using the following logic. All
that one needs to do is to set the appropriate memory value, {M00, M01,
M10, M11} to the desired output. This design is not necessarily efficient, but
it will perform any
2-input function.
Decoder M
Selector
A
Moreover, it is
0
B
M
easy to generalize
A
1
Y
this design to any
B
M
A
number of inputs.
2
00
A decoder
generates
all possible
product
terms for
a set of
inputs
Such, reconfigurable
gates are called memories.
01
10
B
A
B
M11
3
RAMs and ROMs are two types.
Comp 101 - Spring 2008
1/21/08
L04 - Computing via Table Lookup
Summary
1. Computer’s process information that takes on discrete values
- If the number of discrete values is limited to 2, the system is called “digital” or
“binary”
- Limiting computation to discrete values makes the system more robust and
reliable
- Fewer discrete values that we have to discriminate between, leads to the simplest
and most reliable systems
2. A large fraction of computation is done using simple table look up
- Humans also learned to compute using tables that we memorized in childhood
- Because data is represented as discrete values, we are able to enumerate every
possible input, and store every possible output in a look-up table
- Much of the logic in a computer is merely implementing look-up tables
Comp 101 - Spring 2008
1/21/08
L04 - Computing via Table Lookup