1
i206: Lecture 3:
Boolean Logic, Logic Circuits
Marti Hearst
Spring 2012
2
Today
•
Quick Binary Exercise
•
Boolean Logic
–
deMorgan’s Law
•
How Bits are Made
•
Relating Boolean Logic to Circuits
–
Logic gates
3
Review Binary
•
Binary used as a code
•
Can use binary to encode a game
representation.
Shapes
Actions
Colors
4
Review Binary
•
Two approaches: modular and efficient
•
Modular:
–
Shape bits, then color bits, then action bits
•
00 111 1 (circle, blue, jump)
–
Allows for expansion if you allow extra bits
–
Make it work for 7 colors:
•
00 1000 1
•
Efficient:
–
Minimize the number of bits needed
–
There are a total of 5 * 3 * 2 = 30 unique states
•
So minimum number of bits = 5?
–
6 bits can ?support 7 * 3 * 2 = 42, for 7 colors, but
would not be modular representation.
Shape: 3 (2
bits)
Color: 5 (3 bits)
Action: 2 (1 bit)
5
Boolean Logic
CPU Operation
Bits & Bytes
Binary
Numbers
Number
Systems
Gates
Boolean
Logic
Circuits
CPU
Machine
Instructions
Assembly
Instructions
Program
Algorithms
Application
Memory
Data
compression
Compiler/
Interpreter
Operating
System
Data Structures
Analysis
I/O
Memory
hierarchy
Design
Methodologies/
Tools
Process
Truth table
Venn Diagram
DeMorgan
’
s Law
Numbers, text,
audio, video,
image, …
Decimal,
Hexadecimal,
Binary
AND, OR, NOT,
XOR, NAND, NOR,
etc.
Op

code, operands
Instruction set arch
Lossless v.
lossy
Info entropy &
Huffman code
Adders, decoders,
Memory latches,
ALUs, etc.
Data
Representation
Data
Data
storage
Principles
Network
Distributed
Systems
Security
Cryptography
Standards
& Protocols
Inter

process
Communication
Formal models
Finite automata
regex
6
How Do Computers Work?
We are going bottom

up, with digressions
Bits & Bytes
Binary Numbers
Number Systems
Orders of Magnitude
Gates
Boolean Logic
Circuits
CPU
Machine Instructions
Assembly Language
Programming Languages
Address Space
Code vs. Data
7
Boolean Logic
•
Boolean algebra
= logical calculation of truth values
(true/false values)
•
Basic boolean operators
–
AND
–
OR
–
XOR (exclusive

OR)
–
NOT
8
Boolean Logic
•
Where else have we seen this?
•
Queries!
–
(“apple” OR “pear” OR “berry”) AND (“juice” OR “smoothie”)
•
Venn Diagrams show Union and Intersection
9
Simple query language:
Boolean
–
Terms + Connectors (or
operators
)
–
terms
•
words
•
normalized (stemmed) words
•
phrases
•
thesaurus terms
–
connectors
•
AND
•
OR
•
NOT
10
Boolean Queries
•
Cat
•
Cat
OR
Dog
•
Cat
AND
Dog
•
(Cat
AND
Dog)
•
(Cat
AND
Dog)
OR
Collar
•
(Cat
AND
Dog)
OR
(Collar
AND
Leash)
•
(Cat
OR
Dog)
AND
(Collar
OR
Leash)
11
Boolean Queries
•
(Cat OR Dog) AND (Collar OR Leash)
–
Documents (columns) containing each of the following
combinations satisfy this Boolean query:
•
Cat
x
x
x
x
•
Dog
x
x
x
x
x
•
Collar
x
x
x
x
•
Leash
x
x
x
x
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Boolean Queries
•
(Cat OR Dog) AND (Collar OR Leash)
–
None of the following combinations work; no documents
returned:
•
Cat
x
x
•
Dog
x
x
•
Collar
x
x
•
Leash
x
x
13
Boolean Notation
U
Operator
Symbol
Example
AND
*
or • or
A*B = A•B = A B = AB
OR
+ or U
A+B = A U B
NOT
¬ or ¯ or

¬(A+B) = (A+B) =

(A+B)
XOR
A B = AB + AB
+
U
+
14
Boolean Queries
–
Usually expressed as INFIX operators in IR
•
((a AND b) OR (c AND b))
–
NOT is UNARY PREFIX operator
•
((a AND b) OR (c AND (NOT b)))
–
AND and OR can be n

ary operators
•
(a AND b AND c AND d)
–
Some rules

(De Morgan revisited)
•
NOT(a) AND NOT(b) = NOT(a OR b)
•
NOT(a) OR NOT(b)= NOT(a AND b)
•
NOT(NOT(a)) = a
15
Boolean Logic Relationships?
A
B
C
A
B
D
A
B
D
e
M
o
r
g
a
n
'
s
L
a
w
s
A
B
A
B
A
B
A
B
C
?
D
Union (OR)
Intersection (AND)
Venn Diagram
16
Boolean Searching
“
Measurement of the
width of cracks in
prestressed
concrete beams
”
Formal Query:
cracks
AND
beams
AND
Width_measurement
AND
Prestressed_concrete
C
racks
B
eams
W
idth
measurement
P
restressed
concrete
17
Boolean Logic and Venn Diagrams
t
3
t
1
t
2
D
1
D
2
D
3
D
4
D
5
D
6
D
8
D
7
D
9
D
10
D
11
m
1
m
2
m
3
m
5
m
4
m
7
m
8
m
6
m
2
=
t
1
t
2
t
3
m
1
=
t
1
t
2
t
3
m
4
=
t
1
t
2
t
3
m
3
=
t
1
t
2
t
3
m
6
=
t
1
t
2
t
3
m
5
=
t
1
t
2
t
3
m
8
=
t
1
t
2
t
3
m
7
=
t
1
t
2
t
3
18
Boolean Logic and Logic Gates
•
AND, OR, NOT, NOR, NAND, XOR
•
Boolean Logic can be expressed as a Logic Gate
•
Gates are devices that produce the outputs of Boolean
operations when given the operations
’
input values
–
Often implemented as electronic circuits called transistors
–
Provide the building blocks from which computers are
constructed
•
Gates operate on inputs of 0 and 1
•
Gates are used to build up circuits that perform computations (e.g.,
addition, subtraction), or store values to be used later
19
AND, OR, XOR, and NOT:
Gates and Truth Tables
20
Exercise
•
Let
’
s solve a problem using Boolean Logic
–
First express the problem using Boolean expressions and/or truth
table
–
Then implement solution using Boolean gates
21
Making Bits
http://www2.electronicproducts.com/Debugging_common_digital_problems_with_a_scope

article

farc_tek_jul2011

html.aspx
22
Making Bits
•
Digital signals are interpreted by a system as
high and low states,
–
above and below a defined voltage threshold.
•
In practice, especially at higher speeds, digital
signals are rarely ideal with flat high and low
levels and clean edge transitions.
http://www2.electronicproducts.com/Debugging_common_digital_problems_with_a_scope

article

farc_tek_jul2011

html.aspx
23
NAND, NOR, XNOR
•
NAND and NOR gates easier to implement in hardware than
AND and OR gates
•
Example: AND logic can be implemented with two NOTs and
a NOR (DeMorgan
’
s Law)
A
B
=
A
B
24
Inverter (NOT Gate)
•
Implements the simplest Boolean operator
“
NOT
”
using two transistors
•
Photolithography techniques used in
fabrication of integrated circuit (IC) chips
http://www.csee.umbc.edu/~plusquel/vlsi/slides/chap3_2.html
Image and further reading at:
http://
www.lithoguru.com
/scientist/
lithobasics.html
25
NAND Gate
•
NAND gate implemented using 4 transistors
26
Integrated Circuit (IC) Chips Contain
Gates & Connectors
Images from http://www.ee.ed.ac.uk/~gaa/DigilabNotes/Digilab/Components/node7.html
27
LEDs (light

emitting diodes)
Images from http://www.cores.ro/electronic/Kit.html
The circuit can turn
the LED on or off.
28
Interactive demo of Boolean logic
http://courses.cs.vt.edu/~csonline/MachineArchitecture/Lessons/Gates/index.html
29
Circuits for Binary Addition
Half Adder
Images from http://www.play

hookey.com/digital/adder.html
30
Circuit Board for Half

Adder
http://www.seas.upenn.edu/~ee111/half

adder/Half

adder.html
31
Circuits for Binary Addition
Full Adder
Images from http://www.play

hookey.com/digital/adder.html
32
Circuits for Binary Addition
Full Adder, 4 bit output + carry
•
Can link up an arbitrary number of
input bits
•
Can modify this easily to do
subtraction
•
Can also modify to do
multiplication
–
How?
–
Usually division is done with a special
floating point chip
Images from http://www.play

hookey.com/digital/adder.html
33
Flip

Flops
•
Also called a latch
•
Used to
“
remember
”
values
•
Keep the same value until the input signals a change
•
Example
–
Circuit with two inputs
–
Setting input A to 1 means
“
set output to 1
”
–
Setting input B to 1 means
“
set output to 0
”
output
Input A
Input B
34
Simple Flip

Flop: The Problem
–
If both inputs change simultaneously, the final state of
the latch cannot be determined ahead of time. This is
called:
•
A Race Condition: the result depends on which input got
changed first.
•
Nondeterministic: Not being able to predict what will happen
with certainty.
–
Nondeterminism
is very, very
bad
from the perspective
of computer science (most of the time).
35
The clock
–
CPS (cycles per second):
•
The measure of how frequently an alternating
current changes direction.
•
This term has been replaced by the term hertz
(Hz).
–
gigahertz (GHz):
•
A unit of frequency denoting 10
9
Hz.
36
Clock cycles
•
Logic gates introduce propagation delays as the signal
goes through them.
•
Therefore, a clock cycle is introduced into the circuit.
–
It’s like a heart beat for the system.
•
It connects up to all the circuits that need it, so they
can all be told to update at the same time.
–
The clock cycle can be no faster than the slowest set of logic
gates.
•
The clock keeps the circuit running in a predictable
fashion, by making sure every part of the circuit gets
changes to input at the same time.
37
Clocked D Latch
Solves the problem:
Clock line allows the system to
control when to read the inputs.
BUT only allows for one input.
Image from http://www.play

hookey.com/digital/
38
Example; Linking Boolean to Circuits
•
Lighting up a digital clock
How are the LEDS
for the digital clock
controlled?
39
7 Segment LEDs
Images from http://www.ee.ed.ac.uk/~gaa/DigilabNotes/Digilab/Components/node7.html
Need to control the inputs a through g
Which ones are turned on to make a 5?
40
7 Segment LEDs: How to set the digits
Images from http://www.ee.ed.ac.uk/~gaa/DigilabNotes/Digilab/Components/node7.html
41
7 segment Digital Display
How to turn on only 1 of the 4 digits at a time?
Images from http://
www.ee.ed.ac.uk
/~
gaa
/
DigilabNotes
/
Digilab
/Components/node7.html
a b c d e f g p
i1 i2 i3 i4
42
n

bit Decoder:
Takes as input
n
inputs; turns on 1 (and only 1) of outputs
Righthand image from http://courses.cs.vt.edu/~csonline/MachineArchitecture/Lessons/Circuits/index.html
n
2
A
B
W
X
Y
Z
a b c d e f g p
? ? ? ?
43
n

bit Decoder:
Takes as input
n
inputs; turns on 1 (and only 1) of outputs
Righthand image from http://courses.cs.vt.edu/~csonline/MachineArchitecture/Lessons/Circuits/index.html
Notice where the invertors are.
They look like the inputs of a truth table.
n
2
A
B
W
X
Y
Z
44
Summary
•
Quick Binary Exercise
•
Boolean Logic
–
deMorgan’s Law
•
How Bits are Made
•
Relating Boolean Logic to Circuits
–
Logic gates
•
Special gates:
–
Flip Flops
–
Adders
–
Decoders
•
Next time: The CPU and instruction sets!?
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