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CSCI N301:

Fundamental Computer

Science Concepts


Copyright
©2004


De灡ptment映䍯m灵ter & In景fm慴i潮 Science

Introducing Gates

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Goals

By the end of this lecture, you should
understand …


How modern computers use transistors as
switches.


What a gate is.


How to construct a pump model of a gate.


How to work with the basic Boolean
operators AND, OR and NOT.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Where We’ve Been


We’ve considered switches, and devised
ways to encode information in switch
math.


We’ve learned that the switch is the
fundamental computer component.


Today, we’ll learn that modern computers
use a specific kind of switch, called a
transistor
.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Transistors


The
transistor

is a solid
-
state binary device (the
phrase “solid
-
state” means that an electrical
current follows a path through solid semi
-
conductive material).


Transistors replaced vacuum tubes in
computers, beginning in the 1960s.


Transistors switch between two states


on and
off


electronically instead of mechanically.


With the movement to transistors came
advances in size (transistors are much smaller)
and speed (transistors are much faster).

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Where do we go from here?


We have switches, which we now know
are transistors. Transistors are either at a
low or high voltage setting.


We need to consider how to
set/manipulate switches.


To understand this, we’ll move up a level
of complexity, to the world of gates …

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Gates


A
gate

is an electronic device that operates on a
collection of binary inputs to produce a binary
output.


This is exactly what we need


the ability to take
binary inputs (which is all we have inside the
computer), manipulate them according to some
set of processing rules and produce results
which a computer can store in binary form.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Gates: The Tool Set


In the same way that binary math was the
toolset we needed for binary encoding, we
need to use a toolset for manipulating
gates.


The toolset we’ll use is Boolean logic,
named after George Boole, the English
mathematician and pioneer of modern
symbolic logic.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Boolean Logic


Boole’s observed that truth values can be
mapped to binary math.


The mapping is straightforward:


We can represent TRUE by using a 1.


We can represent FALSE by using a 0.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Boolean Logic, the Other Half


We are half way there … we know how to
map True/False to 1/0.


Now we need a branch of mathematics
that presents a set of rules for
manipulating truth values.


That’s the second half of Boolean logic …

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Truth Tables


Boole mapped truth tables that tell us how
to connect truth values.


There are three primary operations that we
can perform on truth values:


Two procedures that combine two truth values
(Combinational Procedures)


One procedure that reverses a truth value
(Complementary Procedure)

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Boolean Combinations


There are two forms of Boolean
combinations (
both are binary operators,
they combine two values
):


Or


And


We need one other piece of information to
discuss this


we need to refine our
definition of a Boolean Value …

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Boolean Values


There are two Boolean values: True and
False.


We can combine these two values to
produce Boolean expressions.


A Boolean expression is one which we can
evaluate as being either True or False.


Let’s try an example …

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Boolean Values and
Expressions


Consider the following sentence:
“It is raining,
and

the radio is playing.”


I could assign a value, either True or False, to each of
the two phrases in that sentence.


It is raining right now, so I will replace that part of the
sentence with a “T” for True.


The radio is not playing, however, so I will replace
that part of the sentence with an “F” for false.


If only I knew what to do with the connector in that
sentence, the word AND….


And that’s what Boole’s truth tables gave us!

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Back to Combinations


There are two ways to connect truth values:


Using the OR operator


Using the AND operator


Possible TRUE outcomes of the OR operator:


Condition A is TRUE


Condition B is TRUE


BOTH Condition A & Condition B are TRUE


There is only one TRUE outcome of the AND
operator:
both conditions
MUST

be true!

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Back to Our Example …


Again, consider the sentence:
“It is raining,
and

the radio
is playing.”


Let’s evaluate the sentence using Boolean logic:


It is raining, so I get a “T” for the first phrase.


My radio is not playing, so I get an “F” for the second phrase.


The connector is an AND, which means both phrases have to be
true for the entire expression to be true, so …


Since the second phrase is FALSE, and since the operator
connecting the two phrases is the AND operator, we can
evaluate the entire sentence as having a truth value of FALSE.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Truth Table for AND

1
st

Phrase

2
nd

Phrase

Overall
Expression

T

T

T

T

F

F

F

T

F

F

F

F

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

AND Operator Summary


There is only one possible way to produce
an overall TRUE expression from two
Boolean phrases connected with the AND
operator: both phrases have to be TRUE
for the entire combination to be TRUE.


All other combinations result in a FALSE.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Truth Table for OR

1
st

Phrase

2
nd

Phrase

Overall
Expression

T

T

T

T

F

T

F

T

T

F

F

F

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

OR Operator Summary


There is only one possible way to produce
an overall FALSE expression from two
Boolean phrases connected with the OR
operator: both phrases have to be FALSE
for the entire combination to be FALSE.


All other combinations result in a TRUE.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

NOT Operator Summary


Unlike AND and OR (which are both binary
operators), NOT is a unary operator (it works
with only 1 value).


NOT “reverses the charge” of a value:


A TRUE (1) becomes a FALSE (0). The
complement
of TRUE is FALSE.


A FALSE (0) becomes a TRUE (1). The
complement

of FALSE is TRUE.


NOT is also known as the
complementary

operator.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Truth Table for NOT

Phrase

Overall
Expression

T

F

F

T

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Complex Comparisons


If are testing more than two phrases, then you’ll need to
use the Boolean Operator Hierarchy:


NOT


AND


OR


A better idea is to use parentheses to indicate which
conditions your program should evaluate first.


When using these truth tables, you can combine
switches in two ways (using AND or using OR) to
produce a single binary output from multiple binary
inputs. The NOT operator also produces a single binary
output, the complement of the input.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Moving from Comparisons to Gates


Using Boolean operators, we have the
capability to take individual switches, and
connect them in
gates
.


We can then wire gates to perform
Boolean logic


AND comparisons, OR
comparisons or NOT complements


base
on the switch settings.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

The AND Gate


To construct an AND gate, we take two
switches in series.


If either switch is set to zero, voltage is
impeded from flowing, and the output will
be a zero.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

The OR Gate


To construct an OR gate, we connect two
switches in parallel.


As long as voltage exists on at least one
input line, the output line will be high.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

The NOT Gate


Only one switch is required to construct a NOT
gate. Why?


A resistor is placed on the output line to force
voltage flow through the transistor.


If the input line is low, the transistor delivers
current to the output line, which is high.


If the input line is high, the transistor blocks flow
to the output line, and the output line remains
low.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Getting the Picture


This all sounds reasonable, but a diagram
method would be helpful.


Computer engineers developed several
models as visual teaching tools for gate
operation.


We will use the “pump” model …

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Pump Model: Off

Imagine two water pipes with pump
headers, as illustrated to the left. The
vertical pipe is open, and has water in
it, which unless impeded, will flow
downward (the path of least
resistance). The horizontal pump is off,
offering no impedance to the water in
the vertical pipe. So, we get output at
the bottom. Look at the horizontal
pump again. See the curly wire? That is
a spring… if we turn the pipe on, it
pushes the spring, which moves the
black block into the vertical pipe, and
flow would be impeded.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Pump Model: On

Here is the vertical pipe
again; this time, the
horizontal pump is
turned on. Notice that
the spring has pushed
the black block into the
horizontal pipe, and
water can’t flow
through.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Complex Pumps

Voltage 2

Voltage 1

Input 1

Input 2

Some key symbols: the voltage
lines, sometimes called Control
Lines, ALWAYS have initial
voltage. The Input Lines can be
high or low. Points along the way
can be measured to see if
voltage exists or not. When an
input line is HIGH, it’s “pump” is
on, and it moves a block to
impede. When an input line is
LOW, it’s “pump” if off, and it
offers no impedance to the
voltage line.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Complex Pumps

Voltage 2

Voltage 1

Input 1
-

High

Now, there is voltage on the
control lines, so points A and D
are high. Input 1 is on (pump
on), input 2 is off (pump off) .
Input 1 closes and impedes, so
current is trapped and can’t pass
to B; B is off. Input 2 is off, so
the “0” at point B passes
unimpeded to point C.
Remember, D had voltage, and
now that C is low, it can’t impede
the flow of D across, so the
voltage at point E is high.

Input 2
-

Low

A

B

C

D

E

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Complex Pumps

Voltage 2

Voltage 1

Input 1
-

Low

Input 2
-

High

A

B

C

D

E

Still voltage on the control lines,
so points A and D are high. Input
1 is off (pump off), input 2 is on
(pump on) . Input 1stays open
and doesn’t impede, so current
passes to B. Input 2 is on, so the
“1” at point B can’t pass to point
C. Remember, D had voltage, and
now that C is low, it can’t impede
the flow of D across, so the
voltage at point E is still high.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Complex Pumps

Voltage 2

Voltage 1

Input 1
-

Low

Input 2
-

Low

A

B

C

D

E

Still voltage on the control lines,
so points A and D are high. Input
1 is off , input 2 is off . Input 1
stays open and doesn’t impede,
so current passes to B. Input 2 is
off, so the “1” at point B passes
to point C. Remember, D had
voltage, but the high voltage at C
means that Voltage 2 closes the
flow to D, impeding it. Thus the
voltage at E is off.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Complex Pumps

Voltage 2

Voltage 1

Input 1
-

High

Input 2
-

High

A

B

C

D

E

Still voltage on the control lines,
so points A and D are high. Input
1 is high , input 2 is high. Input 1
closes and impedes, so no
current passes to B. Input 2
closes, so is downstream of it has
no current, either. Thus C is low.
Remember, D had voltage, which
passes unimpeded to E. Thus
the voltage at E is high.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

That Was an OR Gate!

What we just illustrated using the Pump Model, of course,
was an Or gate. Every combination of two inputs in which
at least one of the inputs was high produced a 1. The case
of two low inputs was the only case which resulted in a O
for an output. Notice that we took two binary outputs,
performed Boolean logic processing, and produced a single
binary output.

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Up a Level of Abstraction


As you might imagine, while the Pump
Model is a good teaching tool, it is a
cumbersome notation for experienced
electrical engineers


A more abstract notation exists for
constructing and evaluating Boolean gates

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Boolean Gate Notation

GATE

SYMBOLS

CIRCUIT
DIAGRAM

NOT

~

AND



OR

+

Q

A

B

A

B

Q

Q

A



N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

What’s Next?


In the next unit, we’ll expand our
knowledge of Boolean logic a bit by
combining gates in a higher level of
complexity to create circuits ...

N301: Fundamental Computer Science Concepts

Copyright
©2004


Department of Computer & Information Science

Questions?