Download Logic From Switches - HWS Department of Mathematics and

Two Great Ideas Meet:
Logic and Arithmetic
CPSC 120
Principles of Computer Science
September 30, 2012
From Switches to Devices
Brief Version
 Recall
our SPST, pushbutton switches
 Let's replace the mechanical activation
with electronic control
 Switch output can activate other switches!
 Coordinate inputs with outputs to define
various devices/logic functions
 Package each device and build more
complex circuits using these packages
Conventions Connecting
Logic, Voltage, and Numbers
Truth
TRUE
FALSE
Voltage
Vdd, Vcc, V+, +5V
Vss, GND, 0V
Switch
On
Off
Number
1
0
Note: This is a type of encoding that is
found everywhere in computing!
Logic From Switches
Mechanical switches
Transistor circuit (Inverter)
Truth table for this circuit
Truth Table for Inverter Circuit
Transistor Logic:
It’s Not What You Think
A
B
0
0
0
1
1
0
1
1
Output
What is the truth table of this circuit?
It’s backwards from our usual
expectation!
A
B
Output
0
0
1
0
1
1
0
1
1
1
0
This is an inverted AND circuit!
We will call it a NAND and package it
as a basic logical package or gate.
Question: How do we get a basic
AND gate?
Aside on making transistors
The first transistor
CMOS diagram
A dozen transistors
50 million transistors
A Catalog of the Basic Gates
Second Idea:
Binary Numbers/Arithmetic






We are used to dealing with numbers in decimal,
also known as base 10.
312 = 3 x 102 + 1 X 101 + 2 x 100 (place value)
We can use other bases, such as 2, to also
represent numbers. Base two is also called
binary arithmetic.
10112 = 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20
10112 = 8
+0
+2
+1
= 1110
This gives us a way to represent numbers using
only 0 and 1 as symbols! Two digits is simpler
than 10 digits!
The Great Idea Now Arrives
 By
using transistors, we can create logic
gates to perform logical operations on
inputs such as AND, OR, NOT, etc
 By using binary arithmetic, we can
represent numbers using 0/1 values
 Therefore, we can perform a numerical
computation provided we know how to…?
(fill in this space)
The Great Idea Now Arrives
 By
using transistors, we can create logic
gates to perform logical operations on
inputs such as AND, OR, NOT, etc
 By using binary arithmetic, we can
represent numbers using 0/1 values
 Therefore, we can perform a numerical
computation provided we know how to…
 Make a logic circuit that adds (or
subtracts) using gates! We will call it…an
adder!
Addition Interlude
A typical decimal addition:
312
+ 129
---------441
(Notice we may have to carry between positions)
A typical binary addition:
10110
+ 1101
-----------100011
22 (Here is equivalent decimal form)
+13
---------35
Requirements for our binary
addition circuit

Add two binary digits, called bits, in the right
most column correctly
 Determine if an addition of bits produces a carry
going to the next column on left
 Also, we need to handle carries from the right
 General situation: Correctly add two bits in a
column, taking into account there may be a carry
in from previous column and determine if there is
a carry out bit as well
Lots more needed to create a
modern electronic computer





We have the idea so far for a Calculator, rather
than a modern computer
Doing arithmetic by mechanical/electronics
devices is not the same as programmed control
We need a way to control/automate the actions
of the calculator we are about to invent
Computers deal with characters A-Z as often as
numbers. How do we code those?
Input and output of our computation needs
design as well. Coming attractions!