great ideas alan turing – what is computable? a task is computable if one can specify a sequence...

Great Ideas Alan Turing –

What is computable? A task is computable if one can specify a sequence of

instructions which when followed will result in the completion of the task.

John Von Neumann – Stored program concept Developed the concept of storing a program in the computer’s

memory rather than it’s circuitry

Copyright © 2008 by Helene G. Kershner

Great Ideas Technological advances

Vacuum tube transistor integrated circuits (multiple transistors on a chip) VLSI (very large scale integration)

Microprocessor chip – computer on a chip High level programming language – people can build “user

friendly” software” Changes in memory device hardware

Storing information digitally, making use of the concept that computers work with two states: on/off, 1/0, yes/no, high current/low current

The computer is a binary machine


Great Ideas The computer is a binary machine

Perform mathematics Perform operations on letters that form words Represent images in grayscale and color Represent sound


The computer is a binary machine


Boolean Logic / Boolean Algebra Boolean Algebra (Boolean Logic) is an algebra for

symbolically representing problems in logic & analyzing them mathematically.

Based on work of George Boole English mathematician An Investigation of the Laws of Thought Published in 1854 Reduced logic of human thought to mathematical operations

An analysis of how natural language works if it were logical


Boolean Logic / Boolean Algebra In 1938 Claude E. Shannon of MIT

Showed that Boolean logic could be applied to the design of relay networks in telephone systems (the ability to switch signals from one place to another automatically)

Invented branch of mathematics called Information Theory

Information Theory applied Boolean algebra to certain engineering problems systems

Today, boolean algebra, as applied to computer hardware design is also known as Switching Theory.


Boolean Logic / Boolean Algebra Boolean Logic is an abstraction.

Definition -- Abstraction: Is the process of taking away or removing characteristics

from something in order to reduce it to a set of essential

characteristics (Whatis.com)

not concrete: not relating to concrete objects but expressing something that can only be appreciated

intellectually (Encarta.msn.com)

Considered apart from concrete existence: an abstract

concept (www.thefreedictionary.com)


Boolean Logic / Boolean AlgebraApplying Boolean Logic to computers allows them to

handle very complex problems using complicated

connections of simple components. Designing hardware and software is very

complicated because we are using them to deal

with complex tasks.


Boolean Logic / Boolean AlgebraAbstraction allows the designer to be separated from the

machine to get a clearer picture of what needs to be done. allows us to drive a car without knowing how the

engine works allows computer users to work with the hardware

without knowing exactly how the machine does what it does.

Add a flash stick, the software will recognize it Plug in a digital camera and the software will find it Move music onto my iPod from the hard-drive without

having any real idea what the computer is doing.

Computers: Complex tasks with Simple Components

Basic Computer Components Switches

On/off, high current / low current, 1/0

Connectors Wires that connect switches – pipes Must be able to branch, one or more paths


Boolean Logic / Boolean Algebra

Logic Gates Hardware interpretation of boolean logic Universal building blocks, special circuits that perform

the operation of:

AND

OR

NOT From these basic elements, computer engineers can

design “anything”


Boolean Logic / Boolean Algebra Different and equal ways to represent this kind of logic

Using the English works

AND, OR, NOT Using Mathematical symbols

Λ, ۰ (means AND)

V , + (means OR)

~, ¯ (means NOT)



Engineering symbols

inverter/flipperhttp://www.ee.surrey.ac.uk/Projects/Labview/gatesfunc/

index.html#andgate


Boolean Logic / Boolean Algebra

Look at statements that can be determined to

be true or false. I think therefore I am.

This can be either true or false Yesterday my daughter in college called needing money.

This can be either true or false My son lives in Washington DC.

This can also be either true or false


Boolean Logic / Boolean AlgebraLook at statements that cannot be to be true or false. A Question is not a statement:

What time is it? While an answer to a question may be Yes or No, the statement is

neither true or false: Is it 3 o’clock? A command is not a statement:

You MUST get an A in this class! Turn at the next corner.

Wishes are not statements: I’d like to make a million dollars. Have a Happy Thanksgiving.

http://www.informatik.htw-dresden.de/~nestleri/logic/01/index.html



Truth Tables Visually shows true/false values and the results (inputs

and outputs) of a logic example. Describes what is happening in a logic gate or logic

statement Uses T/F, 1/0


Boolean Logic / Boolean Algebra Logic can be applied to statements. Statements have the property of either being True (T, 1)

or False (F, 0). Not everything we say is a statement because it cannot

be thought of as having the value of either True or False.


Venn Diagrams: A, BBoolean Logic

The set A is mutually exclusive from set B They have nothing in common

http://cs.uni.edu/~campbell/stat/venn.html


Venn Diagrams: A AND B Boolean Logic

The set A and the set B have a common area. The pink area labeled 2, is represented by A AND B

A B

http://www.purplemath.com/modules/venndiag2.htm


Logic -- AND

AND combines two statements/inputs either one of which can be True or False A statement using AND is only true if both parts of

the sentence are true

John has a sub-prime mortgage AND John cannot pay his mortgage.


Logic -- AND

John has a sub-prime mortgage AND John cannot pay his mortgage.

Truth table identifies the possibilities for AND:

sub-prime mortgage AND cannot pay mortgage Result

F F F

F T F

T F F

T T T


Logic -- AND

A: sub-prime mortgage

B: cannot pay mortgage

A AND B Result

F F F

F T F

T F F

T T T


Venn Diagram: A OR B Boolean Logic A OR B represents the area covered by all of A as well as all of

B.

A B



Logic – OR

OR combines two statements/inputs either one of which can be True or False A statement using OR is true if either parts of the

sentence is true. I like mango OR chocolate ice cream. John has a sub-prime mortgage OR John has a traditional

mortgage. John can afford his mortgage OR John is unable to pay his

mortgage. . John has a sub-prime mortgage OR John is unable to pay his

mortgage.


Logic – OR I like mango OR chocolate ice cream. John has a sub-prime mortgage OR John has a traditional mortgage. John can afford his mortgage OR John is unable to pay his mortgage. . John has a sub-prime mortgage OR John is unable to pay his

mortgage. A OR B

A OR B R

F F F F T T T F T T T T


Logic – OR

A: John has a sub-prime mortgageB: John is unable to pay his mortgage.

A OR B Result A OR B R F F F 0 0 0 F T T 0 1 1 T F T 1 0 1 T T T 1 1 1


Venn Diagram: NOT A Boolean Logic

NOT A is the same as ~ A NOT A is everything that is outside of A.

A B



Logic – NOT/Invert The NOT command, flips or Inverts the value it is given. The operation switches that statement between True and False

I have a traditional mortgage. I do NOT have a sub-prime mortgage. NOT a sub-prime mortgage INVERTs or is the opposite sub-prime

mortgage which is a traditional mortgage. A = I eat peanut-butter

NOT A means I do NOT eat peanut-butter If A is True, then NOT A is False If A is False, then NOT A is True

A = I love peanut-butter For me A is False, so NOT A would be True

NOTA R


Logic – NOT/Invert Truth Table

A NOT A = ~A

0 1

1 0

NOTA R


Logic Symbols – Order of Operations Order of Operations: In a logic statement using the three

operators we have learned the order is Parenthesis ()

NOT NOT, ~ ¯¯

AND AND, ۰, Λ

OR OR, +, V


Venn Diagram: NOT (A AND B) Boolean Logic

NOT (A OR B) is the same as area labeled 4. It is the area that is NOT or outside of (A OR B).

A B



VENN Diagrams: NOT (A OR B) Boolean Logic NOT (A AND B) is the pink area. (A AND B) is the white area,

so NOT (A AND B) is all the area outside of (A AND B).

A B



Using Boolean Algebra in the “Real World”

Boolean Operators http://www.youtube.com/watch?v=xsSZps3NH-M

Logic

Logic problems can relate back to English statements:

A newly constructed building has three types of security alarms. It has an intrusion alarm, a fire alarm and a flood alarm. If the intrusion or fire alarm goes off then the police department is called. If the fire or flood alarm goes off then the fire department is called. But if the flood alarm goes off don’t call the police. And if the intrusion alarm goes off don’t call the fire department.

A = intrusion alarm A OR B = Police

B = fire alarm

C = flood alarm B OR C = Fire Department


great ideas alan turing – what is computable? a task is computable if one can specify a sequence...

Documents

kershner slide

binary machine copyright

sound copyright

logical copyright

circuitry copyright

computer hardware design

reduced logic of human

computer users