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Binary Numbers

- Decimal numbers
- Binary numbers

http://faculty.mc3.edu/pvetere/Applets/APPLETS/NUMSYS/applet_frame.htm

Text

Computers have revolutionized our world. They have changed the course of our daily lives, the way we do science, the way we entertain ourselves, the way that business is conducted, and the way we protect our security.

Text

Computers have revolutionized our world. They have changed the course of our daily lives, the way we do science, the way we entertain ourselves, the way that business is conducted, and the way we protect our security.

Les ordinateurs ont révolutionné notre monde. Ils ont changé le cours de notre vie quotidienne, notre façon de faire la science, la façon dont nous nous divertissons, la façon dont les affaires sont menées, et la façon dont nous protégeons notre sécurité.

Text

Computers have revolutionized our world. They have changed the course of our daily lives, the way we do science, the way we entertain ourselves, the way that business is conducted, and the way we protect our security.

Les ordinateurs ont révolutionné notre monde. Ils ont changé le cours de notre vie quotidienne, notre façon de faire la science, la façon dont nous nous divertissons, la façon dont les affaires sont menées, et la façon dont nous protégeons notre sécurité.

計算機已經徹底改變我們的世界。當然，他們已經改變了我們的日常生活中，我們這樣做科研，我們自娛自樂的方式，經營的方式進行的方式，以及我們保護我們的安全。

Representing Text

- Decide how many characters we need to represent.
- Determine the required number of bits.
- Ascii: 7 bits. Can encode 27 = 128 different symbols.

Ascii

http://www.krisl.net/cgi-bin/ascbin.pl

When We Need More Characters

What about things like:

简体字

Answer: Unicode: 32 bits. Over 4 million characters.

http://www.unicode.org/charts/

A conversion applet:

http://www.pinyin.info/tools/converter/chars2uninumbers.html

But What Do Symbols Look Like?

Computers have revolutionized our world.

Computers have revolutionized our world.

Computers have revolutionized our world.

Computers have revolutionized our world.

Computers have revolutionized our world.

The Basic Idea

results = google(text, query)

The Basic Idea

results = google(text, query)

if word_count(text) > 5000:

return(“Done!!”)

else:

return(“No sleep yet.”)

The Basic Idea

results = google(text, query)

if word_count(text) > 5000:

return(“Done!!”)

else:

return(“No sleep yet.”

display = render(text, font)

The Basic Idea

Computers have revolutionized our world.

Pixels

Now we must turn this 2-dimensional bit matrix into a string of bits.

Pixels

0000110000 0001111000 0011111100

0111111110 0111111110 0111111110

0111001110 0111001110 0111001110 0111001110

RGB

The red channel

RGB

The green channel

RGB

Red Green Blue

Experimenting with RGB

http://www.jgiesen.de/ColorTheory/RGBColorApplet/rgbcolorapplet.html

Representing Programs

public static TreeMap<String, Integer> create() throws IOException

public static TreeMap<String, Integer> create() throws IOException

{ Integer freq;

String word;

TreeMap<String, Integer> result = new TreeMap<String, Integer>();

JFileChooser c = new JFileChooser();

int retval = c.showOpenDialog(null);

if (retval == JFileChooser.APPROVE_OPTION)

{ Scanner s = new Scanner( c.getSelectedFile());

while( s.hasNext() )

{ word = s.next().toLowerCase();

freq = result.get(word);

result.put(word, (freq == null ? 1 : freq + 1));

}

}

return result;

}

}

Chess Boards

Forsythe-Edwards Notation

rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1

http://en.wikipedia.org/wiki/Forsyth-Edwards_Notation

The Roots of Modern Technology

1834 Charles Babbage’s

Analytical Engine

Ada writes of the engine, “The Analytical Engine has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform.”

The picture is of a model built in the late 1800s by Babbage’s son from Babbage’s drawings.

Using Logic

- TaiShanHasTail
- SmokyHasTail
- PuffyHasTail
- ChumpyHasTail
- SnowflakeHasTail

Using Logic

- Panda(TaiShan).
- Bear(Smoky).
- x (Panda(x) Bear(x).
- x (Bear(x) HasPart(x, Tail)).
- x (Bear(x) Animal(x)).
- x (Animal(x) Bear(x)).
- x (Animal(x) y (Mother-of(y, x))).
- x ((Animal(x) Dead(x)) Alive(x)).

Does TaiShan have a tail?

What is a Heuristic?

The word heuristic comes from the Greek word (heuriskein), meaning “to discover”, which is also the origin of eureka, derived from Archimedes’ reputed exclamation, heurika (“I have found”), uttered when he had discovered that the volume of water displaced in the bath equals the volume of whatever (him) got put in the water. This could be used as a method for determining the purity of gold.

What is a Heuristic?

The word heuristic comes from the Greek word (heuriskein), meaning “to discover”, which is also the origin of eureka, derived from Archimedes’ reputed exclamation, heurika (“I have found”), uttered when he had discovered that the volume of water displaced in the bath equals the volume of whatever (him) got put in the water. This could be used as a method for determining the purity of gold.

A heuristic is a rule that helps us find something.

An Aside on Checking Facts on the Web

Who invented the 15-puzzle?

Sam Loyd did: (http://www.jimloy.com/puzz/15.htm)

Did he or didn’t he:

(http://www.archimedes-lab.org/game_slide15/slide15_puzzle.html)

No he didn’t: (http://www.cut-the-knot.org/pythagoras/fifteen.shtml)

Breadth-First Search

Is this a good idea?

More Interesting Problems

The 20 legal initial moves

Scalability

Solving hard problems requires search in a large space.

To play master-level chess requires searching about 8 ply deep. So about 358 or 21012 nodes must be examined.

Yet This Exists

How?

A Heuristic Function for Chess

c1 * material +

c2 * mobility +

c3 * king safety +

c4 * center control + ...

Computing material:

Pawn 100 Knight 320 Bishop 325 Rook 500 Queen 975 King 32767

The Advent of the Computer

1945 ENIAC The first electronic digital computer

1948 Modified to be a stored program machine

1949 EDVAC

Possibly the first stored program computer

Moore’s Law

http://www.intel.com/technology/mooreslaw/

Can This Trend Continue?

http://www.nytimes.com/2010/08/31/science/31compute.html?_r=1

How Much Compute Power Might It Take?

http://www.frc.ri.cmu.edu/~hpm/book97/ch3/index.html

How Much Compute Power is There?

Hans Moravec: http://www.frc.ri.cmu.edu/~hpm/talks/revo.slides/power.aug.curve/power.aug.gif

Kurweil’s Vision

http://www.pocket-lint.co.uk/news/news.phtml/12920/13944/Computers-match-humans-by-2030.phtml

Some Other People Agree

http://www.networkworld.com/news/2009/092109-intel-cto-interview.html

Limits to What We Can Compute

Are there fundamentally uncomputable things?

- Does God exist?
- What’s the best way to run a country?
- Does this puzzle have a solution?

What Can We Do?

- Can we make all true statements theorems?
- Can we decide whether a statement is a theorem?

Another Example

read number

set result to 1

set counter to 2

until counter > number do

set result to result * counter

add 1 to counter

print result

Programs Debug Programs

Given an arbitrary program, can it be guaranteed to halt?

read number

set result to 1

set counter to 2

until counter > number do

set result to result * counter

add 1 to counter

print result

Suppose number = 5:

resultnumber counter

1 5 2

2 5 3

6 5 4

24 5 5

120 5 6

Changing It a Bit

Given an arbitrary program, can it be guaranteed to halt?

read number

set result to 1

set counter to 2

until counter > number do

set number to number * counter

add 1 to counter

print result

Suppose number = 5:

resultnumber counter

1 5 2

1 10 3

1 30 4

1 120 5

1 600 6

How About this One?

Does this program halt on all inputs?

times3(x: positive integer) =

While x 1 do:

If x is even then x = x/2.

Else x = 3x + 1.

Let’s try it.

Another Undecidable Problem

The Post Correspondence Problem

A PCP Instance With a Simple Solution

Solution: 3, 4, 1

A PCP Instance With No Simple Solution

Shortest solution has length 252.

Can A Program Do This?

Can we write a program to answer the following question:

Given a PCP instance P, decide whether or not P has a solution. Return:

True if it does.

False if it does not.

What is a Program?

A procedure that can be performed by a computer.

The Post Correspondence Problem

A program to solve this problem:

Until a solution or a dead end is found do:

If dead end, halt and report no.

Generate the next candidate solution.

Test it. If it is a solution, halt and report yes.

So, if there are say 4 rows in the table, we’ll try:

1 2 3 4

1,1 1,2 1,3 1,4 1,5

2,1 ……

1,1,1 ….

Will This Work?

- If there is a solution:
- If there is no solution:

Is the Tiling Problem Decidable?

Wang’s conjecture: If a given set of tiles can be used to tile an

arbitrary surface, then it can always do so periodically. In other

words, there must exist a finite area that can be tiled and then

repeated infinitely often to cover any desired surface.

But Wang’s conjecture is false.

Important Issues

- The halting problem is undecidable.
- There’s no black box reasoning engine for standard logic.
- Would quantum computing change the picture?
- Does undecidability doom our attempt to make artificial copies of ourselves?

The Traveling Salesman Problem

15

25

10

28

20

4

8

40

9

7

3

23

Given n cities and the distances between each pair of

them, find the shortest tour that returns to its starting point

and visits each other city exactly once along the way.

The Traveling Salesman Problem

15

25

10

28

20

4

8

40

9

7

3

23

Given n cities:

Choose a first city n

Choose a second n-1

Choose a third n-2

… n!

The Traveling Salesman Problem

Can we do better than n!

● First city doesn’t matter.

● Order doesn’t matter.

So we get (n-1!)/2.

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