Computation. Binary Numbers Decimal numbers Binary numbers.

Computation

Binary Numbers

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

• Decimal numbers

• Binary numbers

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

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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.

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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.

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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。

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.

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

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

Representing Text

F o u r

01000110 01101111 01110101 01110010

Representing Text

T h e n u m b e r i s 1 7 .

54 68 65 20 6E 75 6D 62 65 72 20 69 73 20 31 37 2E

When We Need More Characters

简体字

What about things like:

When We Need More Characters

简体字

What about things like:

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

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

A conversion applet:http://www.pinyin.info/tools/converter/chars2uninumbers.html

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

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

But What Do Symbols Look Like?

Computers have revolutionized our world.





The Basic Idea

results = google(text, query)

The Basic Idea


if word_count(text) > 5000: return(“Done!!”)else:

return(“No sleep yet.”)

The Basic Idea


if word_count(text) > 5000: return(“Done!!”)else:

return(“No sleep yet.”

display = render(text, font)

The Basic Idea


Digital Images

Pixels

Pixels

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

Pixels

0000110000 0001111000 0011111100 0111111110 0111111110 01111111100111001110 0111001110 0111001110 0111001110

Digital Images

Two Color Models

RGB

The red channel

RGB

The green channel

RGB

Red Green Blue

Experimenting with RGB

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

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

Representing Sounds

Digitizing Sound

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

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

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

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

Molecules

It’s just a string:

AUGACGGAGCUUCGGAGCUAG

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?

Search

http://www.javaonthebrain.com/java/puzz15/

Start state Goal state

http://www.javaonthebrain.com/java/puzz15/

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.


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?

Depth-First Search

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.

Growth Rates of Functions

Scalability

To play one master-level game

21012 nodes

Seconds since Big Bang 3 1017

Number of sequential games since Big Bang

150,000

Yet This Exists

How?

A Heuristic Function for Chessc1 * 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/

How It Has Happened

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

How Much Compute Power Is There?

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

Countdown to Singularity

Law of Accelerating Returns

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?

1. Can we make all true statements theorems?

2. Can we decide whether a statement is a theorem?

The Halting Problem

Program, M

input string, w

Does M halt on w?Yes

No

A Simple Example

read nameif name = “Elaine” then print “You win!!” else print “You lose ”

Another Example

read numberset result to 1set counter to 2until counter > number do

set result to result * counteradd 1 to counter

print result

Programs Debug Programs


set result to result * counteradd 1 to counter

print result

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

Suppose number = 5:

result number counter 1 5 2 2 5 3 6 5 4 24 5 5120 5 6

Changing It a Bit


set number to number * counteradd 1 to counter

print result

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

Suppose number = 5:

result number 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.

The Halting Problem Is Undecidable

Program, M

input string, w

Does M halt on w?Yes

No

Another Undecidable Problem

The Post Correspondence Problem

A PCP Instance With a Simple Solution

i X Y

1 b aab

2 abb b

3 aba a

4 baaa baba

A PCP Instance With a Simple Solution

i X Y

1 b aab

2 abb b

3 aba a

4 baaa baba

Solution: 3, 4, 1

Another PCP Instance

i X Y

1 11 011

2 01 0

3 001 110


List 1 List 2

1 ba bab

2 abb bb

3 bab abb

A PCP Instance With No Simple Solution

i X Y

1 1101 1

2 0110 11

3 1 110

A PCP Instance With No Simple Solution

i X Y

1 1101 1

2 0110 11

3 1 110

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?

What is a Program?

A procedure that can be performed by a computer.


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:

A Tiling Problem

Try This One

Another Tiling Problem

Is the Tiling Problem Decidable?

Wang’s conjecture: If a given set of tiles can be used to tile anarbitrary surface, then it can always do so periodically. In otherwords, there must exist a finite area that can be tiled and thenrepeated 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?

Is Decidability Enough?

The Traveling Salesman Problem

Given n cities and the distances between each pair ofthem, find the shortest tour that returns to its starting pointand visits each other city exactly once along the way.

15

20

25

89

23

40

10

4

73

28


15

20

25

89

23

40

10

4

73

28

Given n cities:

Choose a first city nChoose a second n-1Choose a third n-2

… n!


Can we do better than n!

● First city doesn’t matter. ● Order doesn’t matter.

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

The Growth Rate of n!

2 2 11 479001600

3 6 12 6227020800

4 24 13 87178291200

5 120 14 1307674368000

6 720 15 20922789888000

7 5040 16 355687428096000

8 40320 17 6402373705728000

9 362880 18 121645100408832000

10 3628800 19 2432902008176640000

11 39916800 36 3.61041

Growth Rates of Functions, Again

Putting it into Perspective

Speed of light 3108 m/sec

Width of a proton 10-15 m

At one operation in the time it takes light to cross a proton

31023 ops/sec

Since Big Bang 31017 sec

Ops since Big Bang 91040 ops 36! = 3.61041

Neurons in brain 1011

Parallel ops since Big Bang

91051 43! = 61052

Does Complexity Doom AI?

Computation. Binary Numbers Decimal numbers Binary numbers.

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