Post on 18-Dec-2015
Computation
Binary Numbers
http://faculty.mc3.edu/pvetere/Applets/APPLETS/NUMSYS/applet_frame.htm
• Decimal numbers
• Binary numbers
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.
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.
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
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.
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
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
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
What is a Heuristic?
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?
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
read numberset result to 1set counter to 2until counter > number do
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
read numberset result to 1set counter to 2until counter > number do
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
Another PCP Instance
i X Y
1 11 011
2 01 0
3 001 110
The Post Correspondence Problem
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.
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:
A Tiling Problem
A Tiling Problem
A Tiling Problem
A Tiling Problem
A Tiling Problem
A Tiling Problem
A Tiling Problem
A Tiling Problem
A Tiling Problem
A Tiling Problem
Try This One
Another Tiling Problem
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
The Traveling Salesman Problem
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!
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.
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?