david evans class 15: golden ages and astrophysics cs200: computer science university of virginia...

David Evanshttp://www.cs.virginia.edu/evans

Class 15: Golden Ages and Astrophysics CS200: Computer Science

University of VirginiaComputer Science

21 February 2003 CS 200 Spring 2003 2

Science’s Endless Golden Age


Astrophysics• “If you’re going to use your computer to simulate

some phenomenon in the universe, then it only becomes interesting if you change the scale of that phenomenon by at least a factor of 10. … For a 3D simulation, an increase by a factor of 10 in each of the three dimensions increases your volume by a factor of 1000.”

• How much work is astrophysics simulation (in notation)?

(n3)When we double the size of the simulation, the work octuples! (Just like oceanography octopi simulations)


Orders of Growth

0

200

400

600

800

1000

1200

1 2 3 4 5 6 7 8 9 10

logn n

nlogn n̂ 2

n̂ 3

bubblesort

simulatinguniverse

insertsort-tree


Astrophysics and Moore’s Law• Simulating universe is (n3)• Moore’s law: computing power

doubles every 18 months• Tyson: to understand something

new about the universe, need to scale by 10x

• How long does it take to know twice as much about the universe?


;;; doubling every 18 months = ~1.587 * every 12 months(define (computing-power nyears) (if (= nyears 0) 1 (* 1.587 (computing-power (- nyears 1)))))

;;; Simulation is (n3) work(define (simulation-work scale) (* scale scale scale))

(define (log10 x) (/ (log x) (log 10))) ;;; log is base e;;; knowledge of the universe is log 10 the scale of universe

;;; we can simulate(define (knowledge-of-universe scale) (log10 scale))

Knowledge of the Universe


(define (computing-power nyears) (if (= nyears 0) 1 (* 1.587 (computing-power (- nyears 1))))) ;;; doubling every 18 months = ~1.587 * every 12 months(define (simulation-work scale) (* scale scale scale)) ;;; Simulation is O(n^3) work(define (log10 x) (/ (log x) (log 10))) ;;; primitive log is natural (base e)(define (knowledge-of-universe scale) (log10 scale)) ;;; knowledge of the universe is log 10 the scale of universe we can simulate

(define (find-knowledge-of-universe nyears) (define (find-biggest-scale scale) ;;; today, can simulate size 10 universe = 1000 work (if (> (/ (simulation-work scale) 1000) (computing-power nyears)) (- scale 1) (find-biggest-scale (+ scale 1)))) (knowledge-of-universe (find-biggest-scale 1)))

Knowledge of the Universe


> (find-knowledge-of-universe 0)1.0> (find-knowledge-of-universe 1)1.041392685158225> (find-knowledge-of-universe 2)1.1139433523068367> (find-knowledge-of-universe 5)1.322219294733919> (find-knowledge-of-universe 10)1.6627578316815739> (find-knowledge-of-universe 15)2.0> (find-knowledge-of-universe 30)3.00560944536028> (find-knowledge-of-universe 60)5.0115366121349325> (find-knowledge-of-universe 80)6.348717927935257

Will there be any mystery left in the Universe when you die?

Only two things are infinite, the universe and human stupidity, and I'm not sure about the former.

Albert Einstein


Any Harder Problems?

• Understanding the universe is (n3)

• Are there any harder problems?


Who’s the real genius?


Solving the Peg Board Game

• Try all possible moves

• Try all possible moves from the positions you get after each possible first move

• Try all possible moves from the positions you get after trying each possible move from the positions you get after each possible first move

• …


Possible Moves

StartPeg board gamen = number of holesInitially, there are n-1 pegs.

Cracker Barrel’s game hasn = 15

Assume there arealways exactly 2 possible moves,how many possiblegames are there?


Cracker Barrel Game

• Each move removes one peg, so if you start with n-1 pegs, there are up to n-2 moves

• Assume (conservatively) there are just two possible choices for every move.

2 * 2 * 2 * 2 * … * 2 = 2n-2

• For n = 15, there are 213 = 8192


All Cracker Barrel Games(starting with peg 2 1 missing)

Pegs Left

Number of Ways

Fraction of Games

IQ Rating

1 1550 0.01 “You’re Genius”2 20686 0.15 “You’re Purty Smart”3 62736 0.46 “Just Plain Dumb”4 46728 0.33

“Just Plain

Eg-no-ra-moose”

5 5688 0.04

6 374 0.0027

7 82 0.00058

10 2 0.00001


How much work is our straightforward peg board

solving procedure?

Note: I don’t know if this is the best possible procedure for solving the peg board puzzle. So the peg board puzzle problem might not be harder than understanding the Universe (but it probably is.)

(2n)


True Genius?“Genius is one percent inspiration, and ninety-nine percent perspiration.”

Thomas Alva Edison

“Genius is one percent sheer luck, but it takes real brilliance to be a true eg-no-ra-moose.”

Cracker Barrel

“80% of life is just showing up.”Woody Allen


Orders of Growth

0

200

400

600

800

1000

1200

1 2 3 4 5 6 7 8 9 10

logn n

nlogn n̂ 2

n̂ 3 2̂ n

bubblesort

simulatinguniverse

insertsort-tree

peg board game


Orders of Growth

0

10000

20000

30000

40000

50000

60000

70000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

logn n

nlogn n̂ 2

n̂ 3 2̂ n

bubblesort

simulatinguniverse

insertsort-tree

peg board game

Orders of Growth

0

200000

400000

600000

800000

1000000

1200000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

logn n

nlogn n̂ 2

n̂ 3 2̂ n

simulating universe

peg board game

“tractable”

“intractable”

I do nothing that a man of unlimited funds, superb physical endurance, and maximum scientific knowledge could not do. – Batman (may be able to solve intractable problems, but computer scientists can only solve tractable ones for large n)


Any other procedures we’ve seen that are more work than

simulating the Universe?


Break Lorenz Cipher Procedure

• Try all possible wheel settings

• How many possible wheel settings: 5 choices for first wheel

* 5 choices for second wheel

* 5 choices for third wheel

• What is n?– The number of wheels

• There are 5n possible wheel settings


Lorenz Deciphering

• For PS4: you had 3 wheels, each with 5 possible settings:

53 = 125 possible combinations to try

• For WWII: Nazis has 12 wheels, each with more than 5 settings (up to 61 settings)

512 = 244 140 625 possible combinations


PS4• Bletchley Park’s cryptographers had to

solve a problem that is 1 953 125 times harder than PS4!– And they also had to figure out the structure

of the Lorenz machine themselves!

• But…having bombs dropping on you is at least 1 million times more motivating than getting a gold star!


The Endless Golden Age

• Golden Age – period in which knowledge/quality of something doubles quickly

• At any point in history, half of what is known about astrophysics was discovered in the previous 15 years!

• Moore’s law today, but other advances previously: telescopes, photocopiers, clocks, etc.


The Real Golden Rule?Why do fields like astrophysics, medicine, biology and computer science (?) have “endless golden ages”, but fields like– music (1775-1825)– rock n’ roll (1962-1973, or whatever was popular when you

were 16)– philosophy (400BC-350BC?)– art (1875-1925?)– soccer (1950-1974)– baseball (1925-1950)– movies (1930-1940)

have short golden ages?

0

1

2

3

4

5

6

1930

1934

1938

1950

1954

1958

1962

1966

1970

1974

1978

1982

1986

1990

1994

1998

2002

Ave

rage

Goa

ls p

er G

ame,

FIF

A W

orld

Cup

s

Changed goalkeeperpassback rule

Goal-den age


Endless Golden Age and “Grade Inflation”

• Average student gets twice as smart and well-prepared every 15 years– You had grade school teachers (maybe

even parents) who went to college!

• If average GPA in 1970 is 2.00 what should it be today (if grading standards didn’t change)?


Grade Inflation or Scale Compression?

2.00 average GPA in 1970 (“gentleman’s C”?)* 2 better students 1970-1985* 2 better students 1985-2003* 3 admitting women, non-whites (1971)* 1.54 population increase* 0.58 increase in enrollment

Virginia 1970 4,648,494

Virginia 2000 7,078,515

Average GPA today should be:

21.4 CS200 has only the best of the best students, and onlythe best 35/40 of them stayed in the course after PS1, so the

average grade in CS200 should be 21.4*2*2*40/35 = 98.0

Students 1970 11,000Students 2002 18,848

(12,595 UG)


The Liberal Arts

Trivium (3 roads)

language

Quadrivium (4 roads)

numbers

Grammar Rhetoric Logic Arithmetic

Geometry

Music

Astronomy

From Lecture 1:


Liberal Arts• Grammar: study of meaning in written

expression

• Rhetoric: comprehension of verbal and written discourse

• Logic: argumentative discourse for discovering truth

• Arithmetic: understanding numbers

• Geometry: quantification of space

• Music: number in time

• Astronomy

BNF replacement rules for describing languages, rules of evaluation for meaning

Not yet… Interfaces between components, program and user

Rules of evaluation, if, recursive definitions

Not much yet…wait until April

Curves as procedures, fractals

Yes, even if we can’t figure out how to play“Hey Jude!”

Yes: Neil deGrasse Tyson says so

Triv

ium

Qua

driv

ium

From Lecture 1:


Charge• Enough with all the liberal arts stuff, every

problem on Exam 1 is about money!

• No problem in Exam 1 is as hard as simulating the universe

• If you want to do something important and be remembered, work in a field that has a short golden age from 2003-2018– Shakespeare will be known a thousand years

from now, but no one will have heard of any 21st century playwright

david evans class 15: golden ages and astrophysics cs200: computer science university of virginia...

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