Transcript Slides

Class 15:
Golden Ages and
Astrophysics
CS200: Computer Science
University of Virginia
Computer Science
David Evans
http://www.cs.virginia.edu/evans
Science’s Endless Golden Age
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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)?
When we double the size of the
3
(n )
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simulation, the work octuples!
(Just like oceanography octopi
simulations)
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Orders of Growth
1200
1000
simulating
universe
logn
n
nlogn
n^2
800
600
400
n^3
200
bubblesort
insertsort-tree
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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?
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Knowledge of 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))
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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))))
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(knowledge-of-universe
(find-biggest-scale
1)))
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> (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
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Only two things are
infinite, the universe
and human
stupidity, and I'm
not sure about the
former.
Albert Einstein
Will there be any mystery left in
the Universe when you die?
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Any Harder Problems?
• Understanding the
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universe is (n )
• Are there any harder
problems?
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Who’s the real genius?
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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
• …
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Possible Moves
Start
Peg board game
n = number of holes
Initially, there are n-1 pegs.
Cracker Barrel’s game has
n = 15
Assume there are
always exactly
2 possible moves,
how many possible
games are there?
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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
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All Cracker Barrel Games
(starting with peg 2 1 missing)
Pegs
Left
Number
of Ways
1
1550
2
20686
3
62736
4
46728
5
5688
6
374
7
82
10
2
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Fraction of
Games
IQ Rating
0.01 “You’re Genius”
0.15 “You’re Purty Smart”
0.46 “Just Plain Dumb”
0.33
0.04
“Just Plain
0.0027
Eg-no-ra-moose”
0.00058
0.00001
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How much work is our
straightforward peg board
solving procedure?

n
(2 )
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.)
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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
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Orders of Growth
1200
1000
simulating
universe
logn
n
nlogn
n^2
800
600
peg board
game
400
n^3
2^n
200
bubblesort
insertsort-tree
0
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Orders of Growth
70000
peg board
game
60000
logn
n
nlogn
n^2
n^3
2^n
50000
40000
30000
20000
10000
0
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simulating
universe
bubblesort
insertsort-tree
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Orders of Growth
1200000
1000000
logn
n
nlogn
n^2
peg
board
game
800000
“intractable”
600000
400000
n^3
2^n
“tractable”
200000
simulating universe
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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?
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Break Lorenz Cipher Procedure
• Try all possible wheel settings
• How many possible wheel settings:
5
*5
*5
choices for first wheel
choices for second wheel
choices for third wheel
• What is n?
– The number of wheels
• There are 5n possible wheel settings
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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
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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!
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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.
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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?
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5
4
3
2
Changed goalkeeper
passback rule
1
Average Goals per Game, FIFA World Cups
Goal-den age
6
0
2002
1998
1994
1990
1986
1982
1978
1974
1970
1966
1962
1958
1954
1950
1938
1934
1930
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)?
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Grade Inflation or Scale Compression?
average GPA in 1970 (“gentleman’s C”?)
better students 1970-1985
better students 1985-2003
admitting women, non-whites (1971)
Virginia 1970 4,648,494
population increase
increase in enrollment Virginia 2000 7,078,515
2.00
*2
*2
*3
* 1.54
* 0.58
Average GPA today should be:
21.4
Students 1970
Students 2002
11,000
18,848
(12,595 UG)
CS200 has only the best of the best students, and only
the 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
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From Lecture 1:
The Liberal Arts
Trivium (3 roads)
Grammar
Rhetoric
Quadrivium (4 roads)
Logic Arithmetic
Music
Geometry
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Astronomy
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From Lecture 1:
•
Liberal
Arts
Grammar: study of meaning in written
Quadrivium
Trivium
expression
BNF replacement rules for describing languages,
rules of evaluation for meaning
• Rhetoric: comprehension of
verbal and written discourse
Not yet… Interfaces between
components, program and user
• Logic: argumentative discourse
for discovering truth
Rules of evaluation, if,
recursive definitions
Not much yet…
wait until April
• Arithmetic: understanding numbers
• Geometry: quantification of space
• Music: number in time
• Astronomy
Curves as procedures,
fractals
Yes, even if we can’t figure out how to play
“Hey Jude!”
Yes: Neil deGrasse Tyson says so
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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
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