Problems and Procedures P=NP Recap Endless Golden Ages Golden Catastrophes Optimism

1. Class 30:
Sex, Relig
ion, and
Politics
cs1120 Fall 2011
David Evans
2 November 2011

2. Plan
Recap big ideas from Monday (without
Halloweenization)
Tyson’s Science’s Endless Golden Age
2

3. What is a problem?
3

4. Procedures vs. Problems
Procedure: Problem:
a well-defined description of an
sequence of steps input and desired
that can be executed output
mechanically (i.e., by
a Turing Machine)
4

5. Example: Factoring
Factoring Procedure Factoring Problem
def factors(n): Input: a number, n, that is
res = [] a product of two primes
for d in range(2, n): numbers
if n % d == 0: Output: two prime
res.append(d) numbers, p and q, such
return res that pq = n
What is the running time of What is the time complexity
this factoring procedure? of the factoring problem?
5

6. P=NP ?
P = set of all problems that can be solved by
algorithms with running times in O(Nk)
where N is the input size and k is any
constant.
NP = set of all problems whose answer can be
checked by algorithms with running times
in O(Nk) where N is the input size and k is
any constant.

7. What about factoring?
How hard is the factoring problem?
Input: a number, n
Output: two integers > 1, p and q, such that pq = n
How hard is the CHECK-FACTORS problem?
Input: p, q, n (three integers, all greater than 1)
Output: true if n = pq, false otherwise

8. ACM1: Beyond Cyberspace (2001)
Neil de Grasse Tyson: "Awash in Data, Awash in Discoveries:
Defining the Golden Age of Science“
Rice Hall Dedication Speaker!
Alan Kay: "The Computer Revolution Hasn't Happened Yet"
Friday,President and Disney Fellow, Walt Disney
(Vice Nov 18, 3pm: “The Future
Belongs to the Innovators”
Imagineering)
Dean Kamen: "Engineering the Future Depends on Future
Engineers" (President and Owner, DEKA Research &
Development Corporation and Founder of FIRST)

9. Science’s Endless
Golden Age

10. “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.”
What is the asymptotic running time
for simulating the universe?

11. “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.”

12. Simulating the Universe
Work scales linearly with volume of
simulation: scales cubically with scale
(n3)
When we double the scale of the
simulation, the work octuples! (Just like
oceanography octopi simulations)

13. Orders of Growth
140
n3 simulating
120 universe
100
80
60
40
n2
20
n
0
1 2 3 4 5

14. Orders of Growth
1400000
simulating
1200000
universe
1000000
800000
600000
400000
200000
0
1 11 21 31 41 51 61 71 81 91 101

15. Orders of Growth
7E+32 factors
2n
6E+32
5E+32
4E+32
3E+32
2E+32
1E+32
0 simulating
1 11 21 31 41 51 61 71 81 91 101 universe

16. Astrophysics and Moore’s Law
Simulating universe is (n 3)
Moore’s “law”: computing power
doubles every 18 months
Dr. 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?

17. Knowledge of the Universe
import math
# 18 months * 2 = 12 months * 3
yearlyrate = math.pow(4, 1.0/3.0) # cube root
def simulation_work(scale):
return scale ** 3
def knowledge_of_universe(scale):
return math.log(scale, 10) # log base 10
def computing_power(nyears):

18. Knowledge of the Universe
import math
# 18 months * 2 = 12 months * 3
yearlyrate = math.pow(4, 1.0/3.0) # cube root
def simulation_work(scale):
return scale ** 3
def knowledge_of_universe(scale):
return math.log(scale, 10) # log base 10
def computing_power(nyears):
def computing_power(nyears): return yearlyrate ** nyears
if nyears == 0: return 1
else: return yearlyrate * computing_power(nyears - 1)

19. Knowledge of the Universe
def computing_power(nyears):
return yearlyrate ** nyears
def simulation_work(scale):
return scale ** 3
def knowledge_of_universe(scale):
return math.log(scale, 10) # log base 10
def relative_knowledge_of_universe(nyears):
scale = 1
while simulation_work(scale + 1) <= 1000 * computing_power(nyears):
scale = scale + 1
return knowledge_of_universe(scale)

20. While Loop
Statement ::= while Expression: Block
(define (while pred body)
x=1 (if (pred)
x =res = 0
1 (begin
(body)
reswhile x < 100:
=0 (while pred body))))
while x < 100:+ x
res = res (define x 1)
resx= res + x
=x+1 (define sum 0)
print resres
print (while (lambda () (< x 100))
(lambda ()
(set! sum (+ sum x))
(set! x (+ x 1))))

21. Knowledge of the Universe
def computing_power(nyears):
return yearlyrate ** nyears
def simulation_work(scale):
return scale ** 3
def knowledge_of_universe(scale):
return math.log(scale, 10) # log base 10
def relative_knowledge_of_universe(nyears):
scale = 1
while simulation_work(scale + 1) <= 1000 * computing_power(nyears):
scale = scale + 1
return knowledge_of_universe(scale)
(Note: with a little bit of math, could
compute this directly using a log instead.)

22. > >>> relative_knowledge_of_universe(0)
1.0
>>> relative_knowledge_of_universe(1)
1.0413926851582249
>>> relative_knowledge_of_universe(2)
1.1139433523068367
>>> relative_knowledge_of_universe(10)
1.6627578316815739
>>> relative_knowledge_of_universe(15)
2.0
>>> relative_knowledge_of_universe(30)
3.0064660422492313
>>> relative_knowledge_of_universe(60)
5.0137301937137719
>>> relative_knowledge_of_universe(80)
6.351644238569782
Will there be any mystery left in the Universe when you die?

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

24. The Endless Golden Age
Golden Age – period in which knowledge/quality
of something improves exponentially
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, agriculture, etc.
Accumulating 4% per year => doubling every 15 years!

25. Endless/Short Golden Ages
Endless golden age: at any point in history, the
amount known is twice what was known 15
years ago
Continuous exponential growth: (kn)
k is some constant (e.g., 1.04), n is number of years
Short golden age: knowledge doubles during a
short, “golden” period, but only improves
linearly most of the time
Mostly linear growth: (n)
n is number of years

26. Average Goals per Game, FIFA World Cups
3
4
5
6
2
1930
1934
1938
1950
1954
1958
1962
Goal-den age
1966
1970
1974
1978
1982
1986
1990
passback rule
1994
Changed goalkeeper
1998
2002
2006
2010

27. 27

28. Computing Power 1969-2011
(in Apollo Control Computer Units)
300,000,000
250,000,000
200,000,000
150,000,000
100,000,000
50,000,000
0

29. Computing Power 1969-1990
(in Apollo Control Computer Units)
18,000
16,000
14,000
12,000
10,000
8,000
6,000
4,000
2,000
0
.5
.5
.5
.5
.5
.5
.5
69
72
75
78
81
84
87
90
70
73
76
79
82
85
88
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19

30. Computing Power 2008-2050?
450,000
400,000
“There’s an unwritten rule in
350,000
astrophysics: your computer
30 years from now?!
300,000
simulation must end before you die.”
250,000
Neil deGrasse Tyson
200,000
150,000 Human brain:
Brain simulator today (IBM Blue Brain): ~100 Billion
100,000
10,000 neurons neurons
30 Million connections
50,000 100 Trillion
connections
0
2008 2011 2014 2017 2020 2023 2026 2029 2032 2035 2038 2041 2044 2047 2050
2011 = 1 computing unit

31. More Moore’s Law?
“Any physical quantity that’s growing
Current technologies are already
exponentially predicts a disaster, you simply
slowing down...
can’t go beyond certain major limits.”
Gordon Moore (2007)
An analysis of the history of technology shows that technological change is
exponential, contrary to the common-sense 'intuitive linear' view. So we won't
experience 100 years of progress in the 21st century-it will be more like 20,000
years of progress (at today’s rate). The 'returns,' such as chip speed and cost-
effectiveness, also increase exponentially. There’s even exponential growth in
the rate of exponential growth. Within a few decades, machine intelligence will
surpass human intelligence, leading to the Singularity — technological change
so rapid and profound it represents a rupture in the fabric of human history.
The implications include the merger of biological and non-biological
intelligence, immortal software-based humans, and ultra-high levels of
intelligence that expand outward in the universe at the speed of light.
but advances won’t come from more transistors with Ray Kurzweil
current technologies...new
technologies, algorithms, parallelization, architectures,

32. Log scale: straight line = exponential curve!
32

33. 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)*
philosophy (400BC-350BC?)
art (1875-1925?)
soccer (1950-1966)
baseball (1925-1950?)
movies (1920-1940?)
have short golden ages?
* or whatever was popular when you were 16.

34. Golden Ages
or
Golden Catastrophes?

35. PS5, Question 1e
Question 1: For each f and g pair below, argue
convincingly whether or not g is (1) O(f), (2)
Ω(f), and (3) Θ(g) …
(e) g: the federal debt n years from today,
f: the US population n years from today

36. Malthusian Catastrophe
Reverend Thomas
Robert
Malthus, Essay on
the Principle of
Population, 1798

37. 37

38. Malthusian Catastrophe
“The great and unlooked for discoveries that have taken
place of late years in natural philosophy, the increasing
diffusion of general knowledge from the extension of
the art of printing, the ardent and unshackled spirit of
inquiry that prevails throughout the lettered and even
unlettered world, … have all concurred to lead many
able men into the opinion that we were touching on a
period big with the most important changes, changes
that would in some measure be decisive of the future
fate of mankind.”

39. Malthus’ Postulates
“I think I may fairly make two postulata.
First, that food is necessary to the existence of
man.
Secondly, that the passion between the sexes
is necessary and will remain nearly in its
present state.
These two laws, ever since we have had any
knowledge of mankind, appear to have been
fixed laws of our nature, and, as we have not
hitherto seen any alteration in them, we have
no right to conclude that they will ever cease to
be what they now are…”

40. Malthus’ Conclusion
“Assuming then my postulata as granted, I
say, that the power of population is indefinitely
greater than the power in the earth to produce
subsistence for man.
Population, when unchecked, increases in a
geometrical ratio. Subsistence increases only in
an arithmetical ratio. A slight acquaintance
with numbers will show the immensity of the
first power in comparison of the second.”

41. Malthusian Catastrophe
Population growth is geometric: (kn) (k > 1)
Food supply growth is linear: (n)
What does this mean as n ?
Food per person = food supply / population
= (n) / (kn)
As n approaches infinity, food per person
approaches zero!

42. Malthus’
Fallacy?

43. Malthus’ Fallacy
He forgot how he started:
“The great and unlooked for discoveries that
have taken place of late years in natural
philosophy, the increasing diffusion of general
knowledge from the extension of the art of
printing, the ardent and unshackled spirit of
inquiry that prevails throughout the lettered
and even unlettered world…”

44. Golden Age of Food Production
Agriculture is an “endless golden age”
field: production from the same land increases
as ~ (1.02n)
Increasing knowledge of
farming, weather forecasting, plant
domestication, genetic engineering, pest
repellants, distribution
channels, preservatives, etc.

45. Growing Corn
1906: < 1,000 2006: 10,000
pounds per acre pounds per acre
Michael Pollan’s The Omnivore’s Dilemma

46. Note: Log axis!
Corn Yield
http://www.agbioforum.org/v2n1/v2n1a10-ruttan.htm

47. 47

48. Green Revolution
Norman Borlaug (1914-2009)

49. “At a time when doom-sayers were hopping around saying
everyone was going to starve, Norman was working. He moved
to Mexico and lived among the people there until he figured
out how to improve the output of the farmers. So that saved a
million lives. Then he packed up his family and moved to
India, where in spite of a war with Pakistan, he managed to
introduce new wheat strains that quadrupled their food
output. So that saved another million. You get it? But he wasn't
done. He did the same thing with a new rice in China. He's
doing the same thing in Africa -- as much of Africa as he's
allowed to visit. When he won the Nobel Prize in 1970, they
said he had saved a billion people. That's BILLION! BUH! That's
Carl Sagan BILLION with a "B"! And most of them were a
different race from him. Norman is the greatest human
being, and you probably never heard of him.”
Penn Jillette (Penn & Teller)

50. Malthus was wrong about #2 Also
Advances in science (birth
control), medicine (higher life
expectancy), education, and societal
and political changes (e.g., regulation
in China) have reduced k (it is < 1 in
many countries now!)

51. Upcoming Malthusian Catastrophes?
Human consumption
of fossil fuels grows
as (kn) (fairly
large k like 1.08?)
Available fuel is
constant (?)
http://wwwwp.mext.go.jp/hakusyo/book/hpag200001/hpag200001_2_006.html

52. PS4, Question 1e
g: the federal debt n years from today,
f: the US population n years from today
Debt increases:
Spending – Revenues
this varies, but usually positive
+ Interest on the previous debt (exponential)
= (kn)
Population increase is not exponential:
rate continues to decrease
=> as n increases, debt per person approaches infinity!
This will eventually be a problem, but growth analysis doesn’t say when.

53. “Cornucopian View”
Few resources are really finite
All scientific things have endless golden ages
Knowledge accumulates
Knowledge makes it easier to acquire more
(We hope) Human ingenuity and economics and
politics will continue solve problems before
they become catastrophes
No one will sell the last gallon of gas for $3.35

54. “Today, of Americans
officially designated as
‘poor’, 99 percent have
electricity, running
water, flush toilets, and a
refrigerator; 95 percent have
a television, 88 percent a
telephone, 71 percent a car
and 70 percent air
conditioning. Cornelius
Vanderbilt had none of
these.” Matt Ridley
54

55. Charles Vest’s slide idea!
America will
always do the
right thing
but only
after exhausting
all other options.
Winston Churchill
55

56. “Kay”-sian View
The best way to
predict the
future is to
invent it.
— Alan Kay

57. Charge
When picking majors:
• Pick a short golden age field that is about to
enter its short golden age: this requires vision
and luck!
• Or, play it safe by picking an endless golden
age field (CS is a good choice for this!)
Friday:
Python Dictionaries
History of Object-Oriented Programming