A world view through the computational lens

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A world view through the computational lens
I Algorithm the common language of
nature, humans and computer II Time, space and
the cosmology of computational problems III
Secrets and lies, knowledge and trust

• Avi Wigderson
• Institute for Advanced Study

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Computer Science
Math
Theory of computing
Biology, Physics Economics,
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Intelligence Man versus Termite

• Patterns vs. brain size
• SURVEY
• Are termites intelligent?
• Humans (1011 neurons)
  Termites (105 neurons)

2 3 5 7 11 13 17 19 23
Voyager face plate
2 3 5 7 11 13 17 19 23
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• Lecture I - plan
• -- Computation is everywhere
• -- Algorithms in Mathematics
• -- The Turing Machine
• -- Limits on CS and Math knowledge
• -- Algorithms in Nature
• -- von Neumann cellular automata
• -- Limits on scientific knowledge

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Computation is everywhere- Long
list of natural phenomena and intellectual
challenges- All have an essential computational
component
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What is computation ?
before
after
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What is computation ?
before
?
after
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9876543 555555
input
function
output
What is being computed? Function. What are
possible inputs? Representation? How to describe
a computational process? What is being
manipulated? Cells/digits
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2 pm
1 month
4 pm
3 months
What laws govern these processes? Good theories
are predictive Nature computes the outcome can
we?
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15h
4/11/03
24h
4/30/03
Will it spread, or die out?
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X2 Y2 Z2
Xn Yn Zn ngt2
X3 Y4 Z5
Theorem no solution! Proof Does not fit on
this slide (200 pages)
Can we automate Andrew Wiles? Is there a program
to solve all equations? to prove all
provable theorems?
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Indonesian 737-400 feared lost with 102
aboard.Indonesia's transportation minister said
Tuesday that rescuers had not found the wreckage
of a missing passenger jetliner, despite earlier
statements from aviation and police officials
that it had been located.
my aunt Esther
sadness
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How do they do it? Is there a better way?
public lecture princeton 07
Start 9th av. New York, NY
End Nassau st, Princeton, NJ
Public Lectures at Princeton 2006-2007
Lectures Lectures are free and open to the
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800 p.m. unless ... lectures.princeton.edu/?cat
Cached - Similar pages
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toward I-280 / NEWARK / I-78 (Portions toll).
6.3 miles 6.
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How to describe computation? Algorithms in
Mathematics
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Algorithms in Mathematics
12345 6789
Function input ? output ALGORITHM (intuitive
def) Step-by-step, simple mechanical procedure,
to compute a function on every possible input
input
addition algorithm
19134
output
History Heroes (millennia scale) -2,300 years
Euclid proofs and algorithms -1,100
years al-Khwarizmi namesake of algorithms
-70 years Turing defined algorithms
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Father of Geometry
Euclid 330-275 BC Employment Library of
Alexandria Selected achievements The Elements
13 volumes on Geometry and Number Theory Most
popular math book for centuries Math proofs
step-by-step deduction from axioms The GCD
algorithm e.g. GCD(12,15)3 Math proof
algorithms always walked hand in hand
function GCD (a, b) while a ? b
if a gt b then a a b
else b b - a return a
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Father of Algebra

• al-Khwarizmi ( latin algorithmi )
• Employment House of Wisdom,
• Baghdad, 813-846 AD
• Selected Publications
• Geography On the appearance of the earth
• Astronomy Astronomical tables
• Algebra Calculation by completion and balancing
• Arithmetic On the Hindu art of reckoning
• Describes the positional number system (digits)
• Gives algorithms for arithmetic operations,
• and for solving linear and quadratic equations

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Father of Computing
Alan Mathison Turing 1912-1954 Selected
achievements 1936 On computable numbers, with
an application to the entscheindungsproblem
Formal definition of an algorithm Foundations
of Computer Science 1939-1945 Blechley Park,
breaking Enigma 1945-1949 building ACE, MARK-I
Early electronic general purpose computers 1950
Computing machinery and intelligence
Foundations of Artificial Intelligence
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Long addition algorithm
1
1
1
5
4
3
2
1
9
8
7
6
4

1
3
1
9

• Scan column. If empty, stop.
• Add digits. Write answer, retain carry.
• Move one column left, write carry.
• Go to 1

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ALGORITHM Step-by-step, local, simple,
mechanical procedure, which evolves an
environment
1
1
1
5
4
3
2
1
9
8
7
6
4

1
3
1
9
Environment infinitely many cells, regular Cell
can hold one symbol from a finite alphabet Head
local moves, read/write symbol, has a
state which remember a few symbols ALGORITHM
finite table of instructions Can handle
infinite number of different inputs
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Turing machine Demo
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• On computable numbers,
• with an application to the
• entscheindungsproblem 1936
• Turings insights
• What is computation what is computed
• Duality of program and input
• Universality the computer revolution
• The power of computation
• The limits of computation

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What is computation Formal definition of an
algorithm A Turing machine which halts in finite
time on every possible (finite) input. Machine M
on input x computes M(x) Duality Input a
finite sequence of symbols x Program a finite
sequence of symbols M Program and input are
interchangeable! A program can be input to
another program
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Universality Universal Turing machine U input
(M,x) output M(x) Computers can be
programmed! U hardware M software
Computer revolution Practice after Theory
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The power of computation Church-Turing Thesis
Every function computable by any reasonable
device, is computable by a Turing machine
Thesis stood unchallenged for 70
years! Corollary Java, C, CRAY,.. Can
be Corollary Every natural process can be
simulated by a Turing machine. THINK
ABOUT IT!
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The limits of computation Are there limits ??

• Turing 36 no algorithm can solve these
• Given a program, does it have a bug?
• Given a math statement, is it provable?
• 36-06 and many other natural ones

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An incomputable problem

• Does a given computer program P
• halt on all inputs?
• Typical
• program
• X8 8, 4, 2, 1
• X6 6, 3, 10, 5, 16, 8, 4, 2, 1
• - So far, Math cannot answer this for P0
• No algorithm can answer this for all P
• Turings proof uses duality universality

Input x (integer) Program P0 (1)
if x1 halt (2) if x is even, set x ? x/2
and go to (1) (3) if x is odd, set x ? 3x1
and go to (1)
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The limits of computation

• Many natural incomputable functions!!
• Is a given computer program bug-free?
• Is a math statement provable?
• Is a given equation solvable?
• Absolute limits on what can be known in
  Mathematics and Computer Science!
• What about the Natural Sciences?

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Algorithms In Nature
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Life, the Universe, and Everything

• Computation evolution of an environment via
• repeated application of simple, local
  rules
• (Almost) all Physics and Biology theories
  satisfy!
• Weather - Proteins in a cell - magnetization
• Ant hills - Fish schools - fission
• Brain - Populations - burning fire
• Epidemics Regeneration - growth

applied simultaneously everywhere
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Natures algorithmsvon Neumanns Cellular
Automata

• A environment of cells
• e.g. a (large) grid
• A neighborhood structure
• e.g cells touching you
• Every cell has finite state
• e.g yellow or green
• representing biological, chemical, physical,
  info.
• Update rule e.g. Majority
• - Initial configuration

TM sequential update CA parallel update
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Evolution Majority rule
Majority assume color of majority of
neighbors Will the Green population ever die
out? What happens if we replace the Majority by
another local rule?
Time
0
1
2
3
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Artificial Life? Intelligence?

• Some rules simulate a universal Turing machine
  (eg Conways Game of Life).
• Conclusions
• - Incomputable to predict evolution in CA
• CA can self reproducing (is it alive?)
• CA can list prime numbers (intelligent?)
• Termites brain can implement any CA rule
• They can list primes, and generate any structure
  computers humans can !

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Algorithms can explain nature

• Synchrony self stabilization
• Fireflies coordinating their flashing
• Heart muscles contracting in rhythm
• Neurons firing in unison

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Synchrony self stabilization

• Programming challenge design termite to
• Put any number of termites in a row.
• Kick any one of them (gently)
• After finite time steps, they march

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Beauty from algorithms
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Summary

• Computation is everywhere
• Turing machine capture all computation
• Algorithmic thinking and modeling reveals new
  aspects of natural phenomena,
• mathematical structures, and our limits
• to understanding in math science

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Plan for the coming lectures
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• I Algorithm the common language of
• nature, humans and computer
• II Time, space and the cosmology of
  computational problems
• III Secrets and lies, knowledge and trust
• Hard and easy problems.
• The importance of efficient algorithms
• The P vs. NP problem, and why is it
• so important to science mathematics
• - The ubiquity of NP-complete problems

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Computation is everywhere
Solvable
Unsolvable
Game Strategies
Graph Isomorphism
Integer Factoring
SAT
Pattern Matching
NP-complete
Shortest Route
Theorem Proving
Shortest Route
P
Map Coloring
Multiplication
Addition
FFT
NP
US
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• I Algorithm the common language of
• nature, humans and computer
• II Time, space and the cosmology of
  computational problems
• III Secrets and lies, knowledge and trust
• - The amazing utility of hard problems
• The assumptions and magic behind security
• of the Internet E-commerce
• How to play Poker, but
• without the cards?