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A world view through the computational lens
Avi WigdersonInstitute for Advanced Study
I Algorithm: the common language of nature, humans and computerII Time, space and the cosmology
of computational problemsIII Secrets and lies, knowledge and trust
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Theory of
computing
Math ComputerScience
Biology, PhysicsEconomics,…
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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
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Voyager face plate
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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
Hair implantprocess
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What is computation ?
before
after
Hair lossprocess
Long additionprocess
12345+ 6789
19134
input
output
?
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What is being computed? Function.What are possible inputs?
Representation?How to describe a computational
process?What is being manipulated? Cells/digits
before
after
Hair loss Long addition
12345
+ 6789
19134
9876543+
555555
input
function
output
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Fetal development
Weatherevolution
1 month
3 months
2 pm
4 pm
What laws govern these processes?Good theories are predictive:Nature computes the outcome – can we?
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SARS infection(in the world)
SARS infection(in the cell)
4/11/03
4/30/03
+15h
+24h
Will it spread, or die out?
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Solving integer equations
Proving theorems
Can we automate Andrew Wiles?Is there a program to solve all equations? to prove all provable theorems?
X2 + Y2 = Z2 Xn + Yn = Zn n>2
X=3 Y=4 Z=5 Theorem: no solution!Proof: Does not fit on this slide (200 pages)
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Face recognition
Emotional reactions
“my aunt Esther”
sadness
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.
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Web search
Shortest route
Public Lectures at Princeton » 2006-2007 Lectures
Lectures are free and open to the public. Lectures are
in McCosh 50 and begin at 8:00 p.m. unless ...
lectures.princeton.edu/?cat Cached - Similar pages
Start: 9th av. New York, NY
public lecture princeton 07
End: Nassau st, Princeton, NJ
1. Start out going SOUTHWEST on 9TH AVE toward W 57TH ST. 1.0 miles
2. Take the LINCOLN TUN ramp toward NEW JERSEY. 0.1 miles
3. Merge onto I-495 W (Crossing into NEW JERSEY). 0.9 miles
4. I-495 W becomes NJ-495 W. 3.2 miles
5. Merge onto I-95 S / NEW JERSEY TURNPIKE S via the exit
on the LEFT toward I-280 / NEWARK / I-78 (Portions toll). 6.3 miles
6. …
How do they do it? Is there a better way?
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How to describe computation?
Algorithms in Mathematics
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Function: input outputALGORITHM (intuitive
def):Step-by-step, simplemechanical procedure, to compute a function on every possible input
addition algorithm
12345
+ 6789
19134
input
output
Algorithms in Mathematics
History & Heroes (millennia scale)-2,300 years: Euclid [proofs and
algorithms]
-1,100 years: al-Khwārizmī [namesake of algorithms]
-70 years: Turing [defined algorithms]
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Euclid ~330-275 BCEmployment: Library of Alexandria Selected achievements:The Elements: 13 volumes on Geometry and Number TheoryMost popular math book for centuriesMath proofs: step-by-step deduction from
axiomsThe GCD algorithm: e.g. GCD(12,15)=3Math proof & algorithms alwayswalked hand in hand
Father of Geometry
function GCD (a, b) while a ≠ b if a > b then a := a – b else b := b - a return a
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al-Khwārizmī ( latin algorithmi )Employment: House of Wisdom, Baghdad, 813-846 ADSelected Publications:Geography: On the appearance of the earthAstronomy: Astronomical tablesAlgebra: Calculation by completion and
balancingArithmetic: On the Hindu art of reckoning- Describes the positional number system
(digits)- Gives algorithms for arithmetic
operations, and for solving linear and quadratic
equations
Father of Algebra
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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 Science1939-1945: Blechley Park, breaking Enigma1945-1949: building ACE, MARK-I Early electronic general purpose
computers1950: “Computing machinery and
intelligence” Foundations of Artificial Intelligence
Father of Computing
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“Long addition” algorithm
1. Scan column. If empty, stop.2. Add digits. Write answer, retain
carry.3. Move one column left, write
carry.4. Go to 1
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9 8 7 6
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ALGORITHM:Step-by-step, local,simple, mechanicalprocedure, which evolves an environment
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 theentscheindungsproblem”
1936
Turing’s 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)
DualityInput: a finite sequence of symbols xProgram: a finite sequence of symbols
M Program and input are
interchangeable!A program can be input to another
program
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UniversalityUniversal Turing machine U:input: (M,x)output: M(x) Computers can be programmed!U: hardware M: software
M1: Spell check a fileM2: Calculate salariesM3: Run computer game
M4: Play musicM5: Show movieM6: Surf the Web
20061946
…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 therelimits ??
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 problemDoes a given computer program P
halt on all inputs?
Typical program
X=8: 8, 4, 2, 1 X=6: 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
Turing’s proof: uses duality & universality
Input: x (integer) Program P0
(1) if x=1 halt
(2) if x is even, set x x/2 and go to (1)
(3) if x is odd, set x 3x+1 and go to (1)
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The limits of computationMany 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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Nature’s algorithms:von Neumann’s 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 updateCA: parallel update
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Evolution: Majority rule
Time 0123
Majority: assume colorof majority of neighbors
Will the Green populationever die out?
What happens if we replace the Majorityby another local rule?
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Artificial Life? Intelligence?
Some rules simulate a universal Turing machine (eg Conway’s “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 ruleThey 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 computerII Time, space and the cosmology
of computational problemsIII 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
US
ShortestRoute
Unsolvable
Solvable
Game Strategies
Multiplication
Addition
PatternMatching
ShortestRoute
Theorem Proving
MapColoring
NP-complete
SATIntegerFactoring
GraphIsomorphism
P
FFTNP
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I Algorithm: the common language of nature, humans and computerII Time, space and the cosmology
of computational problemsIII 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?