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

Theory of

computing

Math ComputerScience

Biology, PhysicsEconomics,…

Intelligence: Man versus Termite

Patterns vs. brain size SURVEY Are termites intelligent?

Humans (~1011 neurons) Termites (105 neurons)

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Voyager face plate

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

Computation is everywhere

- Long list of natural phenomena and intellectual challenges- All have an essential computational component

What is computation ?

before

after

Hair implantprocess

What is computation ?

before

after

Hair lossprocess

Long additionprocess

12345+ 6789

19134

input

output

?

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

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?

SARS infection(in the world)

SARS infection(in the cell)

4/11/03

4/30/03

+15h

+24h

Will it spread, or die out?

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)

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.

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?

How to describe computation?

Algorithms in Mathematics

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]

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

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

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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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

Turing machine

Demo

“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

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

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

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!

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

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)

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?

Algorithms In

Nature

“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

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

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?

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 !

Algorithms can explain nature

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

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

Beauty from algorithms

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

Plan for the coming lectures

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

Computation is everywhere

US

ShortestRoute

Unsolvable

Solvable

Game Strategies

Multiplication

Addition

PatternMatching

ShortestRoute

Theorem Proving

MapColoring

NP-complete

SATIntegerFactoring

GraphIsomorphism

P

FFTNP

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?