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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. A world view through the computational lens. Avi Wigderson Institute for Advanced Study. Computer Science.

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A world view through the computational lens

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  1. 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 A world view through the computational lens Avi Wigderson Institute for Advanced Study

  2. Computer Science Math Theory of computing Biology, Physics Economics,…

  3. 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

  4. 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

  5. Computation is everywhere-Long list of natural phenomena and intellectual challenges- All have an essential computational component

  6. Hair implant process What is computation ? before after

  7. 12345 + 6789 input Hair loss process Long addition process 19134 output What is computation ? before ? after

  8. 12345 + 6789 before Hair loss Long addition 19134 after 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

  9. Fetal development Weather evolution 2 pm 1 month 4 pm 3 months What lawsgovern these processes? Good theories are predictive: Nature computes the outcome – can we?

  10. SARS infection (in the world) SARS infection (in the cell) +15h 4/11/03 +24h 4/30/03 Will it spread, or die out?

  11. Solving integer equations Proving 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) Can we automate Andrew Wiles? Is there a program to solve all equations? to prove all provable theorems?

  12. Face recognition Emotional reactions 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

  13. Web search Shortest route 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 public. Lectures are in McCosh 50 and begin at 8:00 p.m. unless ... lectures.princeton.edu/?catCached - Similar pages 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. …

  14. How to describe computation? Algorithms in Mathematics

  15. 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-Khwārizmī[namesake of algorithms] -70 years:Turing [defined algorithms]

  16. 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 > b then a := a – b else b := b - a return a

  17. Father of Algebra • al-Khwārizmī ( latinalgorithmi ) • 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

  18. 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:buildingACE, MARK-I Early electronic general purpose computers 1950:“Computing machinery and intelligence” Foundations of Artificial Intelligence

  19. “Long addition” algorithm 1 1 1 1 2 3 4 5 6 7 8 9 4 1 9 1 3 • Scan column. If empty, stop. • Add digits. Write answer, retain carry. • Move one column left, write carry. • Go to 1

  20. ALGORITHM: Step-by-step, local, simple, mechanical procedure, which evolves an environment 1 1 1 1 2 3 4 5 6 7 8 9 4 1 9 1 3 Environment: infinitely many cells, regular Cell: can hold one symbol from a finite alphabet Headlocal moves, read/write symbol, has a state which “remember” a few symbols ALGORITHM: finite table of instructions Can handle infinite number of different inputs

  21. Turing machine Demo

  22. “On computable numbers, • with an application to the • entscheindungsproblem”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

  23. 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

  24. 1946 M4: Play music M5: Show movie M6: Surf the Web M1: Spell check a file M2: Calculate salaries M3: Run computer game 2006 Universality Universal Turing machine U: input: (M,x) output: M(x) Computers can be programmed! U: hardware M: software …Computer revolution… Practice after Theory

  25. The power of computation Church-Turing Thesis: Every functioncomputable 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!

  26. 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

  27. An incomputable problem • Does 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 forP0 • 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)

  28. 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?

  29. Algorithms In Nature

  30. “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

  31. 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 update CA: parallel update

  32. Evolution: Majority rule Majority: assume color of majority of neighbors Will the Greenpopulation ever die out? What happens if we replace the Majority by another local rule? Time 0 2 3 1

  33. 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 rule They can list primes, and generate any structure computers & humans can !

  34. Algorithms can explain nature Synchrony & self stabilization • Fireflies coordinating their flashing • Heart muscles contracting in rhythm • Neurons firing in unison • …

  35. 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

  36. Beauty from algorithms

  37. 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

  38. Plan for the coming lectures

  39. 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

  40. Solvable Unsolvable Game Strategies Graph Isomorphism Integer Factoring SAT Computation is everywhere Pattern Matching NP-complete Shortest Route Theorem Proving Shortest Route P Map Coloring Multiplication Addition NP FFT US

  41. 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?

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