1. Computation

2. Binary Numbers http://schoolscience.rice.edu/duker/robots/binarynumber.html Decimal numbers Binary numbers

3. Representing Text Decide how many characters we need to represent. Determine the required number of bits. Ascii: 7 bits. Can encode 2 7 = 128 different symbols.

4. Ascii http://www.neurophys.wisc.edu/comp/docs/ascii.html

5. Representing Text Four … 01000110 01101111 01110101 01110010

6. When We Need More Characters 简体字 What about things like:

7. When We Need More Characters Answer: Unicode: 32 bits. Over 4 million characters. http://www.unicode.org/charts/ 简体字 What about things like:

8. Digital Images

9. RGB The red channel

10. RGB The green channel

11. RGB Red GreenBlue

12. Representing Pictures

13. Representing Sounds

14. Representing Programs public static TreeMap create() throws IOException { Integer freq; String word; TreeMap result = new TreeMap (); JFileChooser c = new JFileChooser(); int retval = c.showOpenDialog(null); if (retval == JFileChooser.APPROVE_OPTION) { Scanner s = new Scanner( c.getSelectedFile()); while( s.hasNext() ) { word = s.next().toLowerCase(); freq = result.get(word); result.put(word, (freq == null ? 1 : freq + 1)); } return result; }

15. The Roots of Modern Technology 5 th c B.C. Aristotelian logic invented 1642 Pascal built an adding machine 1694 Leibnitz reckoning machine

16. The Roots, continued 1834 Charles Babbage’s Analytical Engine Ada writes of the engine, “The Analytical Engine has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform.” The picture is of a model built in the late 1800s by Babbage’s son from Babbage’s drawings.

17. The Roots: Logic 1848 George Boole The Calculus of Logic chocolate nuts mint chocolate and  nuts and mint

18. Boolean Logic PQ PPP  QP  QP  QP  QP  QP  QP  Q True FalseTrue False TrueFalse True FalseTrueFalse TrueFalse True

19. Using Boolean Logic PQ PPP  QP  QP  Q(P  Q)  Q  P  ((P  Q)  Q) True FalseTrue False TrueFalse TrueFalse True False TrueFalseTrue

20. How Big Are the Truth Tables? PQR True False TrueFalseTrue False True FalseTrueFalse True False

21. 2n2n

22. Increased Power of Quantified Logic Bear(Smoky).  x (Bear(x)  Animal(x)).  x (Animal(x)  Bear(x)).  x (Animal(x)   y (Mother-of(y, x))).  x ((Animal(x)   Dead(x))  Alive(x)).

23. Using Quantified Logic (FOL) We can represent what we know as a set of logical formulas and then manipulate them.

24. Facts in English (1) Marcus was a man. (2) Marcus was a Pompeian. (3) All Pompeians were Romans. (4) Caesar was a ruler. (5) All Romans were either loyal to Caesar or hated him. (6) Everyone is loyal to someone. (7) People only try to assassinate rulers they are not loyal to. (8) Marcus tried to assassinate Caesar.

25. Facts in First Order Logic (FOL) (1) Marcus was a man. man(Marcus) (2) Marcus was a Pompeian. Pompeian(Marcus) (3) All Pompeians were Romans.  x Pompeian(x)  Roman(x) (4) Caesar was a ruler. ruler(Caesar) (5) All Romans were either loyal to Caesar or hated him.  x Roman(x)  loyalto(x, Caesar)  hate(x, Caesar) (6) Everyone is loyal to someone.  x  y loyalto(x, y) (7) People only try to assassinate rulers they are not loyal to.  x  y person(x)  ruler(y)  tryassassinate(x, y)   loyalto(x, y) (8) Marcus tried to assassinate Caesar. tryassassinate(Marcus, Caesar)

26. Search

27. Breadth-First Search Is this a good idea?

28. Depth-First Search

29. Scalability Solving hard problems requires search in a large space. To play master-level chess requires searching about 8 ply deep. So about 35 8 or 2  10 12 nodes must be examined.

30. Growth Rates of Functions

31. The Advent of the Computer 1945 ENIAC The first electronic digital computer

32. 1949 EDVAC The first stored program computer

33. Moore’s Law http://www.intel.com/technology/mooreslaw/

34. How Much Computer Power Might It Take? http://www.frc.ri.cmu.edu/~hpm/book97/ch3/index.html

35. How Much Compute Power is There? From Hans Moravec, Robot Mere Machine to Transcendent Mind 1998.

36. Kurweil’s Vision http://www.pocket-lint.co.uk/news/news.phtml/12920/13944/Computers- match-humans-by-2030.phtml

37. Limits to What We Can Compute Are there fundamentally uncomputable things? Does God exist? What’s the best way to run a country? Does this puzzle have a solution?

38. Mathematics in the Early 20 th Century 1900 Hilbert’s program and the effort to formalize mathematics 1931 Kurt Gödel’s paper, On Formally Undecidable Propositions 1936 Alan Turing’s paper, On Computable Numbers with an application to the Entscheidungs problem

39. What Can Algorithms Do? 1.Can we make all true statements theorems? 2.Can we decide whether a statement is a theorem?

40. Gödel’s Incompleteness Theorem Kurt Gödel showed, in the proof of his Incompleteness Theorem [Gödel 1931], that the answer to question 1 is no. In particular, he showed that there exists no decidable axiomatization of Peano arithmetic that is both consistent and complete.

41. The Entscheidungsproblem Does there exist an algorithm to decide, given an arbitrary sentence w in first order logic, whether w is valid? Given a set of axioms A and a sentence w, does there exist an algorithm to decide whether w is entailed by A? Given a set of axioms, A, and a sentence, w, does there exist an algorithm to decide whether w can be proved from A?

42. Turing Machines At each step, the machine must: ● choose its next state, ● write on the current square, and ● move left or right.

43. An Example

44. The Church-Turing Thesis Turing machines Lambda calculus Standard programming languages like Java Conway’s game of life DNA computing All formalisms powerful enough to describe everything we think of as a computational algorithm are equivalent.

45. The Game of Life Playing the game The rules: ● A dead cell with exactly three live neighbors becomes a live cell (birth). ● A live cell with two or three live neighbors stays alive (survival). ● In all other cases, a cell dies or remains dead (overcrowding or loneliness). A game halts iff it reaches some stable configuration.

46. Back to the Entscheidungsproblem 1.Given a Turing machine M, we can construct a logical formula F that is true iff M ever halts. 2.If there were a solution to the Entscheidungsproblem, then we could determine the truth of F and thus be able to decide whether M ever halts. 3.But there is no procedure for determining (in the general case) whether M halts. 4.So there is no solution to the Entscheidungsproblem. Theorem: The Entscheidungsproblem is unsolvable. Proof: (A variant of Turing’s proof)

47. The Halting Problem Turing machine, M input string, w Does M halt on w? Accept Reject

48. An Example of a Similar Question Does this program halt on all inputs? times3(x: positive integer) = While x  1 do: If x is even then x = x/2. Else x = 3x + 1. Let’s try it.

49. The Halting Problem Is Undecidable Proof: If it were decidable, then some TM M H would decide it. M H would implement the specification: halts( ) = If is a Turing machine description and M halts on input w then accept. Else reject.

50. Trouble Trouble(x: string) = if halts(x, x) then loop forever, else halt. What is Trouble( )? What is halts( )? ● If halts reports that Trouble( ) halts, Trouble ________. ● If halts reports that Trouble( ) does not halt, Trouble __________.

51. Another Undecidable Problem The Post Correspondence Problem

52. A PCP Instance With a Simple Solution iXY 1 baab 2 abbb 3 abaa 4 baaababa

53. A PCP Instance With a Simple Solution iXY 1 baab 2 abbb 3 abaa 4 baaababa Solution: 3, 4, 1

54. Another PCP Instance iXY 1 11011 2 010 3 001110

55. A PCP Instance With No Simple Solution iXY 1 11011 2 011011 3 1110

56. A PCP Instance With No Simple Solution iXY 1 11011 2 011011 3 1110 Shortest solution has length 252.

57. A Tiling Problem Given a finite set T of tiles of the form: Is it possible to tile an arbitrary surface in the plane?

58. A Set of TilesThat Cannot Tile the Plane

59. Is the Tiling Problem Decidable? Wang’s conjecture: If a given set of tiles can be used to tile an arbitrary surface, then it can always do so periodically. In other words, there must exist a finite area that can be tiled and then repeated infinitely often to cover any desired surface. But Wang’s conjecture is false.

60. The Implications The Entscheidungs problem is undecidable. There’s no black box reasoning engine for FOL. Would quantum computing change the picture? Does undecidability doom our attempt to make artificial copies of ourselves?

61. Is Decidability Enough?

62. Boolean Logic, Again PQ PPP  QP  QP  Q(P  Q)  Q  P  ((P  Q)  Q) True FalseTrue False TrueFalse TrueFalse True False TrueFalseTrue How many steps would it take a deterministic Turing machine to examine this table?

63. Nondeterminism

64. Nondeterministically Deciding SAT PQ PPP  QP  QP  Q(P  Q)  Q  P  ((P  Q)  Q) True FalseTrue False TrueFalse TrueFalse True False TrueFalseTrue How many steps in a single path of the process of examining this table?

65. P and NP P – problems that are solvable deterministically in polynomial time. NP – problems that are solvable nondeterministically in polynomial time.

66. The Traveling Salesman Problem Given n cities and the distances between each pair of them, find the shortest tour that returns to its starting point and visits each other city exactly once along the way. 15 20 25 8 9 23 40 10 4 7 3 28

67. The Traveling Salesman Problem 15 20 25 8 9 23 40 10 4 7 3 28 Given n cities: Choose a first cityn Choose a secondn-1 Choose a thirdn-2 … n!

68. The Traveling Salesman Problem Can we do better than n! ● First city doesn’t matter. ● Order doesn’t matter. So we get (n-1!)/2.

69. The Growth Rate of n! 2211479001600 36126227020800 4241387178291200 5120141307674368000 67201520922789888000 7504016355687428096000 840320176402373705728000 936288018121645100408832000 103628800192432902008176640000 113991680036 3.6  10 41

70. The Traveling Salesman Problem

71. Solving it Nondeterministically 15 20 25 8 9 23 40 10 4 7 3 28

72. SAT and Traveling Salesman DND SAT2n2n n Traveling Salesmann!n!n

73. P and NP P – problems that are solvable deterministically in polynomial time. NP – problems that are solvable nondeterministically in polynomial time. NP-complete – NP problems that are at least as hard as every other NP-complete problem.

74. Growth Rates of Functions, Again

75. Does Complexity Doom AI?