IST 4 Information and Logic. Claude Shannon April 30,

Transcription

1 IST 4 Information and Logic Claude Shannon April 30,

2 T = today x= hw#x out x= hw#x due mon tue wed thr 28 M1 oh 1 4 oh M1 oh oh 1 2 M2 18 oh oh 2 fri oh oh = office hours oh 25 oh M2 2 3 T oh midterms oh Mx= MQx out 9 oh 3 4 oh Mx= MQx due 16 oh oh oh 5 oh oh oh

3 Boolean Proofs: Axioms + Fun

4 Proof Dependency A1-A4 L1 L2 T0 T1 T2 T3 T4

5 Syllogism to Algebra George Boole, 1847 George Boole

6 George Boole George Boole Early Days Born in Lincoln, England an industrial town His father was a shoemaker with a passion for mathematics and science When George was 8 he surpassed his father s knowledge in mathematics By age 14 he was fluent in Latin, German, French, Italian and English and algebra When his was 15 he had to go to work to support his family, he became a math teacher in the Wesleyan Methodist academy in Doncaster (40 miles ) Lost his job after two years. Lost two more teaching jobs When he was 20 he opened his own school in his hometown - Lincoln

7 George Boole George Boole Early Career Born in Lincoln, England an industrial town In 1841 (26) he published three papers in the newly established Cambridge Mathematical Journal (edited by DG). In 1844 (29) he published On a General Method of Analysis ; he considered it to be his best paper. This paper won the first (newly established) Gold medal for Mathematics awarded by Royal Society In 1846 (31) he applied for a professor position in the newly established Queen s College 3 campuses in Ireland In 1847 (32) while waiting to hear from Ireland, he published The Mathematical Analysis of Logic source: wikipedia

8 George Boole George Boole Ireland 8848 m 29,029 ft Born in Lincoln, England an industrial town In 1847 (32) while waiting to hear from Ireland, he published The Mathematical Analysis of Logic In 1849 (34), his was offered a position of the first professor of mathematics at Queen s college at Cork He married Mary Everest ( ) in 1855 (23,40) and they had five daughters Niece of George Everest (Mt. Everest ) led the expedition to map the Himalayas In 1864, died of pneumonia (49) source: wikipedia

9 , taught at at Queen s college in Cork George Boole Geography Born in Lincoln, England an industrial town source: wikipedia

10 : lived here until got married Cork, Ireland Source:

11 Gottfried Leibniz Leibniz never held an academic position George Boole Boole was never a student at a university only a professor

12 Algorizmi AD It is all about Binary Gottfried Leibniz ??? Boolean George Boole

13 Boolean Algebra Boolean is not Binary...

14 Two-valued Boolean Algebra Boolean Algebra: set of elements B={0,1}, two binary operations OR and AND xy OR(x,y) xy AND(x,y) iff both x and y are 0 1 iff both x and y are 1 We proved it is a Boolean algebra

15 Four-valued Boolean Algebra Four-valued Boolean Algebra: set of elements?? two binary operations?? and?? Two-valued Boolean Algebra: set of elements B={0,1}, two binary operations OR and AND

16 0-1 vectors: operations? Elements: () () () ()

17 Elements are: (), (), (), () +

18 Elements are: (), (), (), () Bitwise OR +

19 Elements are: (), (), (), () xy OR(x,y) Bitwise OR

20 Elements are: (), (), (), ()

21 Elements are: (), (), (), () Bitwise AND

22 Elements are: (), (), (), () Bitwise AND xy AND(x,y)

23 Is it a Boolean algebra? Elements: () () () () 0 1 Operations: Bitwise OR Bitwise AND Bitwise Complement

24 +

25 +

26 +

27 + We can prove it!

28 In general: 0-1 vectors a Boolean algebra? Elements: It is a Boolean algebra! () () () () 0 1 Operations: Bitwise OR Bitwise AND Bitwise Complement

29 0-1 vectors Boolean algebra Elements: n=3 Operations: (0) (1) (0) (1) (1) (1) (0) (1) 0 1 Bitwise OR Bitwise AND Bitwise Complement True for a two-valued Boolean algebra n=1 True for any Boolean algebra with 0-1 vectors and bitwise OR and AND?? arbitrary finite n

30 Idea 1: Any identity that is true for 0-1 is true for 0-1 vectors with bitwise OR/AND Idea 2: Axioms are identities True for 0-1 à true for 0-1 vectors s

31 The 0-1 Theorem 0-1 Theorem: An identity is true for 0-1 vectors with bitwise OR and AND if and only if it is true for two valued (0-1) with bitwise OR and AND Proof: The easy direction Assume an identity true for 0-1 vectors True for 0-1

32 0-1 Theorem: The 0-1 Theorem An identity is true for 0-1 vectors with bitwise OR and AND if and only if it is true for two valued (0-1) with bitwise OR and AND Proof: The non-obvious direction Assume an identity true for 0-1 Need to prove true for any 0-1 vectors

33 Example: Theorem 2: Proof: The identity is true for = = = = 1 Need to prove it for 0-1 vectors

34 Example: By contradiction Theorem 2: Proof (for 0-1 vectors): If an identity is not true in general; then there is an assignment of elements that violates the equality Assume true for all 0-1 assignments and not true for some other assignment Hence, there must be a position in the binary vector that is violated There exists a 0-1 assignment to the identity that violates the equality, CONTRADICTION!!

35 Recap: The 0-1 Theorem An identity is true for 0-1 vectors with bitwise OR/AND if and only if it is true for two valued (0-1) with OR/AND Proof: The easy direction Assume an identity true for 0-1 vectors 0-1 is a special case: True for 0-1 The non-obvious direction CONTRADICTION!! Assume an identity true for 0-1 Q Need to prove true for 0-1 vectors Assume there exists a general identity violating assignment Show that there is a 0-1 identity violating assignment

36 Application of the the 0-1 Theorem Elements: 0-1 vectors of length n, there are 2 n vectors Operations: Bitwise OR Bitwise AND Bitwise Complement It is a Boolean algebra with for n=1 By the 0-1 Theorem: It is a Boolean algebra for any finite n

37 Boolean Algebra Two-value? Or not?

38 Prove or Disprove At least one of the following is true:

39 xy True for two-valued Boolean algebra Prove or Disprove OR(x,y) At least one of the following is true: Is it true for 0-1 vectors Boolean algebras?

40 + Prove or Disprove At least one of the following is true: Is it true for 0-1 vectors Boolean algebras? NO

41 The 0-1 Theorem: True only for identities!!! If Claim is NOT TRUE in general! + Then or

42 Examples other Boolean Algebras 0-1 vectors Arithmetic Boolean algebras Algebra of subsets union / intersection next next

43 Boolean algebra Boolean integers s

44 Euclid, 3BC Greatest Common Divisor 297 = 3x3x3x 405 = 3x3x3x3x5 884 = 2x2x13x = 2x2x3x3x17 27 = 3x3x3 68 = 2 x 2 x 17 Application: Simplifying fractions

45 Euclid, 3BC Least Common Multiple 297 = 3x3x3x 405 = 3x3x3x3x5 884 = 2x2x13x = 2x2x3x3x = 3x3x3x3x5x 7956 = 2x2x3x3x13x17 Application: Adding fractions

46 Euclid, 3BC Greatest Common Divisor Least Common Multiple gcd(297,405) = 27 lcm(297,405) = = 3x3x3x 405 = 3x3x3x3x5 27x4455 = 120, x405 = 120, = 3x3x3x 405 = 3x3x3x3x5 gcd(a,b) x lcm(a,b) = a x b

47 George Boole Arithmetic Boolean Algebra The set of elements: {1,2,3,6} The operations: lcm and gcd 1 is Boolean 0 6 is Boolean 1 lcm = lowest common multiple gcd = greatest common divisor

48 The set of elements: {1,2,3,6} The operations: lcm and gcd 1 is Boolean 0 6 is Boolean 1 What is the complement?

49 The set of elements: {1,2,3,6} The operations: lcm and gcd 1 is Boolean 0 6 is Boolean 1 Elements: Set of all the divisors of an integer n For which n does it work?

50 The set of elements: {1,2,4,8} The operations: lcm and gcd 1 is Boolean 0 8 is Boolean 1 The complement?

51 The set of elements: {1,2,3,6} The operations: lcm and gcd 1 is Boolean 0 6 is Boolean 1 Elements: Set of all the divisors of an integer n For which n does it work? Prime factors appear at most once in n

52 George Boole Bunitskiy Algebra 1899 Boolean Integers 2 x 3 x 5 = 30 2 x 3 x 7 = 42 Euclid, 3BC Every prime in the prime factorization is a power of one (square-free integer) Elements: The set of divisors of a Boolean integer {1,2,3,5,6,,15,30} The operations: lcm and gcd The 0 and 1 elements: 1 and 30

53 Is Bunitskiy Algebra a Boolean Algebra? YES LCM GCD 6 = 3x2 3 = 3 2 = 2 Bitwise OR Bitwise AND 1 = 1

54 + lcm gcd Bunitskiy algebra is isomorphic to the algebra of 0-1 vectors, it is a Boolean algebra! syntax the same different semantics

55 Boolean algebra subsets of a set s

56 Algebra of Subsets S is the set of all points Elements: all possible subsets of a set S + is union of sets: is intersection of sets How many elements? 2 s

57 Example: S = Operations: union and intersection Elements: Complement:

58 Is the algebra of subsets a Boolean Algebra? YES Corresponding 0-1 vectors: S = () () () () Elements: Algebra of subsets is isomorphic to the algebra of 0-1 vectors, it is a Boolean algebra! syntax the same different semantics

59 Elements are: (), (), (), ()

60 Union Bitwise OR Elements are: (), (), (), ()

61 Intersection Bitwise AND Elements are: (), (), (), ()

62 Boolean algebra an amazing theorem

63 Examples of Boolean Algebras Size 2 k 0-1 (two valued) Boolean algebra OR / AND 0-1 vectors bitwise OR / bitwise AND Arithmetic Boolean algebras lcm / gcd Algebra of subsets union / intersection They are isomorphic!

64 Representation Theorem (Stone 1936): Every finite Boolean algebra is isomorphic to a Boolean algebra of 0-1 vectors Algebra 1 Algebra 2 elements elements operations operations

65 Representation Theorem (Stone 1936): Every finite Boolean algebra is isomorphic to a Boolean algebra with elements being bit vectors of finite length with bitwise operations OR and AND Two Boolean algebras with m elements are isomorphic k Every Boolean algebra has 2 elements Provides intuition beyond the axioms: We can naturally reason about results in Boolean algebra

66 Marshall Stone Marshall Stone Proved in AB = years After Boole The Boolean Syntax invented in 1847 has a unique representative semantic!!! Marshall entered Harvard in 1919 intending to continue his studies at Harvard law school; fell in love with Mathematics, and the rest is history Harlan Fiske Stone 12 th Chief Justice of the US Marshall had a passion for travel. He began traveling when he was young and was on a trip to India when he died...

67 Quiz #3 min What is the complement of Show your work and justify every step! Use the following DeMorgan Laws: And the axioms: Solution in this form:

68 What is the complement of DeMorgan DeMorgan A4 A3 A4 A3 A2 And the axioms: A1 DeMorgan Laws:

69 One week!

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71 d1 d2 majority c 2 symbol adder c s parity

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74 Need to provide a complete proof No duality arguments See the solution to Quiz #3 For (c): Sum of products (no need to expand to DNF)

75 Prove 1 Prove multiply