IST 4 Information and Logic

Transcription

1 IST 4 Information and Logi

2 T = today x= hw#x out x= hw#x due mon tue wed thr fri 3 M 7 oh M 4 oh oh 2M2 2 oh oh 2 oh 28 oh M2 oh oh = offie hours 5 3 Mx= MQx out 2 oh 3 4 oh oh midterms oh Mx= MQx due T oh oh 2 oh oh 5 oh oh

3 4 Milestones in the Development of Information The Babylonian Knew Everything masters of syntax,, base-6 positional system, abaus,... Universal Language/Syntax Dream alulus, l binary systems, logi... Logi to Algebra Boolean algebra for syntax proessing... Algebra to Physis Algebra to Physis automati syntax proessing with relay iruits, information theory...

4 Last Leture -Shannon was a Babylonian Created logi design The two-olor line omposition is a - Boolean algebra - Analysis of relay iruits relay iruits Boolean funtions - Satisfying i assignments and omplexity: P vs NP

5 Shannon 96-2 Claude Elwood Shannon was born in Petoskey, Mihigan, on April 3, 96. The first sixteen years of Shannon's life were spent in Gaylord, Mihigan

6 Shannon 96-2 Shannon s Bakground Claude Elwood Shannon was born in Petoskey, Mihigan, on April 3, 96. The first sixteen years of Shannon's life were spent in Gaylord, Mihigan His father, Claude Sr. ( ), was a businessman and for a period a Judge of Probate His mother, Mabel Wolf Shannon (88-945), 945), was a language teaher and for a number of years prinipal of Gaylord High Shool George Boole (85-864)

7 Shannon 96-2 Shannon s Bakground Father: Claude Sr. ( ) Mother: Mabel Wolf Shannon (88-945) 945) U of Mihigan, undergrad, EE and Math (6): MIT MS and PHD (2) : Institute for Advaned Studies (24) : 94-4 Bell Labs (25) : MIT Professor (42) : Married Elizabeth (Betty) Moore Shannon in 949 Children: Andrew and Margarita Betty and Claude in 98

8 Shannon 96-2 logi design information theory ryptography omputer hess omputational juggling...

9 Shannon 96-2 Posted on the lass web site

10 William Shokley The Modern Relay? Calteh BS: MIT PhD: A o-inventor of the modern relay - the transistor at Bell Labs... Together with Bardeen and Brattain Blaker Hovse Blaker Hovse ame into existene when its initial members met in the Athenaeum for its first meeting on Thursday, May 4, 93 Last Wednesday it was 83 years old! Shokley was a junior in Physis Soure: blaker house web page

11 Connetion Between Boolean Calulus and Physial Ciruits Shannon 938 Analysis (last time): Boolean funtions relay iruits Synthesis (today): Boolean funtions relay iruits

12 Relay Ciruits synthesis s

13 Boolean Calulus and Physial Ciruits Q: Boolean alulus l to relay iruits? it How many relays with series-parallel?

14 Boolean Calulus and Physial Ciruits Q: Boolean alulus l to relay iruits? it How many relays with series-parallel? a d b a e b d e

15 Boolean Calulus and Physial Ciruits Q: Boolean alulus l to relay iruits? it How many relays with series-parallel? a b d a e b d e

16 Boolean Calulus and Physial Ciruits Q: Boolean alulus l to relay iruits? it Less than relays with series-parallel?

17 Boolean Calulus and Physial Ciruits How many relays? 8

18 Boolean Calulus and Physial Ciruits How many relays? 8 a b a b d e

19 Boolean Calulus and Physial Ciruits How many relays? 8 a b a b d e

20 Boolean Calulus and Physial Ciruits How many relays? 8 a b a b d e Can you ompute f with less than 8 relays? Lower bound on the number on the number of relays??? 5 relays?

21 Boolean Calulus and Physial Ciruits 5 relays?

22 Boolean Calulus and Physial Ciruits a b d e 5 relays?

23 Boolean Calulus and Physial Ciruits a b d e 5 relays?

24 Boolean Calulus and Physial Ciruits a b d e 5 relays?

25 Boolean Calulus and Physial Ciruits a b d e 5 relays?

26 Boolean Calulus and Physial Ciruits a b d e This iruit is not series-parallel and we do not have algorithms for synthesis it using the algebra 5 relays?

27 Boolean Calulus and Physial Ciruits a b d e How to effiiently onstrut good relay iruits its is an open problem

28 Synthesis trees s

29 XOR of two Variables with Relays? How many relays? a a b b

30 XOR of More Variables How many relays? 2 a a a a b b b b Can we do better with series-parallel?

31 XOR of More Variables a How many relays? b b a b b

32 Does it look familiar? a b b a b b

33 The proof for magi boxes using sub-funtions Does it look familiar? a b b a b b

34 3-input binary s-box an be divided to two 2-input s-boxes x y z o * * z= then * * z= then * * * *

35 x y x y x y z o * * * * * * * * z z= then o z= then

36 Representing syntax boxes with a

37 A syntax table / Boolean funtion as a binary deision tree A path orresponds to an entry in the syntax table = =???? (,) (,) (,) (,)

38 A syntax table / Boolean funtion as a binary deision tree ab XOR(a,b)???? = = (,) (,) (,) (,)

39 A syntax table / Boolean funtion as a binary deision tree ab XOR(a,b)???? = = (,) (,) (,) (,)

40 A syntax table / Boolean funtion as a binary deision tree ab magi box???? = =

41 A syntax table / Boolean funtion as a binary deision tree = = Can it help with the synthesis of relay iruits?

42 Trees and Relay Ciruits relay iruits? = =

43 Trees and Relay Ciruits relay iruits? = =

44 Trees and Relay Ciruits a b b a b b

45 Trees and Relay Ciruits a b b How many relays? a b b Can we do better? no

46 Trees and Relay Ciruits not series-parallel Can we do better? no

47 Do binary deision trees always provide the best series-parallel solutions?

48 Do binary deision trees always provide the best series-parallel solutions? ab magi box???? = =

49 Do binary deision trees always provide the best series-parallel solutions? ab magi box??? = =

50 Do binary deision trees always provide the best series-parallel solutions? ab magi box? = =

51 Do binary deision trees always provide the best series-parallel solutions? ab magi box? = =

52 Do binary deision trees always provide the best series-parallel solutions? NO ab magi box = =

53 Synthesis symmetri funtions s

54 Two questions about XOR Why are about XOR? Why is XOR easy to implement?

55 The Boolean Funtions of the Adder d d2 2 symbol adder s sum arry

56 sum d d2 arry 2 symbol adder s A new design: d d2 d d2 2 symbol adder 2 symbol adder s How will you help your mom to orretly use the adders? s d d2 2 symbol adder s

57 d d2 2 symbol adder s A new design: d d2 d d2 2 symbol adder 2 symbol adder s How will you help your mom to orretly use the adders? s d d2 2 symbol adder s

58 MAJ and XOR are symmetri Boolean funtions d d2 2 symbol adder s sum arry

59 Symmetri Funtions AND, OR, MAJ and XOR are symmetri Boolean funtions Permuting the inputs does not hange the output SYM Definition: A Boolean funtion f is symmetri if for an arbitrary permutation

60 sum ab XOR(a,b,) ab XOR(a,b,) number of s XOR(a,b,) XOR(a,b,) 2 3

61 arry ab MAJ(a,b,) number of s ab 2 3 MAJ(a,b,)

62 Symmetri Funtions Definition: A Boolean funtion f is symmetri if for an arbitrary permutation Theorem: A Boolean funtion f(x) is symmetri if and only if it is a funtion of the number of s in X, namely X

63 Theorem: A Boolean funtion f(x) is symmetri if and only if it is a funtion of the number of s in X, namely X f symmetri f a funtion of X f a funtion of X f symmetri ab XOR(a,b,) number of s 2 3 ab XOR(a,b,)

64 Proof: f symmetri f a funtion of X Given: Need to prove: However,, Hene, Q ab XOR(a,b,) b) 2 3 ab XOR(a,b,)

65 Proof: f a funtion of X f symmetri Given: Need to prove: However, ab XOR(a,b,) b) 2 3 ab Q XOR(a,b,)

66 Questions on Symmetri Funtions Q: How many symmetri Boolean funtions of n variables? Q2: How an we effiiently implement symmetri funtions with relay iruits? A: Theorem: The number of symmetri funtions of n variables is:

67 Number of Symmetri Funtions Theorem: The number of symmetri funtions of n variables is: Proof: f symmetri f a funtion of X * Symmetri * funtion table 2 * * an be or n * funtions Q

68 Questions on Symmetri Funtions Q: How many symmetri Boolean funtions of n variables? Q2: How an we effiiently implement symmetri funtions with relay iruits? A2: In Shannon s MS thesis and now! A: Theorem: The number of symmetri funtions of y n variables is:

69 Ciruits and Symmetri Funtions What is the idea? Need to ount the number of s 2 ab XOR(a,b,) 3

70 Ciruits and Symmetri Funtions What is the idea? 2 3 Need to ount the number of s 2 ab XOR(a,b,) 3

71 Ciruits and Symmetri Funtions What is the idea? 2 3 XOR Need to ount the number of s 2 ab XOR(a,b,) 3

72 binomial tree? Not a tree The key: Symmetri funtions are defined d by the number of s blue = go down red = go right 2 What are the numbers? What are we hoosing? Number of s

73 binomial tree The key: Symmetri funtions are defined d by the number of s blue = go down red = go right How many edges/relays es/rel s 2 for n variables? 2 3 3(3+)/2 = 6 Number of round nodes is an arithmeti ti sum: n(n+)/2 Number of edges is twie the number of nodes: n(n+)

74 Synthesis summary binary trees and binomial trees s

75 Arbitrary Boolean funtions deision tree relay iruits Number of relays an be very LARGE: 2 n

76 Symmetri Boolean funtions binomial tree relay iruits 2 3 Number of relays about n 2

77 Synthesis binary adder s

78 Bak to the Adder digit digit 2 arry 2 symbol adder arry sum

79 dd d2d The Sum Funtion 2 symbol adder s sum d d2 sum sum

80 dd d2d The Carry Funtion 2 symbol adder arry s arry d d2 arry arry

81 Implementing a 2-bit Adder 2 3 XOR = =

82 Implementing a 2-bit Adder 2 3 XOR = =

83 Implementing a 2-bit Adder 2 3 XOR = = XOR

84 Implementing a 2-bit Adder 2 3 XOR = = XOR

85 Implementing a 2-bit Adder 2 3 XOR 2 3 = = These two nodes are equivalent XOR

86 Implementing a 2-bit Adder 2 3 XOR 2 3 = = XOR

87 Implementing a 2-bit Adder 2 3 XOR = =

88 Implementing a 2-bit Adder 2 3 MAJ 2 = = 2 3

89 Implementing a 2-bit Adder 2 3 MAJ 2 = = 2 3

90 Implementing a 2-bit Adder MAJ = = There is no reason to 2 3 > hek here 2 3

91 Implementing a 2-bit Adder MAJ = = There is no reason to 2 3 > hek here 2 3

92 Implementing a 2-bit Adder MAJ 2 3 > = = 2

93 Implementing a 2-bit Adder 2 3 MAJ = =

94 Now what? XOR MAJ 8 relays 6 relays Combination of the two funtions? = =

95 Implementing a 2-bit Adder Summary XOR MAJ 8 relays 6 relays Combination of the two funtions? = =

96 sum: Relay Ciruits for the Sum and the Carry Funtions arry: d d2 2 symbol adder s How many relays for a 2 symbol adder? XOR MAJ

97 Shannon 96-2 This is Shannon s Design XOR MAJ

98 Shannon 96-2 This is Shannon s Design XOR MAJ XOR MAJ

99 Shannon s Symmetri Funtion Design Shannon

100 Shannon 96-2 The amazing last page of Shannon s Thesis binary arithmeti to Boolean algebra Boolean algebra to optimized relay iruits Key referenes on algebra (in the ontext t of logi)