IST 4 Information and Logic

Transcription

1 IST 4 Information and Logic

2 mon tue wed thr fri sun T = today 3 M oh x= hw#x out oh M 7 oh oh 2 M2 oh oh x= hw#x due 24 oh oh 2 oh = office hours oh oh M2 8 3 oh midterms oh oh Mx= MQx out 5 oh 3 4 oh T Mx= MQx due 22 oh oh 4 5 CP CP = challenge problem 29 CP2 5 oh 5 oh oh CP oh oh

3 Shannon 96-2 Today s lecture is based on Shannon s MSc s thesis

4 Shannon 96-2 George Boole (85-864): generation of Shannon s grandfather Claude Elwood Shannon was born in Petoskey, Michigan, on April 3, 96. The first sixteen years of Shannon's life were spent in Gaylord, Michigan 54 His father, Claude Sr. ( ), was a businessman and for a period a Judge of Probate 36 His mother, Mabel Wolf Shannon (88-945), was a language teacher and for a number of years principal of Gaylord High School

5 Shannon 96-2 U of Michigan, undergrad, EE and Math (6): MIT MSc and PHD (2) : Institute for Advanced Studies (24) : 94-4 Bell Labs (25) : MIT Professor (42) : Married Elizabeth (Betty) Moore Shannon in 949 (33)

6 Married Elizabeth (Betty) Moore Shannon in 949 Betty and Claude in 98 Claude (and Betty?) with Caltech s professor Yaser Abu-Mostafa at the 985 Information Theory Symposium, Brighton, England Children: Andrew and Margarita

7 Shannon 96-2 logic design information theory cryptography computer chess and AI computational juggling...

8 Shannon 96-2 Posted on the class web site

9 William Shockley The Modern Relay? Caltech BS: MIT PhD: A co-inventor of the modern relay - the transistor at Bell Labs... Together with Bardeen and Brattain. 956 Nobel Prize in Physics Blacker Hovse Blacker Hovse came into existence when its initial members met in the Athenaeum for its first meeting on Thursday, May 4, 93 Blacker is 86 years old! Shockley was a junior in Physics Source: Blacker house web page

10 Connection Between Boolean Calculus and Physical Circuits Shannon 938 Analysis (last time): Boolean functions relay circuits Synthesis (today): Boolean functions relay circuits

11 Relay Circuits synthesis s

12 Boolean Calculus and Physical Circuits Q: Boolean calculus to relay circuits? How many relays with series-parallel?

13 Boolean Calculus and Physical Circuits Q: Boolean calculus to relay circuits? How many relays with series-parallel? a b d e a c e b c d

14 Boolean Calculus and Physical Circuits Q: Boolean calculus to relay circuits? How many relays with series-parallel? a b d e a c e b c d

15 Boolean Calculus and Physical Circuits Less than relays with series-parallel? a b d e a c e b c d

16 Boolean Calculus and Physical Circuits Less than relays with series-parallel? a b d c e e b c d

17 Can you compute f with less than 8 relays? Lower bound on the number on the number of relays??? 5 relays? How many relays? 8 a b d b c e c e

18 Boolean Calculus and Physical Circuits 5 relays?

19 Boolean Calculus and Physical Circuits a b c d e 5 relays?

20 Boolean Calculus and Physical Circuits a b c d e 5 relays?

21 Boolean Calculus and Physical Circuits a b c d e 5 relays?

22 Boolean Calculus and Physical Circuits a b c d e 5 relays?

23 Boolean Calculus and Physical Circuits a b c d e This circuit is not series-parallel and we do not have algorithms for synthesis it using the algebra 5 relays?

24 Boolean Calculus and Physical Circuits a b c d e How to efficiently construct good relay circuits is an open problem

25 Synthesis trees s

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

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

28 XOR of More Variables a How many relays? b c b c a b b c c

29 Does it look familiar? a= Idea? a b b c c odd parity a= a b b c c even parity

30 The proof for magic boxes using sub-functions Does it look familiar? a= a= a a b b b b c c c c

31 3-input binary s-box can be divided to two 2-input s-boxes b c a o odd parity a= then a= then even parity Shannon 938

32 b c b c o a b c a o a= then a= then Shannon 938

33 Representing syntax boxes with

34 A syntax table / Boolean function as a binary decision tree A path corresponds to an entry in the syntax table = =???? (,) (,) (,) (,)

35 A syntax table / Boolean function as a binary decision tree ab XOR(a,b) = =???? (,) (,) (,) (,)

36 A syntax table / Boolean function as a binary decision tree ab XOR(a,b)???? = = (,) (,) (,) (,)

37 A syntax table / Boolean function as a binary decision tree ab magic box???? = =

38 A syntax table / Boolean function as a binary decision tree = = Can the tree representation help with the synthesis of relay circuits?

39 Trees and Relay Circuits relay circuits? = =

40 Trees and Relay Circuits relay circuits? = =

41 Trees and Relay Circuits a b b c c a b b c c

42 Trees and Relay Circuits a b b c c b c a b c How many relays? Can we do better?

43 Trees and Relay Circuits not series-parallel 8 relays Can we do better?

44 Simplification of Boolean Functions sub-trees, karnaugh maps simpler tree? Key idea: No need to climb a tree if we know the value earlier = =

45 Simplification of Circuits, sub-trees, karnaugh maps Key idea: No need to climb a tree if we know the value earlier = =

46 Simplification of Circuits, sub-trees, karnaugh maps Observation: the order of variables is important = =

47 Simplification of Circuits, sub-trees, karnaugh maps Observation: the order of variables is important = =

48 Simplification of Circuits, sub-trees, karnaugh maps Observation: the order of variables is important = =

49 No need in Boolean algebra... syntax boxes binary decision trees + simplification relay circuits

50 Do binary decision trees always provide the best series-parallel solutions?

51 Find the simplest series-parallel circuit for the following syntax box ab magic box = =

52 Do binary decision trees always provide the best series-parallel solutions? ab magic box??? = =

53 Do binary decision trees always provide the best series-parallel solutions? ab magic box? = =

54 Do binary decision trees always provide the best series-parallel solutions? ab magic box? = =

55 Do binary decision trees always provide the best series-parallel solutions? ab magic box? = =

56 Do binary decision trees always provide the best series-parallel solutions? ab magic box? = =

57 Do binary decision trees always provide the best series-parallel solutions? ab magic box = =?

58 Do binary decision trees always provide the best series-parallel solutions? NO ab magic box? = =

59 Synthesis symmetric functions s

60 Two questions about XOR Why care about XOR? Why is XOR easy to implement?

61 The Boolean Functions of the Adder d d2 c 2 symbol adder c s sum carry

62 sum d d2 carry c 2 symbol adder s c A new design: d d2 c c d d2 2 symbol adder 2 symbol adder c s How will you help your mom to correctly use the adders? c s d c d2 2 symbol adder c s

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

64 MAJ and XOR are symmetric Boolean functions d d2 c 2 symbol adder c s sum carry

65 Symmetric Functions AND, OR, MAJ and XOR are symmetric Boolean functions Permuting the inputs does not change the output SYM Definition: A Boolean function f is symmetric if for an arbitrary permutation

66 sum abc XOR(a,b,c) number of s 2 3 abc XOR(a,b,c)

67 carry abc MAJ(a,b,c) number of s 2 3 abc MAJ(a,b,c)

68 Symmetric Functions Definition: A Boolean function f is symmetric if for an arbitrary permutation Theorem: A Boolean function f(x) is symmetric if and only if it is a function of the number of s in X, namely X

69 Theorem: A Boolean function f(x) is symmetric if and only if it is a function of the number of s in X, namely X f symmetric f a function of X f a function of X f symmetric abc XOR(a,b,c) number of s 2 3 abc XOR(a,b,c)

70 Proof: f symmetric f a function of X Given: Need to prove: However,, Hence, Q abc XOR(a,b,c) 2 3 abc XOR(a,b,c)

71 Proof: f a function of X f symmetric Given: Need to prove: However, abc XOR(a,b,c) 2 3 abc XOR(a,b,c) Q

72 Q: How many symmetric Boolean functions of n variables? A: Theorem: The number of symmetric functions of n variables is:

73 A: Theorem: The number of symmetric functions of n variables is: Proof: f symmetric f a function of X Symmetric function table 2 * * * * can be or n * functions Q

74 A: Theorem: The number of symmetric functions of n variables is: Q: How many symmetric Boolean functions of n variables? Q2: How can we efficiently implement symmetric functions with relay circuits? A2: In Shannon s MSc thesis and now!

75 Circuits and Symmetric Functions What is the idea? Need to count the number of s 2 3 abc XOR(a,b,c)

76 Circuits and Symmetric Functions What is the idea? Need to count the number of s abc XOR(a,b,c)

77 Circuits and Symmetric Functions What is the idea? Need to count the number of s 2 3 XOR 2 3 abc XOR(a,b,c)

78 binomial tree? Not a tree The key: Symmetric functions are defined by the number of s blue = go down red = go right 2 What are the numbers? What are we choosing? Variables that are red =

79 binomial tree The key: Symmetric functions are defined by the number of s blue = go down red = go right How many edges/relays for n variables? (3+)/2 = 6 Number of round nodes is an arithmetic sum: n(n+)/2 Number of edges is twice the number of nodes: n(n+)

80 Synthesis summary binary trees and binomial trees s

81 Arbitrary Boolean functions decision tree relay circuits Number of relays can be very LARGE: 2 n

82 Symmetric Boolean functions binomial tree relay circuits 2 3 Number of relays about n 2

83 Synthesis binary adder s

84 Back to the Adder digit digit 2 carry 2 symbol adder carry MAJ XOR sum

85 Implementing a 2-bit Adder XOR = =

86 Implementing a 2-bit Adder XOR = =

87 Implementing a 2-bit Adder XOR = = XOR

88 Implementing a 2-bit Adder XOR = = XOR

89 Implementing a 2-bit Adder XOR = = These two nodes are equivalent c= results in XOR= XOR

90 Implementing a 2-bit Adder XOR = = XOR

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

92 Implementing a 2-bit Adder MAJ = =

93 Implementing a 2-bit Adder MAJ = =

94 Implementing a 2-bit Adder 2 3 MAJ = = There is no reason to check c here: It is for both values 2 3

95 Implementing a 2-bit Adder MAJ = =

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

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

98 Now what? XOR MAJ 8 relays 6 relays Multi-terminal: Combination of the two functions? = =

99 Now what? XOR MAJ 8 relays 6 relays Multi-terminal: Combination of the two functions? = =

100 Now what? XOR MAJ 8 relays 6 relays Multi-terminal: Combination of the two functions? = =

101 Relay Circuits for the Sum and the Carry Functions sum: carry: d d2 c 2 symbol adder c s How many relays for a 2 symbol adder? XOR MAJ

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

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

104 Shannon 96-2 The amazing last page of Shannon s Thesis binary arithmetic to Boolean algebra Boolean algebra to optimized relay circuits Key references on algebra (in the context of logic)

105 Quiz time

106 Quiz #8 min. What is the Boolean function that is computed by the following relay circuit? Boolean formula 2. Show the simplest relay circuit (smallest number of relays) that can compute this function. Relay circuit Why is it the simplest circuit? Explanation Show your work! a b c d d