DOC PREVIEW
CMU CS 15251 - Lecture

This preview shows page 1-2-3-4-5-6-7-52-53-54-55-56-57-58-59-104-105-106-107-108-109-110 out of 110 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 110 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Unary, Binary, and BeyondPowerPoint PresentationA Frequently Arising CalculationThe Geometric SeriesSlide 5nth Triangular Numbernth Square NumberSlide 8Slide 9Slide 10Slide 11Strings Of Symbols.Strings over the alphabet .Slide 14*Slide 16The Natural Numbers“God Made The Natural Numbers. Everything Else Is The Work Of Man.”Last Time: Unary NotationSlide 20Peano Unary (PU)Each number is a sequence of symbols in {S, 0}+Peano Unary Representation of Natural NumberInductive Definition of +S0 + S0 = SS0 (i.e., “1+1=2”)Inductive Definition of *Inductive Definition of ^ = { 0, S0, SS0, SSS0, . . .} = { 0, 1, 2, 3, . . .}a = [a DIV b]*b + [a MOD b]45 = [45 DIV 10]*10 + [ 45 MOD 10] = 4*10 + 5Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Initial Cases, length(X)=1Suppose n=length(X)>S0Example X= 238Slide 45Slide 46Slide 47Slide 48One digit cases.Suppose X > 9Slide 51Slide 52Clear when X is a single digit.Slide 54Slide 55X2 NLZ, n=length(Z) > 1Slide 57Slide 58Base X NotationS = an-1, an-2, …, a0 represents the number: an-1 Xn-1 + an-2 Xn-2 + . . . + a0 X0Slide 61Bases In Different CulturesBiggest n “digit” number in base X would be: (X-1)Xn-1 + (X-1)Xn-2 +…+ (X-1)X0Slide 64Biggest n “digit” number in base X would be: S= (X-1)Xn-1 + (X-1)Xn-2 +…+ (X-1)X0 Add 1 to get: Xn + 0 Xn-1 + … + 0 X0 Thus, S = Xn - 1Slide 66(X-1)Xn-1 + (X-1)Xn-2 +…+ (X-1)X0 = Xn - 1The highest n digit number in base X.Slide 69Slide 70Plus/Minus Base XSlide 72Slide 73Slide 74Slide 750 has a unique n-digit plus/minus base X representation as all 0’s.Each of the numbers from 0 to Xn-1 is uniquely represented by an n-digit number in base X.Slide 78Fundamental Theorem For Base X: Each of the numbers from 0 to Xn-1 is uniquely represented by an n-digit number in base X.Slide 80Egyptian MultiplicationEgyptian Multiplication a times b by repeated doublingEgyptian Multiplication 15 times 5 by repeated doublingWhy does that work?Slide 85Slide 86Egyptian Base ConversionSlide 88Start the algorithmSlide 90Slide 91Slide 92Slide 93Slide 94And Keep Going until 0Slide 96Rhind Papyrus (1650 BC) 70*13Slide 98Rhind Papyrus (1650 BC)Slide 100Slide 101Slide 102Standard Binary Multiplication = Egyptian MultiplicationEgyptian Base 3Unique Representation Theorem for Egyptian Base 3 No integer has 2 distinct, n-digit, Egyptian base-3 representations. We can represent all integers from -(3n-1)/2 to (3n-1)/2Slide 106Slide 107Slide 108Slide 109ReferencesUnary, Binary, and BeyondGreat Theoretical Ideas In Computer ScienceSteven RudichCS 15-251 Spring 2004Lecture 3 Jan 20, 2004 Carnegie Mellon UniversityWe are going to need this fundamental sum:The Geometric Series 1 + X1 + X11 + X + X22 + X + X 33 + … + X + … + Xn-2n-2 + X + Xn-1n-1 == XXn n – 1 – 1 XX - - 11A Frequently Arising Calculation(X-1) ( 1 + X1 + X2 + X 3 + … + Xn-2 + Xn-1 )= X1 + X2 + X 3 + … + Xn-1 + Xn - 1 - X1 - X2 - X 3 - … - Xn-2 – Xn-1= - 1 + Xn= Xn - 1The Geometric Series(X-1) ( 1 + X1 + X2 + X 3 + … + Xn-1 ) = Xn - 1 1 + X1 + X11 + X + X22 + X + X 33 + … + X + … + Xn-2n-2 + X + Xn-1n-1 == XXn n – 1 – 1 XX - - 11 when X1Last time we talked about unary notation.nth Triangular Numbern = 1 + 2 + 3 + . . . + n-1 + n = n(n+1)/2nth Square Numbern = 1 + 3 + … + 2n-1 = Sum of first n odd numbersnth Square Numbern = n + n-1 = n2(( nn))2 =2 = (( n-1n-1))2 2 ++nn (( nn))2 2 = + + . . . = + + . . . + + nnWe will define sequences of symbols that give us a unary representation of the Natural number.First we define the general language we use to talk about strings.Strings Of Symbols.We take the idea of symbol and sequence of symbols as primitive.Let  be any fixed finite set of symbols.  is called an alphabet, or a set of symbols. Examples:  = {0,1,2,3,4}= {a,b,c,d, …, z}= all typewriter symbols.Strings over the alphabet A string is a sequence of symbols from . Let s and t be strings. Let st denote the concatenation of s and t, i.e., the string obtained by the string s followed by the string t. Define + by the following inductive rules: x2 ) x2 +s,t 2 + ) st 2 +Intuitively, + is the set of all finite strings that we can make using (at least one) letters from .Define  be the empty string. I.e.,XY= XY for all strings X and Y. is also called the string of length 0.Define 0 = {  }Define * = + [ {}Intuitively, * is the set of all finite strings that we can make using letters from including the empty string.The Natural Numbers = { 0, 1, 2, 3, . . .}Notice that we include 0 as a Natural number.“God Made The Natural Numbers. Everything Else Is The Work Of Man.” = { 0, 1, 2, 3, . . .}KroneckerLast Time: Unary Notation1234To handle the notation for zero, we introduce a small variation on unary.Peano Unary (PU)0 01 S02 SS03 SSS04 SSSS05 SSSSS06 SSSSSS0Giuseppe Peano [1889]Each number is a sequence of symbols in {S, 0}+0 01 S02 SS03 SSS04 SSSS05 SSSSS06 SSSSSS0Peano Unary Representation of Natural Number = { 0, S0, SS0, SSS0, . . .}0 is a natural number called zero.Set notation: 0 2 If X is a natural number, then SX is a natural number called successor of X.Set notation: X 2  ) SX 2 Inductive Definition of + = { 0, S0, SS0, SSS0, . . .}Inductive definition of addition (+):X, Y 2  )X “+” 0 = XX “+” SY = S(X”+”Y)S0 + S0 = SS0 (i.e., “1+1=2”)Proof:S0 + S0 = S(S0 + 0) = S(S0) = SS0 X, Y 2  )X “+” 0 = XX “+” SY = S(X”+”Y)Inductive Definition of * = { 0, S0, SS0, SSS0, . . .}Inductive definition of times (*):X, Y 2  )X “*” 0 = 0X “*” SY = (X”*”Y) + XInductive Definition of ^ = { 0, S0, SS0, SSS0, . . .}Inductive definition of raised to the (^):X, Y 2  )X “^” 0 = 1 [or X0 = 1 ]X “^” SY = (X”^”Y) * X [or XSY = XY * X] = { 0, S0, SS0, SSS0, . . .}Defining < for :8 x,y 2 “x > y” is TRUE  “y < x” is FALSE“x > y” is TRUE  “y > x” is FALSE“x+1 > 0” is TRUE“x+1 > y+1” is TRUE ) “x > y” is TRUE = { 0, 1, 2, 3, . . .}Defining partial minus for :8 x,y 2  x-0 = x x>y ) (x+1) – (y+1) = x-ya = [a DIV b]*b + [a MOD b]Defining DIV and MOD for :8 a,b 2  a<b ) a DIV b = 0 a¸b>0 ) a DIV b = 1 + (a-b) DIV ba MOD b = a – …


View Full Document

CMU CS 15251 - Lecture

Documents in this Course
lecture

lecture

66 pages

lecture

lecture

79 pages

lecture

lecture

111 pages

lecture

lecture

85 pages

lecture17

lecture17

64 pages

Lecture

Lecture

85 pages

Lecture

Lecture

71 pages

Lecture

Lecture

70 pages

Lecture

Lecture

11 pages

Lecture

Lecture

45 pages

Lecture

Lecture

50 pages

Lecture

Lecture

93 pages

Lecture

Lecture

93 pages

Lecture

Lecture

35 pages

Lecture

Lecture

98 pages

Lecture

Lecture

74 pages

Lecture

Lecture

13 pages

Lecture

Lecture

15 pages

Lecture

Lecture

66 pages

Lecture

Lecture

82 pages

Lecture

Lecture

15 pages

Lecture

Lecture

47 pages

Lecture

Lecture

69 pages

Lecture

Lecture

13 pages

Lecture

Lecture

67 pages

Lecture

Lecture

68 pages

Lecture

Lecture

69 pages

lecture03

lecture03

44 pages

Lecture

Lecture

69 pages

Lecture

Lecture

68 pages

Lecture

Lecture

55 pages

Lecture

Lecture

79 pages

Lecture

Lecture

85 pages

Lecture

Lecture

87 pages

Lecture

Lecture

85 pages

Lecture

Lecture

103 pages

Lecture

Lecture

9 pages

Lecture

Lecture

83 pages

Lecture

Lecture

8 pages

lecture03

lecture03

68 pages

lecture24

lecture24

78 pages

lecture03

lecture03

72 pages

Thales

Thales

129 pages

lecture13

lecture13

81 pages

Lecture

Lecture

64 pages

lecture01

lecture01

59 pages

lecture11

lecture11

105 pages

Lecture

Lecture

89 pages

Lecture

Lecture

74 pages

lecture25

lecture25

57 pages

Lecture

Lecture

99 pages

lecture

lecture

50 pages

lecture

lecture

14 pages

Lecture

Lecture

78 pages

lecture

lecture

8 pages

Lecture

Lecture

98 pages

lecture

lecture

83 pages

lecture23

lecture23

88 pages

lecture

lecture

64 pages

lecture

lecture

72 pages

Lecture

Lecture

88 pages

lecture

lecture

79 pages

Lecture

Lecture

60 pages

lecture

lecture

74 pages

lecture19

lecture19

72 pages

lecture25

lecture25

86 pages

lecture

lecture

13 pages

lecture17

lecture17

79 pages

lecture

lecture

91 pages

lecture

lecture

78 pages

Lecture

Lecture

11 pages

Lecture

Lecture

54 pages

lecture

lecture

72 pages

lecture

lecture

119 pages

lecture

lecture

167 pages

lecture

lecture

73 pages

lecture

lecture

73 pages

lecture

lecture

83 pages

lecture

lecture

49 pages

lecture

lecture

16 pages

lecture

lecture

67 pages

lecture

lecture

81 pages

lecture

lecture

72 pages

lecture

lecture

57 pages

lecture16

lecture16

82 pages

lecture21

lecture21

46 pages

Lecture

Lecture

92 pages

Lecture

Lecture

14 pages

Lecture

Lecture

49 pages

Lecture

Lecture

132 pages

Lecture

Lecture

101 pages

Lecture

Lecture

98 pages

Lecture

Lecture

59 pages

Lecture

Lecture

64 pages

Lecture

Lecture

106 pages

Lecture

Lecture

70 pages

Lecture

Lecture

80 pages

Lecture

Lecture

76 pages

Lecture

Lecture

91 pages

Lecture

Lecture

112 pages

Lecture

Lecture

91 pages

Lecture

Lecture

10 pages

Lecture

Lecture

39 pages

Lecture

Lecture

79 pages

Lecture

Lecture

74 pages

Lecture

Lecture

44 pages

Lecture

Lecture

39 pages

Lecture

Lecture

99 pages

Lecture

Lecture

44 pages

Lecture

Lecture

59 pages

Lecture

Lecture

36 pages

lecture17

lecture17

36 pages

lecture

lecture

71 pages

lecture

lecture

79 pages

lecture

lecture

12 pages

lecture

lecture

43 pages

lecture

lecture

87 pages

lecture

lecture

35 pages

lecture03

lecture03

23 pages

lecture

lecture

68 pages

lecture

lecture

74 pages

lecture

lecture

21 pages

lecture

lecture

79 pages

lecture

lecture

15 pages

lecture

lecture

83 pages

lecture

lecture

13 pages

Lecture

Lecture

53 pages

lecture

lecture

55 pages

lecture

lecture

49 pages

lecture

lecture

10 pages

lecture

lecture

70 pages

lecture

lecture

12 pages

Lecture

Lecture

105 pages

Lecture

Lecture

9 pages

Lecture

Lecture

72 pages

Lecture

Lecture

66 pages

Lecture

Lecture

54 pages

Lecture

Lecture

98 pages

Lecture

Lecture

57 pages

Lecture

Lecture

75 pages

Lecture

Lecture

48 pages

lecture

lecture

53 pages

Lecture

Lecture

72 pages

Lecture

Lecture

53 pages

Lecture

Lecture

84 pages

Lecture

Lecture

55 pages

Lecture

Lecture

15 pages

Lecture

Lecture

6 pages

Lecture

Lecture

38 pages

Lecture

Lecture

71 pages

Lecture

Lecture

70 pages

lecture

lecture

48 pages

lecture

lecture

76 pages

lecture

lecture

48 pages

lecture

lecture

52 pages

lecture

lecture

43 pages

lecture

lecture

81 pages

lecture

lecture

82 pages

lecture

lecture

83 pages

lecture

lecture

64 pages

lecture

lecture

71 pages

lecture

lecture

65 pages

lecture

lecture

56 pages

lecture

lecture

12 pages

lecture

lecture

66 pages

lecture

lecture

50 pages

lecture

lecture

86 pages

lecture

lecture

70 pages

Lecture

Lecture

74 pages

Lecture

Lecture

54 pages

Lecture

Lecture

90 pages

lecture

lecture

78 pages

lecture

lecture

87 pages

Lecture

Lecture

55 pages

Lecture

Lecture

12 pages

lecture21

lecture21

66 pages

Lecture

Lecture

11 pages

lecture

lecture

83 pages

Lecture

Lecture

53 pages

Lecture

Lecture

69 pages

Lecture

Lecture

12 pages

lecture04

lecture04

97 pages

Lecture

Lecture

14 pages

lecture

lecture

75 pages

Lecture

Lecture

74 pages

graphs2

graphs2

8 pages

lecture

lecture

82 pages

Lecture

Lecture

8 pages

lecture

lecture

47 pages

lecture

lecture

91 pages

lecture

lecture

76 pages

lecture

lecture

73 pages

lecture

lecture

10 pages

lecture

lecture

63 pages

lecture

lecture

91 pages

lecture

lecture

79 pages

lecture

lecture

9 pages

lecture

lecture

70 pages

lecture

lecture

86 pages

lecture

lecture

102 pages

lecture

lecture

145 pages

lecture

lecture

91 pages

Lecture

Lecture

87 pages

lecture

lecture

87 pages

Notes

Notes

19 pages

Lecture

Lecture

50 pages

Lecture

Lecture

13 pages

Lecture

Lecture

97 pages

Lecture

Lecture

98 pages

Lecture

Lecture

83 pages

Lecture

Lecture

77 pages

Lecture

Lecture

102 pages

Lecture

Lecture

63 pages

Lecture

Lecture

104 pages

lecture

lecture

41 pages

lecture

lecture

14 pages

Lecture

Lecture

87 pages

Lecture

Lecture

94 pages

lecture

lecture

9 pages

Lecture

Lecture

96 pages

Lecture

Lecture

72 pages

Lecture

Lecture

35 pages

Lecture

Lecture

77 pages

Lecture

Lecture

98 pages

Lecture

Lecture

48 pages

Lecture

Lecture

66 pages

Lecture

Lecture

53 pages

lecture18

lecture18

101 pages

Lecture

Lecture

10 pages

Lecture

Lecture

70 pages

Lecture

Lecture

12 pages

Lecture

Lecture

74 pages

graphs

graphs

10 pages

Lecture

Lecture

62 pages

Lecture

Lecture

11 pages

Lecture

Lecture

71 pages

Lecture

Lecture

42 pages

lecture15

lecture15

72 pages

Lecture

Lecture

82 pages

Load more
Download Lecture
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Lecture and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Lecture 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?