Representations count

In your grandparents’ attic you discover a mysterious old computer. You boot it up and discover it runs Ruby, but doesn’t support negative numbers! How can you implement negative numbers in an elegant way? We’ll explore two solutions and discover how important it is to pick the right representation.

Transcript

1. REPRESENTATIONS
   @tomstuart, RubyConf AU 2019
   https://codon.com/representations-count
   COUNT
   

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2. This is a
   fairy tale.
   

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3. View Slide

4. $ irb
   >> 5 * 2
   => 10
   >> 5 - 2
   => 3
   >> 2 - 5
   ImpossibleNumberError (0 is the smallest number)
   

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5. class SignedNumber
   def self.positive(number)
   end
   def self.negative(number)
   end
   def ==(other)
   end
   def +(other)
   end
   def -(other)
   end
   def *(other)
   end
   def <(other)
   end
   end
   

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6. $ irb
   >> require 'signed_number'
   => true
   >> def pos(number)
   SignedNumber.positive(number)
   end
   => :pos
   >> def neg(number)
   SignedNumber.negative(number)
   end
   => :neg
   

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7. >> pos(2) + pos(5) == pos(7)
   => true
   >> pos(2) - pos(5) == neg(3)
   => true
   >> (pos(9) - pos(3)) * (neg(4) + neg(3)) == neg(42)
   => true
   >> pos(2) * neg(3) < pos(4) + neg(9)
   => true
   

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8. Take a sec
   to think.
   

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9. class SignedNumber
   def self.positive(number)
   end
   def self.negative(number)
   end
   def ==(other)
   end
   def +(other)
   end
   def -(other)
   end
   def *(other)
   end
   def <(other)
   end
   end
   

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10. 3
    -
    

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11. 3
    sign (+ or –)
    size
    -
    

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12. class SignedNumber
    def self.positive(number)
    end
    def self.negative(number)
    end
    # …
    end
    protected
    attr_reader :sign, :size
    def initialize(sign, size)
    @sign, @size = sign, size
    end
    new(:negative, number)
    new(:positive, number)
    

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13. ==
    

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14. sign == other.sign && size == other.size
    class SignedNumber
    # …
    def ==(other)
    end
    # …
    end
    

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15. >> pos(1) == pos(1)
    => true
    >> neg(2) == neg(2)
    => true
    >> pos(1) == pos(2)
    => false
    >> pos(1) == neg(1)
    => false
    ✔
    ✔
    ✔
    ✔
    

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16. +
    

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17. SignedNumber.new(sign, size + other.size)
    end
    # …
    end
    class SignedNumber
    # …
    def +(other)
    

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18. >> pos(3) + pos(1) == pos(4)
    => true
    >> neg(3) + neg(1) == neg(4)
    => true
    >> pos(3) + neg(1) == pos(2)
    => false
    >> pos(3) + neg(1)
    => #
    ✔
    ✔
    ✘
    

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19. else
    SignedNumber.new(sign, size - other.size)
    class SignedNumber
    # …
    def +(other)
    if sign == other.sign
    class SignedNumber
    # …
    def +(other)
    SignedNumber.new(sign, size + other.size)
    end
    # …
    end
    end
    

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20. >> pos(3) + neg(1) == pos(2)
    => true
    >> neg(3) + pos(1) == neg(2)
    => true
    >> pos(1) + neg(3) == neg(2)
    ImpossibleNumberError (0 is the smallest number)
    ✔
    ✔
    

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21. else
    SignedNumber.new(
    sign == :positive ? :negative : :positive,
    other.size - size
    )
    end
    end
    end
    # …
    end
    class SignedNumber
    # …
    def +(other)
    if sign == other.sign
    SignedNumber.new(sign, size + other.size)
    else
    SignedNumber.new(sign, size - other.size)
    if size >= other.size
    

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22. >> pos(1) + neg(3) == neg(2)
    => true
    >> neg(1) + pos(3) == pos(2)
    => true
    >> pos(3) + neg(3) == neg(3) + pos(3)
    => false
    >> pos(3) + neg(3)
    => #
    >> neg(3) + pos(3)
    => #
    >> pos(0) == neg(0)
    => false
    ✔
    ✘
    ✘
    ✔
    

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23. class SignedNumber
    # …
    def ==(other)
    end
    # …
    end
    ( || size.zero?) && size == other.size
    sign == other.sign
    

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24. >> pos(3) + neg(3) == neg(3) + pos(3)
    => true
    ✔
    

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25. View Slide

26. class SignedNumber
    # …
    def ==(other)
    (sign == other.sign || size.zero?) && size == other.size
    end
    def +(other)
    if sign == other.sign
    SignedNumber.new(sign, size + other.size)
    else
    if size >= other.size
    SignedNumber.new(sign, size - other.size)
    else
    SignedNumber.new(
    sign == :positive ? :negative : :positive,
    other.size - size
    )a
    end
    end
    end
    def -(other)
    if sign == other.sign
    if size >= other.size
    SignedNumber.new(sign, size - other.size)
    else
    SignedNumber.new(
    sign == :positive ? :negative : :positive,
    other.size - size
    )b
    end
    else
    SignedNumber.new(sign, size + other.size)
    end
    end
    def *(other)
    if sign == other.sign
    SignedNumber.new(:positive, size * other.size)
    else
    SignedNumber.new(:negative, size * other.size)
    end
    end
    def <(other)
    if sign == other.sign
    if sign == :positive
    size < other.size
    else
    other.size < size
    end
    else
    sign == :negative && !(size.zero? && other.size.zero?)
    end
    end
    # …
    end
    

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27. class SignedNumber
    # …
    def ==(other)
    (sign == other.sign || size.zero?) && size == other.size
    end
    def +(other)
    if sign == other.sign
    SignedNumber.new(sign, size + other.size)
    else
    if size >= other.size
    SignedNumber.new(sign, size - other.size)
    else
    SignedNumber.new(
    sign == :positive ? :negative : :positive,
    other.size - size
    )a
    end
    end
    end
    def -(other)
    

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28. )a
    end
    end
    end
    def -(other)
    if sign == other.sign
    if size >= other.size
    SignedNumber.new(sign, size - other.size)
    else
    SignedNumber.new(
    sign == :positive ? :negative : :positive,
    other.size - size
    )b
    end
    else
    SignedNumber.new(sign, size + other.size)
    end
    end
    def *(other)
    if sign == other.sign
    SignedNumber.new(:positive, size * other.size)
    

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29. other.size - size
    )b
    end
    else
    SignedNumber.new(sign, size + other.size)
    end
    end
    def *(other)
    if sign == other.sign
    SignedNumber.new(:positive, size * other.size)
    else
    SignedNumber.new(:negative, size * other.size)
    end
    end
    def <(other)
    if sign == other.sign
    if sign == :positive
    size < other.size
    else
    other.size < size
    

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30. SignedNumber.new(:positive, size * other.size)
    else
    SignedNumber.new(:negative, size * other.size)
    end
    end
    def <(other)
    if sign == other.sign
    if sign == :positive
    size < other.size
    else
    other.size < size
    end
    else
    sign == :negative && !(size.zero? && other.size.zero?)
    end
    end
    # …
    end
    

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31. class SignedNumber
    # …
    def ==(other)
    (sign == other.sign || size.zero?) && size == other.size
    end
    def +(other)
    if sign == other.sign
    SignedNumber.new(sign, size + other.size)
    else
    if size >= other.size
    SignedNumber.new(sign, size - other.size)
    else
    SignedNumber.new(
    sign == :positive ? :negative : :positive,
    other.size - size
    )a
    end
    end
    end
    def -(other)
    if sign == other.sign
    if size >= other.size
    SignedNumber.new(sign, size - other.size)
    else
    SignedNumber.new(
    sign == :positive ? :negative : :positive,
    other.size - size
    )b
    end
    else
    SignedNumber.new(sign, size + other.size)
    end
    end
    def *(other)
    if sign == other.sign
    SignedNumber.new(:positive, size * other.size)
    else
    SignedNumber.new(:negative, size * other.size)
    end
    end
    def <(other)
    if sign == other.sign
    if sign == :positive
    size < other.size
    else
    other.size < size
    end
    else
    sign == :negative && !(size.zero? && other.size.zero?)
    end
    end
    # …
    end
    

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32. class SignedNumber
    # …
    def ==(other)
    (sign == other.sign || size.zero?) && size == other.size
    end
    def +(other)
    if sign == other.sign
    SignedNumber.new(sign, size + other.size)
    else
    if size >= other.size
    SignedNumber.new(sign, size - other.size)
    else
    SignedNumber.new(
    sign == :positive ? :negative : :positive,
    other.size - size
    )
    end
    end
    end
    def -(other)
    if sign == other.sign
    if size >= other.size
    SignedNumber.new(sign, size - other.size)
    else
    SignedNumber.new(
    sign == :positive ? :negative : :positive,
    other.size - size
    )
    end
    else
    SignedNumber.new(sign, size + other.size)
    end
    end
    def *(other)
    if sign == other.sign
    SignedNumber.new(:positive, size * other.size)
    else
    SignedNumber.new(:negative, size * other.size)
    end
    end
    def <(other)
    if sign == other.sign
    if sign == :positive
    size < other.size
    else
    other.size < size
    end
    else
    sign == :negative && !(size.zero? && other.size.zero?)
    end
    end
    # …
    end
    

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33. This was
    too hard.
    

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34. Can we
    do better?
    

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35. class SignedNumber
    def self.positive(number)
    end
    def self.negative(number)
    end
    def ==(other)
    end
    def +(other)
    end
    def -(other)
    end
    def *(other)
    end
    def <(other)
    end
    end
    

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36. 3
    -
    

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37. $3
    -
    

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38. Your balance: -$3
    Your balance: $7
    deposit $10
    

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39. Your balance: -$3
    Your balance: $97
    deposit $100
    

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40. Your balance: $3
    Your balance: $13
    deposit $10
    

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41. +2 +2 +3
    

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42. -2 -2 -3
    

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43. This
    has useful
    properties.
    

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44. Regularity.
    

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45. -2
    :negative, 2 :negative, 1
    :negative, 0
    :positive, 0
    :positive, 1 :positive, 2
    -1 0 +1 +2
    

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46. -2 -1 0 +1 +2
    …
    …
    …
    …
    …
    

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47. Visual
    intuition.
    

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48. +2
    

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49. View Slide

50. View Slide

51. -2
    

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52. +3 +2
    +
    

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53. View Slide

54. +5
    

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55. Compositionality.
    

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56. +3 +2
    -
    

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57. View Slide

58. View Slide

59. +1
    

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60. protected
    attr_reader :left, :right
    def initialize(left, right)
    @left, @right = left, right
    end
    end
    class SignedNumber
    def self.positive(number)
    end
    def self.negative(number)
    end
    # …
    new(number, 0)
    new(0, number)
    

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61. ==
    

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62. +2 +3
    = ?
    

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63. View Slide

64. View Slide

65. +3 +2
    = ?
    

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66. View Slide

67. View Slide

68. = ?
    +2 +2
    

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69. View Slide

70. View Slide

71. +2 = +2
    +2 = +3 +3 = +2
    

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72. +2 = +2
    +2 = +3 +3 = +2
    left
    right
    right
    left
    +
    +
    

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73. left + other.right == other.left + right
    end
    # …
    end
    class SignedNumber
    # …
    def ==(other)
    

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74. ✔
    

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75. <
    

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76. +2 < +2
    +2 < +3 +3 < +2
    

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77. +2 < +2
    +2 < +3 +3 < +2
    left
    right
    right
    left
    +
    +
    

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78. other.left + right < left + other.right
    end
    # …
    end
    class SignedNumber
    # …
    def <(other)
    

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79. ✔
    

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80. +
    

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81. +3 +2
    +
    

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82. View Slide

83. View Slide

84. left
    right
    left
    right
    +
    +
    

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85. SignedNumber.new(left + other.left, right + other.right)
    end
    # …
    end
    class SignedNumber
    # …
    def +(other)
    

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86. ✔
    

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87. -
    

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88. +2 +3
    -
    

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89. View Slide

90. View Slide

91. left
    right
    right
    left
    +
    +
    

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92. SignedNumber.new(left + other.right, other.left + right)
    end
    # …
    end
    class SignedNumber
    # …
    def -(other)
    

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93. ✔
    

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94. *
    

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95. ×
    +4 +5
    

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96. View Slide

97. View Slide

98. View Slide

99. View Slide

100. View Slide

101. =
     

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102. -
     =
     

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103. -
     =
     +
     

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104. -
     + =
     -
     

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105. -
     + =
     -
     48 16
     6 18 20
     

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106. =
     54 34
     20
     -
     

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107. -
     =
     

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108. × =
     +4 +5 +20
     

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109. SignedNumber.new(
     left * other.right + other.left * right,
     left * other.left + right * other.right
     )
     end
     # …
     end
     class SignedNumber
     # …
     def *(other)
     

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110. ✔
     

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111. class SignedNumber
     # …
     def ==(other)
     left + other.right == other.left + right
     end
     def +(other)
     SignedNumber.new(left + other.left, right + other.right)
     end
     def -(other)
     SignedNumber.new(left + other.right, other.left + right)
     end
     def *(other)
     SignedNumber.new(
     left * other.right + other.left * right,
     left * other.left + right * other.right
     )
     end
     def <(other)
     other.left + right < left + other.right
     end
     # …
     end
     

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112. class SignedNumber
     # …
     def ==(other)
     left + other.right == other.left + right
     end
     def +(other)
     SignedNumber.new(left + other.left, right + other.right)
     end
     def -(other)
     SignedNumber.new(left + other.right, other.left + right)
     end
     def *(other)
     SignedNumber.new(
     left * other.right + other.left * right,
     left * other.left + right * other.right
     )
     end
     def <(other)
     other.left + right < left + other.right
     end
     # …
     end
     

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113. class SignedNumber
     # …
     def ==(other)
     left + other.right == other.left + right
     end
     def +(other)
     SignedNumber.new(left + other.left, right + other.right)
     end
     def -(other)
     SignedNumber.new(left + other.right, other.left + right)
     end
     def *(other)
     SignedNumber.new(
     left * other.right + other.left * right,
     left * other.left + right * other.right
     )
     end
     def <(other)
     other.left + right < left + other.right
     end
     # …
     end
     class SignedNumber
     # …
     def ==(other)
     (sign == other.sign || size.zero?) && size == other.size
     end
     def +(other)
     if sign == other.sign
     SignedNumber.new(sign, size + other.size)
     else
     if size >= other.size
     SignedNumber.new(sign, size - other.size)
     else
     SignedNumber.new(
     sign == :positive ? :negative : :positive,
     other.size - size
     )
     end
     end
     end
     def -(other)
     if sign == other.sign
     if size >= other.size
     SignedNumber.new(sign, size - other.size)
     else
     SignedNumber.new(
     sign == :positive ? :negative : :positive,
     other.size - size
     )
     end
     else
     SignedNumber.new(sign, size + other.size)
     end
     end
     def *(other)
     if sign == other.sign
     SignedNumber.new(:positive, size * other.size)
     else
     SignedNumber.new(:negative, size * other.size)
     end
     end
     def <(other)
     if sign == other.sign
     if sign == :positive
     size < other.size
     else
     other.size < size
     end
     else
     sign == :negative && !(size.zero? && other.size.zero?)
     end
     end
     # …
     end
     

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114. View Slide

115. This is a
     fairy tale.
     

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116. Regularity.
     Visual intuition.
     Compositionality.
     

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117. Representations
     count,
     so choose wisely.
     

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118. View Slide

119. tomstuart/negative-numbers
     

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120. @tomstuart
     https://codon.com/representations-count
     THANKS!
     

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