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Representations count

Tom Stuart
February 07, 2019

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.

Tom Stuart

February 07, 2019
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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. View Slide

  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. View Slide

  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. View Slide

  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. View Slide

  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. View Slide

  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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