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
2. This is a fairy tale.
4. $ irb >> 5 * 2 => 10 >> 5 - 2 => 3 >> 2 - 5 ImpossibleNumberError (0 is the smallest number)
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
6. $ irb >> require 'signed_number' => true >> def pos(number) SignedNumber.positive(number) end => :pos >> def neg(number) SignedNumber.negative(number) end => :neg
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
8. Take a sec to think.
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
10. 3 -
11. 3 sign (+ or –) size -
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)
13. ==
14. sign == other.sign && size == other.size class SignedNumber # … def ==(other) end # … end
15. >> pos(1) == pos(1) => true >> neg(2) == neg(2) => true >> pos(1) == pos(2) => false >> pos(1) == neg(1) => false ✔ ✔ ✔ ✔
16. +
17. SignedNumber.new(sign, size + other.size) end # … end class SignedNumber # … def +(other)
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) => #<SignedNumber @sign=:positive, @size=4> ✔ ✔ ✘
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
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) ✔ ✔
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
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) => #<SignedNumber @sign=:positive, @size=0> >> neg(3) + pos(3) => #<SignedNumber @sign=:negative, @size=0> >> pos(0) == neg(0) => false ✔ ✘ ✘ ✔
23. class SignedNumber # … def ==(other) end # … end ( || size.zero?) && size == other.size sign == other.sign
24. >> pos(3) + neg(3) == neg(3) + pos(3) => true ✔
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
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)
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)
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
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
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
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
33. This was too hard.
34. Can we do better?
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
36. 3 -
37. $3 -
38. Your balance: -$3 Your balance: $7 deposit $10
39. Your balance: -$3 Your balance: $97 deposit $100
40. Your balance: $3 Your balance: $13 deposit $10
41. +2 +2 +3
42. -2 -2 -3
43. This has useful properties.
44. Regularity.
45. -2 :negative, 2 :negative, 1 :negative, 0 :positive, 0 :positive, 1 :positive, 2 -1 0 +1 +2
46. -2 -1 0 +1 +2 … … … … …
47. Visual intuition.
48. +2
51. -2
52. +3 +2 +
54. +5
55. Compositionality.
56. +3 +2 -
59. +1
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)
61. ==
62. +2 +3 = ?
65. +3 +2 = ?
68. = ? +2 +2
71. +2 = +2 +2 = +3 +3 = +2
72. +2 = +2 +2 = +3 +3 = +2 left right right left + +
73. left + other.right == other.left + right end # … end class SignedNumber # … def ==(other)
74. ✔
75. <
76. +2 < +2 +2 < +3 +3 < +2
77. +2 < +2 +2 < +3 +3 < +2 left right right left + +
78. other.left + right < left + other.right end # … end class SignedNumber # … def <(other)
79. ✔
80. +
81. +3 +2 +
84. left right left right + +
85. SignedNumber.new(left + other.left, right + other.right) end # … end class SignedNumber # … def +(other)
86. ✔
87. -
88. +2 +3 -
91. left right right left + +
92. SignedNumber.new(left + other.right, other.left + right) end # … end class SignedNumber # … def -(other)
93. ✔
94. *
95. × +4 +5
101. =
102. - =
103. - = +
104. - + = -
105. - + = - 48 16 6 18 20
106. = 54 34 20 -
107. - =
108. × = +4 +5 +20
109. SignedNumber.new( left * other.right + other.left * right, left * other.left + right * other.right ) end # … end class SignedNumber # … def *(other)
110. ✔
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
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
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
115. This is a fairy tale.
116. Regularity. Visual intuition. Compositionality.
117. Representations count, so choose wisely.
119. tomstuart/negative-numbers
120. @tomstuart https://codon.com/representations-count THANKS!