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Your data structures are made of maths!
- 1. Category Theory for Beginners Your data structures are made of maths! Melbourne Scala User Group Mar 2015 @KenScambler
- 2. Data structures More and more “logical” Less low-level memory/hardware connection More and more support for immutability We can use maths to reason about them! Category Theory can reveal even deeper symmetries
- 3. Algebraic Data Types Constants, sums, products, exponents We can directly manipulate them with algebra!
- 5. NinjaTurtles
- 8. =4 × 2 8
- 9. Integers as types? We can actually use integers to represent our types! The integers correspond to the size of the type
- 10. Products in code (NinjaTurtle, BluesBrother)
- 15. case class TurtleAndBrother( foo1: NinjaTurtle, foo2: BluesBrother)
- 16. case class TurtleAndBrother( foo1: NinjaTurtle, foo2: BluesBrother) 4
- 17. case class TurtleAndBrother( foo1: NinjaTurtle, foo2: BluesBrother) 4 2
- 19. case class TurtleAndBrother( foo1: NinjaTurtle, foo2: BluesBrother) 8
- 20. trait Confluster { def defrabulate(): BluesBrother def empontigle(): NinjaTurtle }
- 21. trait Confluster { def defrabulate(): BluesBrother def empontigle(): NinjaTurtle } 2
- 22. trait Confluster { def defrabulate(): BluesBrother def empontigle(): NinjaTurtle } 4 2
- 23. trait Confluster { def; def; } BluesBrother NinjaTurtle 2 4 ×
- 24. trait Confluster { def defrabulate(): BluesBrother def empontigle(): NinjaTurtle } 8
- 25. Sums A + B
- 28. true false + = Boolean Shapes Boolean or Shapes true false + =
- 29. true false true false =2 + 4 6
- 30. sealed trait Holiday case object Christmas extends Holiday case object Easter extends Holiday case object AnzacDay extends Holiday Sums in code
- 31. sealed trait Holiday case object Christmas extends Holiday case object Easter extends Holiday case object AnzacDay extends Holiday 1
- 32. sealed trait Holiday case object Christmas extends Holiday case object Easter extends Holiday case object AnzacDay extends Holiday 11
- 33. sealed trait Holiday case object Christmas extends Holiday case object Easter extends Holiday case object AnzacDay extends Holiday 11 1
- 34. sealed trait Holiday, case, extends Christmas Easter AnzacDay 11 1 +
- 35. sealed trait Holiday case object Christmas extends Holiday case object Easter extends Holiday case object AnzacDay extends Holiday 3
- 36. sealed trait Opt[A] case class Some[A](a: A) extends Opt[A] case class None[A] extends Opt[A]
- 37. sealed trait Opt[A] case class Some[A](a: A) extends Opt[A] case class None[A] extends Opt[A] A
- 38. sealed trait Opt[A] case class Some[A](a: A) extends Opt[A] case class None[A] extends Opt[A] A 1
- 39. sealed trait Opt[A], case class, extends Some[A](a: A) None[A] A 1 +
- 40. sealed trait Opt[A] case class Some[A](a: A) extends Opt[A] case class None[A] extends Opt[A]A + 1
- 48. Exponents BA
- 49. Exponents BA A type to the power of another type?? Huh?
- 52. true false Boolean Shapes = Shapes Boolean true false true false true false true false true false true false true false true false
- 54. Function types = exponents! A B = BA
- 55. Exponents in code def getTurtle(): NinjaTurtles
- 56. def getTurtle: () => NinjaTurtles
- 57. def getTurtle: () => NinjaTurtles 1
- 58. def getTurtle: () => NinjaTurtles 1 4
- 59. def getTurtle: () => NinjaTurtles 1 4
- 60. def getTurtle: () => NinjaTurtles 41
- 61. def getTurtle: () => NinjaTurtles 4
- 62. Functions “with no arguments” are tacitly from a singleton type such as Unit Singleton types carry no information.
- 63. trait State[S, A] { def run(state: S): (A, S) }
- 64. trait State[S, A] { def run(state: S): (A, S) } A×S
- 65. trait State[S, A] { def run(state: S): (A, S) } S A×S
- 66. trait State[S, A] { def run(state: S): (A, S) } (A×S)S
- 67. {•} Zero On e Product s Sum sExponent s Sets Scala Algebra {} A×B A∪B AB (A, B) 0 1 AB A+B A -> B A / B Nothing Unit BA
- 68. Currying def tupled[A, B, C](a: A, b: B): C def curried[A, B, C]: A => (B => C)
- 69. Currying def tupled[A, B, C](a: A, b: B): C def curried[A, B, C]: A => (B => C) A×B
- 70. Currying def tupled[A, B, C](a: A, b: B): C def curried[A, B, C]: A => (B => C) A×B C
- 71. Currying def tupled[A, B, C](a: A, b: B): C def curried[A, B, C]: A => (B => C) CAB
- 72. Currying def tupled[A, B, C](a: A, b: B): C def curried[A, B, C]: A => (B => C) CAB A(B C)
- 73. Currying def tupled[A, B, C](a: A, b: B): C def curried[A, B, C]: A => (B => C) CAB ACB
- 74. Currying def tupled[A, B, C](a: A, b: B): C def curried[A, B, C]: A => (B => C) CAB CAB
- 75. Recursion sealed trait List[+A] case class Cons[A](h: A, t: List[A]) extends List[A] case object Nil extends List[Nothing]
- 76. sealed trait List[+A] case class Cons[A](h: A, t: List[A]) extends List[A] case object Nil extends List[Nothing] A
- 77. sealed trait List[+A] case class Cons[A](h: A, t: List[A]) extends List[A] case object Nil extends List[Nothing] A L(A)
- 78. sealed trait List[+A] case class Cons[A](h: A, t: List[A]) extends List[A] case object Nil extends List[Nothing] A × L(A)
- 79. sealed trait List[+A] case class Cons[A](h: A, t: List[A]) extends List[A] case object Nil extends List[Nothing] A × L(A) 1
- 80. sealed trait List[+A] case class Cons[A](h: A, t: List[A]) extends List[A] case object Nil extends List[Nothing] 1 + A × L(A)
- 81. Expanding a list… L(a) = 1 + a × L(a)
- 82. Expanding a list… L(a) = 1 + a L(a)
- 83. Expanding a list… L(a) = 1 + a L(a) = 1 + a (1 + a L(a))
- 84. Expanding a list… L(a) = 1 + a L(a) = 1 + a (1 + a L(a)) = 1 + a + a2 (1 + a L(a))
- 85. Expanding a list… L(a) = 1 + a L(a) = 1 + a (1 + a L(a)) = 1 + a + a2 (1 + a L(a)) … = 1 + a + a2 + a3 + a4 + a5…
- 86. Expanding a list… L(a) = 1 + a L(a) = 1 + a (1 + a L(a)) = 1 + a + a2 (1 + a L(a)) … = 1 + a + a2 + a3 + a4 + a5… Nil or 1-length or 2-length or 3-length or 4-length etc
- 87. What does it mean for two types to have the same number?
- 92. Terminology isomorphism “Sorta kinda the same-ish” but I want to sound really smart - Programmers
- 93. Terminology isomorphism “Sorta kinda the same-ish” but I want to sound really smart - Programmers
- 94. Terminology isomorphism One-to-one mapping between two objects so you can go back-and-forth without losing information
- 98. There can be lots of isos between two objects! If there’s at least one, we can say they are isomorphic or A ≅ B
- 99. Programming language syntaxProgramming language FEATURES!!! NAMES NAMES NAMES actual structure Programming language syntax Programming language FEATURES!!! NAMES NAMES NAMES actual structure class Furpular { class Diggleton { } }
- 100. Programming language syntaxProgramming language FEATURES!!! NAMES NAMES NAMES actual structure Programming language syntax Programming language FEATURES!!! NAMES NAMES NAMES actual structure class Furpular { class Diggleton { } }
- 101. Programming language syntaxProgramming language FEATURES!!! NAMES NAMES NAMES actual structure Programming language syntax Programming language FEATURES!!! NAMES NAMES NAMES actual structure class Furpular { class Diggleton { } } Looking at data structures algebraically lets us compare the true structure
- 102. actual structure but faster actual structure Knowing an isomorphism, we can rewrite for • Performance • Memory usage • Elegance with proven correctness!
- 103. actual structure but faster actual structure Knowing an isomorphism, we can rewrite for • Performance • Memory usage • Elegance with proven correctness!* *mumble mumble non-termination
- 104. Category Theory object A B object arrow Arrows join objects. They can represent absolutely anything.
- 105. Categories generalise functions over sets A B f
- 106. Arrows compose like functions A B C f g
- 107. Arrows compose like functions A C g ∘ f
- 108. A Every object has an identity arrow, just like the identity function
- 109. That’s all a category is!
- 110. Products in CT A × BA B first seco nd trait Product[A,B] { def first: A def second: B }
- 111. Sums in CT A + BA B Left Right sealed trait Sum[A,B] case class Left[A,B](a: A) extends Sum[A,B] case class Right[A,B](b: B) extends Sum[A,B]
- 112. Opposite categories C Cop A B C g ∘ f f g A B C f ∘ g f g Isomorphic!
- 113. A B C g ∘ f f g A B C f ∘ g f g Just flip the arrows, and reverse composition!
- 114. A A×B B A product in C is a sum in Cop A sum in C is a product in Cop A+B B A C Cop
- 116. Terminology dual An object and its equivalent in the opposite category are to each other.
- 117. Terminology Co-(thing) Often we call something’s dual a
- 118. Terminology Coproducts Sums are also called
- 119. Tightening the definitions A × BA B first seco nd × × trait ProductPlusPlus[A,B] { def first: A def second: B def banana: Banana def brother: BluesBrother }
- 120. A × BA B first seco nd × × Does that still count as A × B?
- 121. A × BA B first seco nd × × No way!
- 122. A × BA B first seco nd × × Umpire theA someB trait Umpire { def theA: A def someB: B }
- 123. A × BA B first seco nd × × Umpire theA someB trait Umpire { def theA: A def someB: B } unique∃
- 124. (a, b,a b Umpire trait Umpire { def theA: A = a def someB: B = b } , )Instances
- 125. (a, b,a b Umpire trait Umpire { def theA: A = a def someB: B = b } , )Instances
- 126. (a, b,a b Umpire trait Umpire { def theA: A = a def someB: B = b } , ) not actually unique Instances
- 127. Requiring a unique arrow from a 3rd object that independently knows A and B proves that there’s no extra gunk.
- 128. But wait!
- 129. What if Umpire has special knowledge about other products?
- 130. A × BA B first seco nd Umpire theA someB trait Umpire { def theA: A def someB: B def specialOtherProd: (A,B) }
- 131. It could introduce strange new things with its special knowledge!
- 132. We need to know nothing about the object other than the two arrows!
- 133. PA B ??? ? ? unique∃ For all objects that 1) have an arrow to A and B 2) there exists a unique arrow to P
- 134. PA B ??? ? ? unique∃ Then P is “the” product of A and B!
- 135. Same with sums!
- 136. SA B ??? ? ? unique∃ For all objects that 1) have an arrow from A and B 2) there exists a unique arrow from S
- 137. SA B ??? ? ? unique∃ Then S is “the” sum of A and B!
- 139. Terminology universal property “The most efficient solution to a problem”
- 140. Terminology universal property Proves that - We generate only what we need - We depend on only what we need
- 141. Compare to programming: trait Monoid[M] { def id: M def compose(a: M, b: M): M } trait Foldable[F[_]] { def foldMap[M: Monoid, A]( fa: F[A], f: A => M): M }
- 142. Like UMPs, type parameters “for all F” “for all A and M where M is a Monoid” don’t just prove what your code is, but what it isn’t.
- 143. Proving what your code isn’t prevents bloat and error, and promotes reuse. Proving what your code is allows you to use it.
- 144. Zero On eProduct s Sum s Exponent s Scala Algebra (A, B) 0 1 AB A+B A -> B A / B Nothing Unit BA Types vs Algebra vs CT CT ? ? ? ? ? ?
- 145. Intermission
- 146. Injections A B A function is injective if it maps 1-to-1 onto a subset of the codomain
- 147. A B All the information in A is preserved…
- 148. A B All the information in A is preserved… A
- 149. B But everything else in B is lost. A B
- 150. Another way of looking at it… C A Bfg h If f ∘ g = f ∘ h, then g = h
- 151. C A Bfg h If f ∘ g = f ∘ h, then g = h
- 152. C A Bfg h If f ∘ g = f ∘ h, then g = h
- 153. Injections in code User( firstName = "Bob", lastName = "Smith", age = 73) User JSON { “firstName”: "Bob", “lastName”: ”Smith”, “age”: 73 }
- 154. User JSON We don’t lose information converting to JSON.
- 155. But from JSON? We lose structure… User JSON
- 156. Surjections A B A function is surjective if it maps onto the whole of the codomain
- 157. A All the information in B is preserved… AB
- 158. A But everything else in A is lost. AB
- 159. A B C If g ∘ f = h ∘ f, then g = h f g h
- 160. A B C f g h If g ∘ f = h ∘ f, then g = h
- 161. A B C f g h If g ∘ f = h ∘ f, then g = h
- 162. Injections generalised C A Bf g h If f ∘ g = f ∘ h, then g = h
- 163. Monomorphisms C A Bf g h f is monic If f ∘ g = f ∘ h, then g = h
- 164. Monomorphisms C A Bf g h If f ∘ g = f ∘ h, then g = h
- 165. Surjections generalised CA Bf g h If g ∘ f = h ∘ f, then g = h
- 166. Epimorphisms CA Bf g h f is epic If g ∘ f = h ∘ f, then g = h
- 167. C A B A mono in C is an epi in Cop A B C Cop C
- 168. Monos are dual to epis.
- 169. Bijections A B A function is bijective if it maps 1-to-1 onto the whole codomain
- 170. Bijections A B We don’t lose any information A B A… A
- 171. Bijections B Nor do we lose information B A B… A B
- 172. The CT equivalent is of course…
- 173. The CT equivalent is of course… Isomorphisms!
- 174. Injection Surjection Bijection Mono Epi Iso
- 175. Motivation So much software is just mapping between things!
- 176. form
- 177. form {…}
- 179. form {…} Order(a,b,c) INSERT INTO…
- 180. form {…} Order(a,b,c) INSERT INTO…
- 181. form {…} Order(a,b,c) INSERT INTO… Receipt(…)
- 182. form {…} Order(a,b,c) INSERT INTO… Receipt(…) Logs on disk
- 183. form {…} Order(a,b,c) INSERT INTO… Receipt(…) Logs on disk <xml>
- 184. form {…} Order(a,b,c) INSERT INTO… Receipt(…)<xml> receipt Logs on disk
- 185. It is essential to understand how information is preserved in flows like this
- 186. Chinese whispers! Bugs proliferate where data is lost!
- 187. Injections and surjections tell us what is preserved and what is lost
- 188. Bijections are especially valuable
- 189. Conclusion Data structures are made out of maths!
- 190. How we map between them is maths too.
- 191. Understanding the underlying shape of data and functions is enormously helpful for more robust, error-free software
- 192. Isomorphism is more interesting than equality! Isomorphic types can be rewritten, optimised without error. Isomorphic mappings allow us to preserve information
- 193. Universal properties exemplify • Depending on the minimum • Producing the minimum ?
- 194. Again, Category Theory shows us deeper, simpler patterns, unifying concepts that otherwise look different ? ? × +
- 195. Further reading Awodey, “Category Theory” Lawvere & Schanuel, “Conceptual Mathematics: an introduction to categories” Jeremy Kun, “Math ∩ Programming” at http://jeremykun.com/ Chris Taylor, “The algebra of algebraic datatypes” http://chris-taylor.github.io/blog/2013/02/10/the-algebra-of- algebraic-data-types/ http://chris-taylor.github.io/blog/2013/02/11/the-algebra-of- algebraic-data-types-part-ii/ http://chris-taylor.github.io/blog/2013/02/13/the-algebra-of- algebraic-data-types-part-iii/
- 196. Further reading Bartosz Milewski “Categories for Programmers” http://bartoszmilewski.com/2014/10/28/category-theory-for- programmers-the-preface/ http://bartoszmilewski.com/2015/03/13/function-types/