Finding the Algebra in Algebraic Data Types

Construction of an algebra using Swift's type system.

Transcript

1. FINDING THE
   ALGEBRA IN
   ALGEBRAIC DATA
   TYPES
   

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2. ALGEBRA?
   Algebra (from Arabic and Farsi "al-jabr" meaning "reunion
   of broken parts") is one of the broad parts of
   mathematics, together with number theory, geometry and
   analysis. In its most general form, algebra is the study of
   mathematical symbols and the rules for manipulating
   these symbols.
   — Wikipedia
   

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3. !
   

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4. DON'T WORRY !
   One algebra you know from primary school
   > Symbols/Elements: !, ", #, $, ...
   > Manipulation/Operators: ➕➖✖➗
   

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5. FANCIER SYMBOLS
   > , ,
   AND RULES
   >
   >
   

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6. THE ALGEBRA
   OF SWIFT TYPES
   

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7. THE ALGEBRA OF SWIFT TYPES
   > Example elements: Int, Bool, ...
   > Example operator: Optional
   

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8. COUNTING THE VALUES
   > Bool : [false, true]
   > UInt : 0, 1, 2 ...
   

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9. GOT !?
   

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10. ()
    > ()
    > Bool
    typealias One = () /* = Void */
    typealias Two = Bool
    let theOne : [One] = [()]
    let allTwos : [Two] = [false, true]
    

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11. NEED ! !
    

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12. enum Zero {}
    

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13. !"
    enum Zero {
    }
    let impossible : Zero = ! /* has no value */
    let possibles : [Zero] = []
    

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14. ADDITION
    enum Add {
    case AddL(L)
    case AddR(R)
    }
    typealias Three = Add /* Add<(), Bool> */
    let allThrees : [Three] = [ .AddL(()),
    .AddR(false),
    .AddR(true)]
    

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15. typealias Five = Add
    let fs : [Five] = [.AddL(.AddL(())),
    .AddL(.AddR(false)),
    .AddL(.AddR(true)),
    .AddR(false),
    .AddR(true)]
    

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16. OPTIONALS
    enum Optional {
    case None
    case Some(T)
    }
    Optional
    

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17. MULTIPLICATION
    struct Mul {
    let l : L
    let r : R
    }
    /* alternative: (L, R) */
    typealias Four = Mul
    let fours : [Four] = [Mul(l:false, r:false),
    Mul(l:false, r:true),
    Mul(l:true, r:false),
    Mul(l:true, r:true)]
    

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18. THE RULES
    typealias Two_ = Add
    let allTwo_s : [Two_] = [.AllL(false), .AddL(true)]
    

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19. typealias ThreeL = Add
    typealias ThreeR = Add
    let l : [ThreeL] = [.AddL(()), .AddR(false), .AddR(true)]
    let r : [ThreeR] = [.AddL(false), .AddL(true), .AddR(())]
    

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20. typealias Two__ = Mul
    let all2__ : [Two__] = [Mul(l:(), r:false),
    Mul(l:(), r:true)]
    

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21. typealias Zero_ = Mul
    let allZero_ : [Zero_] = []
    // Mul(l:.AddL(()), r:!)
    

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22. DISTRIBUTIVE LAW :)
    Mul> === Add, Mul>
    

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23. WHAT ABOUT
    FUNCTION TYPES?
    

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24. FUNCTION TYPES
    enum Colour { case White; case Red }
    enum Fill { case None; case Pattern; case Solid }
    func myFavouriteFillForColour(colour : Colour) -> Fill {
    switch (colour) {
    case .White:
    return .Solid
    case .Red:
    return .Pattern
    }
    }
    How many functions myFavouriteFillForColour?
    

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25. COUNTING (Colour) -> Fill
    

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26. FUNCTION TYPES
    > Colour , Fill
    > (Colour) -> Fill
    turns out:
    > (A) -> B
    !
    

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27. (A, B) -> C ❓
    

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28. > (A, B) -> C
    > (A) -> (B -> C)
    > currying ✅
    func isNiceTuple(colour : Colour, fill : Fill) -> Bool {
    return true
    }
    func isNiceCurried(colour : Colour)(fill : Fill) -> Bool {
    return true
    }
    

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29. THANK YOU
    QUESTIONS?
    @JOHANNESWEISS
    

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