Type Introduction Illustrated for Casual Haskellers

Type introduction illustrated for casual haskellers to get over the FoldableTakenobu T.Rev. 0.01.2 WIP “What is this description ?!” foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b NOTE - This document shows one of the mental model. - Please see also references. - This is written for Haskell, especially ghc7.10/8.0 and later. Contents1. Introduction Appendix I – Various types - Values, Types, Type classes - Bool, Char, Int, Float - Polymorphic types - Maybe, List, Either, Tuple - Type constructors - Polymorphic and type constructors Appendix II – Various type classes - Eq, Ord 2. more, Types and Type classes - Num - Function types - Foldable - <a href="/tags/Type_class/" rel="tag">Type class</a> operations - Functor, Applicative, Monad - Monoid 3. What is this? - TraversableAppendix III – Advanced topicsReferences 1. Introduction 1. IntroductionValues, Types, Type classes Values‘a’ False 1.0 1 2 ‘h’1.5 True ‘5’ 700 3.14References : [B2] Ch.2, [B3] Ch.2 Types“Bool” type “Int” type “Float” type “Char” type1 1.0 ‘a’ False 2 1.5 ‘h’True 700 3.14 ‘5’ ...... A type is a collection of values which have common property.References : [B2] Ch.2, [B3] Ch.2, [B1] Ch.2, [D2], [B5] Ch.8, [H1] Ch.4 Type classes“Num” type classBool Int Float Char1 1.0 ‘a’ False 2 1.5 ‘h’ ... True 700 3.14 ‘5’ ...... A type class is a collection of types which have common operations.References : [B1] Ch.2, [B2] Ch.2, [B3] Ch.3, [B4] Ch.6, [H1] Ch.4 1. IntroductionPolymorphic types Proper typesBool Int Float Char1 1.0 ‘a’ Proper False 2 1.5 ‘h’ types True 700 3.14 ‘z’ ...... References : [B1] Ch.2, [B2] Ch.2, [B3] Ch.3, [B4] Ch.6 Polymorphic types aPolymorphic typesParametric polymorphism...Bool Int Float Char1 1.0 ‘a’ Proper False 2 1.5 ‘h’ types True 700 3.14 ‘z’ ...... References : [B1] Ch.7, [B3] Ch.3, [B4] Ch.6, [D1] Week 2, [H1] Ch.4 Polymorphic types restricted with type classesNum a => aPolymorphic types limited with the type class...Bool Int Float Char1 1.0 ‘a’ Proper False 2 1.5 ‘h’ ... types True 700 3.14 ‘z’ ...... Num type classReferences : [B1] Ch.2, [B2] Ch.2, [B3] Ch.3, [B4] Ch.6, [D1] Week 4 Polymorphic types aPolymorphic typesNum a => a polymorphic types with type classInt Proper typesReferences : [B1] Ch.2, [B2] Ch.2, [B3] Ch.3, [B4] Ch.6, [D1] Week 2, 4 1. IntroductionType constructors Type constructors nullary type constructorIntReferences : [B1] Ch.7 Type constructors unary type constructorMaybe Int“Maybe” type constructor takes one type argument (unary).References : [B1] Ch.7 Type constructors nullary unary binary 3-ary ...Int Maybe Int Either Int Char (,,) Int Int FloatFloat IO Int State Int Char :Int -> Bool [] Int (,) Int Char[ Int ] (Int, Char) Syntactic sugar Syntactic sugar : : : References : [B1] Ch.7, [H1] Ch.4 1. IntroductionPolymorphic types and type constructors Polymorphic types and type constructors aPolymorphic types ?Int Maybe IntType constructorsReferences : [B1] Ch.2, 7, [B2] Ch.2, [B3] Ch.3, [B4] Ch.6 Polymorphic types and type constructors aPolymorphic types t aMaybe aInt Maybe IntType constructorsReferences : [B1] Ch.2, 7, [B2] Ch.2, [B3] Ch.3, [B4] Ch.6 Polymorphic types and type constructors t a...Maybe a [] a Tree a IO a... Polymorphic typesReferences : [B1] Ch.2, 7, [B2] Ch.2, [B3] Ch.3, [B4] Ch.6 Polymorphic types and type constructorsMaybe aPolymorphic types...Maybe Int Maybe Char Maybe Float Maybe BoolProper types ...References : [B1] Ch.2, 7, [B2] Ch.2, [B3] Ch.3, [B4] Ch.6 Polymorphic types and type constructors a nullary unary binary 3-ary ... t a t a b t a b c Polymorphic types ...... Proper Int Maybe Int Either Int Char (,,) Int Int Float types: Float [] Int (,) Int CharInt -> Bool IO Int State Int Char: : : References : [B1] Ch.2, 7, [B2] Ch.2, [B3] Ch.3, [B4] Ch.6 2. more, Types and Type classes 2. more, Types and Type classesFunction types <a href="/tags/Function_type/" rel="tag">Function type</a>Int fun :: Int -> Bool Bool funThe “->” represents the function type.References : [B2] Ch.2, [B1] Ch.5, [B3] Ch.7, [B5] Ch.9, [H1] Ch.4 Function type with multiple arguments equivalent to f :: Int -> (Float -> Bool) Int fun :: Int -> Float -> Bool Bool fun FloatInt fun :: Int -> Float -> Char -> Bool Float Bool funCharReferences : [B2] Ch.2, [B1] Ch.5, [B3] Ch.7 Function type with same typeInt fun :: Int -> Int -> Int funReferences : [B2] Ch.2, [B1] Ch.5, [B3] Ch.7 Function type with function as argumentInt -> IntInt -> Int fun :: (Int -> Int) -> Int -> Bool Bool funIntReferences : [B2] Ch.2, [B1] Ch.5, [B3] Ch.7 Function type with function as resultInt fun :: Int -> (Int -> Int) Int -> Int fun Int -> IntReferences : [B2] Ch.2, [B1] Ch.5, [B3] Ch.7 Function type with polymorphic function a fun :: a -> Bool Bool funPolymorphic types <a href="/tags/Parametric_polymorphism/" rel="tag">Parametric polymorphism</a>... Proper typesInt fun :: Int -> Bool Bool Char fun :: Char -> Bool Bool fun funReferences : [B2] Ch.2, [B1] Ch.5, [B3] Ch.7 Function type for polymorphic function with type class a fun :: Num a => a -> Bool Bool funPolymorphic types limited with the type class...... Proper typesInt fun :: Int -> Bool Bool Char fun :: Char -> Bool Bool fun funReferences : [B2] Ch.2, [B1] Ch.5, [B3] Ch.7 2. more, Types and Type classesType class operations A type class has the class operationsClass a => a class operations (class methods / common operations)?References : [B1] Ch.2, 7, [B2] Ch.2, [B3] Ch.3, [B4] Ch.6 A type class has the class operationsEq a => a Bool Eq a => a -> a -> Bool==Eq class has “==“ (equality) operation.References : [B1] Ch.2, 7, [B2] Ch.2, [B3] Ch.3, [B4] Ch.6 A type class has the class operations a Eq a => a -> a -> Bool Bool ==Polymorphic Ad-hoc polymorphism (overloading) types... Proper typesInt Char Int -> Int -> Bool Bool Char -> Char -> Bool Bool == ==Each type, that belongs to the type class, must be support the overloaded operations.References : [B1] Ch.2, 7, [B2] Ch.2, [B3] Ch.3, [B4] Ch.6 Declaration of a type class and instances class Eq a where (==) :: a -> a -> Bool a Eq a => a -> a -> Bool Bool ==Polymorphic types... Proper typesInt Char Int -> Int -> Bool Bool Char -> Char -> Bool Bool == == instance Eq Int where instance Eq Char where (==) = eqInt (==) = ...References : [B1] Ch.2, 7, [B2] Ch.2, [B3] Ch.3, [B4] Ch.6 3. What is this? What is this ?! foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b foldr = ... ？References : [B1] Ch.12, 5, 7, [B2] Ch.6, 2, [B3] Ch.7, 3, [B4] Ch.4, 6 What is this ?! foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b bFoldable t => t a foldr a -> b -> b a b bReferences : [B1] Ch.12, 5, 7, [B2] Ch.6, 2, [B3] Ch.7, 3, [B4] Ch.4, 6 What is this ?! foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b bFoldable t => t a foldr a -> b -> b a b bReferences : [B1] Ch.12, 5, 7, [B2] Ch.6, 2, [B3] Ch.7, 3, [B4] Ch.4, 6 What is this ?! foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b bFoldable t => t a foldr a -> b -> b a b bReferences : [B1] Ch.12, 5, 7, [B2] Ch.6, 2, [B3] Ch.7, 3, [B4] Ch.4, 6 What is this ?! foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b bFoldable t => t a foldr a -> b -> b a b bReferences : [B1] Ch.12, 5, 7, [B2] Ch.6, 2, [B3] Ch.7, 3, [B4] Ch.4, 6 What is this ?! foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b bFoldable t => t a foldr a -> b -> b a b bReferences : [B1] Ch.12, 5, 7, [B2] Ch.6, 2, [B3] Ch.7, 3, [B4] Ch.4, 6 What is this ?! foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b bFoldable t => t a foldr a -> b -> b a b bReferences : [B1] Ch.12, 5, 7, [B2] Ch.6, 2, [B3] Ch.7, 3, [B4] Ch.4, 6 What is this ?! foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b bFoldable t => t a foldr a -> b -> b a b bReferences : [B1] Ch.12, 5, 7, [B2] Ch.6, 2, [B3] Ch.7, 3, [B4] Ch.4, 6 Example of polymorphism on foldr bFoldable t => t a foldr a -> b -> b Polymorphic types ... b b[a] Tree a foldr foldr a -> b -> b a -> b -> bReferences : [B1] Ch.12, 5, 7, [B2] Ch.6, 2, [B3] Ch.7, 3, [B4] Ch.4, 6 Example of polymorphism on foldr b[a] foldr a -> b -> b Polymorphic types ...Proper types Int Int[Char] [Int] foldr foldr example Char -> Char -> Int Int -> Int -> Int foldr (+) 0 [1, 2, 3] foldr (*) 1 [1, 2, 3] foldr max 0 [1, 2, 3] : References : [B1] Ch.12, 5, 7, [B2] Ch.6, 2, [B3] Ch.7, 3, [B4] Ch.4, 6 Appendix I – Various types Bool, Char, Int, Float typesBool Char Int Float‘a’ 1 1.0 False ‘h’ 2 1.5True ‘z’ 700 3.14 ...... References : [B2] Ch.2, [B3] Ch.2, [B1] Ch.2, [H1] Ch.6, [S1] Maybe typeMaybe aPolymorphic types...Maybe Bool Maybe Int Maybe Float Maybe CharNothing Nothing Nothing Nothing Proper Just 1 Just 1.0 Just ‘a’ Just True types Just 2 Just 1.5 Just ‘h’ Just False Just 700 Just 3.14 Just ‘z’ ...... References : [B1] Ch.7, [S1] List typeSyntactic sugar [a] [] aPolymorphic types...[ Bool ] [ Int ] [ Float ] [ Char ][] [] [] [] [1.0] Proper [True] [1] [‘a’] types [1, 2] [1.5, 20.0] [‘h’] [True, False] [700, ..] [3.14] [‘a’, ‘b’, ‘c’] ...... References : [B1] Ch.1, [B2] Ch.4, [B3] Ch.3, [H1] Ch.6, [S1] Either typeEither a bPolymorphic types...Either Bool Int Either Char Int Either Int Float Either Float CharLeft True Left 1 Proper Left ‘a’ Left 1.0 Left False Right 2 Right 1.5 types Left 20.0 Right 7 Right 700 Right 3.14 Right ‘z’ ...... References : [B1] Ch.7, [S1] Tuple (pair) typeSyntactic sugar (a, b) (,) a bPolymorphic types...(Bool, Int) (Char, Int) (Int, Float) (Float, Char)(True, 1) (‘a’, 2) (1, 1.0) Proper (1.0, ‘A’) (False, 7) (‘b’, 5) (5, 1.5) types (20.0, ‘b’) (True, 10) (‘z’, 700) (100, 3.14) (3.14, ‘z’) ...... References : [B1] Ch.1, [B2] Ch.2, [B3] Ch.3, [H1] Ch.6, [S1] Appendix II – Various type classes Eq class‘s characteristic operationsEq a => a Bool==/=The Eq class has equality operations.References : [B1] Ch.1, [B2] Ch.2, [B3] Ch.3, [H1] Ch.6, [S1] Ord class‘s characteristic operationsOrd a => a Bool<<=>>=The Ord class has comparison operations.References : [B1] Ch.1, [B2] Ch.2, [B3] Ch.3, [H1] Ch.6, [S1] Num class‘s characteristic operationsNum a => a+-* absThe Num class has arithmetic operations.References : [B1] Ch.1, [B2] Ch.2, [B3] Ch.3, [H1] Ch.6, [S1] Foldable class‘s characteristic operations bFoldable t => t a foldr a -> b -> bReferences : [B1] Ch.12, [B2] Ch.6, [B3] Ch.7, [D3], [S1] Functor class‘s characteristic operations example [] b Maybe b Tree b IO b : Functor f => f a Functor f => f b fmap :: Functor f => (a -> b) -> f a -> f b fmap a -> bReferences : [B1] Ch.7, [D3], [H1] Ch.6, [S1] Applicative class‘s characteristic operations a pureApplicative f => f a Applicative f => f b<*>Applicative f => f (a -> b)References : [B1] Ch.11, [D3], [S1] Monad class‘s characteristic operations a returnMonad m => m a Monad m => m b>>=Monad m => a -> m bReferences : [B1] Ch.12, [B2] Ch.10, [D3], [H1] Ch.6, [S1] Monoid class‘s characteristic operationsMonoid a => a mappend memptyReferences : [B1] Ch.12, [D3], [S1] Related topics: monoid lawsMonoid a => a mappend binary operationMonoid a => a mempty mappend unit lawMonoid a => a mappend mappend mappend mappend associative law sameProgrammers should satisfy the monoid laws.References : [B1] Ch.12, [D3], [S1] Traversable class‘s characteristic operationsTraversable t => t a Traversable t => f (t b) traverse a -> f bReferences : [D3], [S1] Appendix III – Advanced topics Universally quantified types forall a . a a implicitly quantified with universal quantification (∀a .) Polymorphic types...Bool Int Float Char1 1.0 ‘a’ Proper False 2 1.5 ‘h’ types True 700 3.14 ‘z’ ...... References : [B5] Ch.23, [H1] Ch.4, [H2] “GHC Language Features” Kinds and type constructors nullary unary binary 3-ary ... <a href="/tags/Kind_(type_theory)/" rel="tag">kind</a> kind kind Maybe :: * -> * Either :: * -> * -> * (,,) :: * -> * -> * -> * kind Int :: * Maybe Int :: * Either Int Char :: * (,,) Int Int Float :: *Int Maybe Int Either Int Char (,,) Int Int FloatFloat IO Int State Int CharInt -> Bool [] Int (,) Int CharReferences : [B1] Ch.7, [B5] Ch.29, [H1] Ch.4 Type systemsTAPL [5] Ch.23 TAPL [5] Ch.29 corresponding to <a href="/tags/System_F/" rel="tag">System F</a> corresponding to System Fω a Polymorphic types t a(Quantified types) Maybe aInt Maybe IntTAPL [5] Ch.30 corresponding to λωType constructors (Type operators)References : [B5] Ch.23, 29, 30 References ReferencesBooks[B1] Learn You a Haskell for Great Good! (LYAH) http://learnyouahaskell.com/[B2] Thinking Functionally with Haskell (IFPH 3rd edition) http://www.cs.ox.ac.uk/publications/books/functional/[B3] Programming in Haskell http://www.cs.nott.ac.uk/~pszgmh/book.html[B4] Real World Haskell (RWH) http://book.realworldhaskell.org/[B5] Types and Programming Languages (TAPL) https://mitpress.mit.edu/books/types-and-programming-languagesDocuments[D1] CIS 194: Introduction to Haskell http://www.seas.upenn.edu/~cis194/lectures.html[D2] Type Systems http://dev.stephendiehl.com/fun/004_type_systems.html[D3] Typeclassopedia http://www.cs.tufts.edu/comp/150FP/archive/brent-yorgey/tc.pdf https://wiki.haskell.org/Typeclassopedia ReferencesSearch[S1] Hoogle https://www.haskell.org/hoogleSpecifications[H1] Haskell 2010 Language Report https://www.haskell.org/definition/haskell2010.pdf[H2] The Glorious Glasgow Haskell Compilation System (GHC user’s guide) https://downloads.haskell.org/~ghc/latest/docs/html/users_guide/index.html https://downloads.haskell.org/~ghc/latest/docs/users_guide.pdf ReferencesFurthermore readings[A1] What I Wish I Knew When Learning Haskell http://dev.stephendiehl.com/hask/[A2] How to learn Haskell https://github.com/bitemyapp/learnhaskell[A3] Documentation https://www.haskell.org/documentation[A4] A Haskell Implementation Reading List http://www.stephendiehl.com/posts/essential_compilers.html[A5] The GHC reading list https://ghc.haskell.org/trac/ghc/wiki/ReadingList

Type Introduction Illustrated for Casual Haskellers

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