Type introduction illustrated for casual haskellers

to get over the Foldable

Takenobu 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. Contents

1. 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 - Type class operations - Functor, Applicative, Monad - Monoid 3. What is this? - Traversable

Appendix III – Advanced topics

References 1. Introduction 1. Introduction

Values, Types, Type classes Values

‘a’ False 1.0 1 2 ‘h’

1.5 True ‘5’ 700 3.14

References : [B2] Ch.2, [B3] Ch.2 Types

“Bool” type “Int” type “Float” type “Char” type

1 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 class

Bool Int Float Char

1 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. Introduction

Polymorphic types Proper types

Bool Int Float Char

1 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

a

Polymorphic types

Parametric polymorphism

...

Bool Int Float Char

1 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 classes

Num a => a

Polymorphic types

limited with the type class

...

Bool Int Float Char

1 1.0 ‘a’ Proper False 2 1.5 ‘h’ ... types True 700 3.14 ‘z’ ......

Num type class

References : [B1] Ch.2, [B2] Ch.2, [B3] Ch.3, [B4] Ch.6, [D1] Week 4 Polymorphic types

a

Polymorphic types

Num a => a polymorphic types with type class

Int Proper types

References : [B1] Ch.2, [B2] Ch.2, [B3] Ch.3, [B4] Ch.6, [D1] Week 2, 4 1. Introduction

Type constructors Type constructors

nullary type constructor

Int

References : [B1] Ch.7 Type constructors

unary type constructor

Maybe 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 Float

Float 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. Introduction

Polymorphic types and type constructors Polymorphic types and type constructors

a

Polymorphic types ?

Int Maybe Int

Type constructors

References : [B1] Ch.2, 7, [B2] Ch.2, [B3] Ch.3, [B4] Ch.6 Polymorphic types and type constructors

a

Polymorphic types t a

Maybe a

Int Maybe Int

Type constructors

References : [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 types

References : [B1] Ch.2, 7, [B2] Ch.2, [B3] Ch.3, [B4] Ch.6 Polymorphic types and type constructors

Maybe a

Polymorphic types

...

Maybe Int Maybe Char Maybe Float Maybe Bool

Proper 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 Char

Int -> 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 classes

Function types Function type

Int fun :: Int -> Bool Bool

fun

The “->” 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 Float

Int fun :: Int -> Float -> Char -> Bool Float Bool fun

Char

References : [B2] Ch.2, [B1] Ch.5, [B3] Ch.7 Function type with same type

Int fun :: Int -> Int -> Int

fun

References : [B2] Ch.2, [B1] Ch.5, [B3] Ch.7 Function type with function as argument

Int -> Int

Int -> Int fun :: (Int -> Int) -> Int -> Bool Bool

fun

Int

References : [B2] Ch.2, [B1] Ch.5, [B3] Ch.7 Function type with function as result

Int fun :: Int -> (Int -> Int) Int -> Int

fun Int -> Int

References : [B2] Ch.2, [B1] Ch.5, [B3] Ch.7 Function type with polymorphic function

a fun :: a -> Bool Bool fun

Polymorphic types Parametric polymorphism

... Proper types

Int fun :: Int -> Bool Bool Char fun :: Char -> Bool Bool fun fun

References : [B2] Ch.2, [B1] Ch.5, [B3] Ch.7 Function type for polymorphic function with type class

a fun :: Num a => a -> Bool Bool fun

Polymorphic types limited with the type class

...... Proper types

Int fun :: Int -> Bool Bool Char fun :: Char -> Bool Bool fun fun

References : [B2] Ch.2, [B1] Ch.5, [B3] Ch.7 2. more, Types and Type classes

Type class operations A type class has the class operations

Class 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 operations

Eq 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 types

Int 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 types

Int 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

b

Foldable t => t a

foldr

a -> b -> b

a b b

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

b

Foldable t => t a

foldr

a -> b -> b

a b b

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

b

Foldable t => t a

foldr

a -> b -> b

a b b

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

b

Foldable t => t a

foldr

a -> b -> b

a b b

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

b

Foldable t => t a

foldr

a -> b -> b

a b b

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

b

Foldable t => t a

foldr

a -> b -> b

a b b

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

b

Foldable t => t a

foldr

a -> b -> b

a b b

References : [B1] Ch.12, 5, 7, [B2] Ch.6, 2, [B3] Ch.7, 3, [B4] Ch.4, 6 Example of polymorphism on foldr b

Foldable t => t a foldr

a -> b -> b Polymorphic types ...

b b

[a] Tree a foldr foldr

a -> b -> b a -> b -> b

References : [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 types

Bool Char Int Float

‘a’ 1 1.0 False ‘h’ 2 1.5

True ‘z’ 700 3.14 ......

References : [B2] Ch.2, [B3] Ch.2, [B1] Ch.2, [H1] Ch.6, [S1] Maybe type

Maybe a

Polymorphic types

...

Maybe Bool Maybe Int Maybe Float Maybe Char

Nothing 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 type

Syntactic sugar [a] [] a

Polymorphic 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 type

Either a b

Polymorphic types

...

Either Bool Int Either Char Int Either Int Float Either Float Char

Left 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) type

Syntactic sugar (a, b) (,) a b

Polymorphic 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 operations

Eq 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 operations

Ord 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 operations

Num a => a

+

-

*

abs

The Num class has arithmetic operations.

References : [B1] Ch.1, [B2] Ch.2, [B3] Ch.3, [H1] Ch.6, [S1] Foldable class‘s characteristic operations

b

Foldable t => t a

foldr

a -> b -> b

References : [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 -> b

References : [B1] Ch.7, [D3], [H1] Ch.6, [S1] Applicative class‘s characteristic operations a

pure

Applicative f => f a Applicative f => f b

<*>

Applicative f => f (a -> b)

References : [B1] Ch.11, [D3], [S1] Monad class‘s characteristic operations a

return

Monad m => m a Monad m => m b

>>=

Monad m => a -> m b

References : [B1] Ch.12, [B2] Ch.10, [D3], [H1] Ch.6, [S1] Monoid class‘s characteristic operations

Monoid a => a

mappend

mempty

References : [B1] Ch.12, [D3], [S1] Related topics: monoid laws

Monoid a => a

mappend binary operation

Monoid a => a

mempty mappend unit law

Monoid a => a mappend mappend mappend mappend associative law

same

Programmers should satisfy the monoid laws.

References : [B1] Ch.12, [D3], [S1] Traversable class‘s characteristic operations

Traversable t => t a Traversable t => f (t b)

traverse

a -> f b

References : [D3], [S1] Appendix III – Advanced topics Universally quantified types

forall a . a a

implicitly quantified with universal quantification (∀a .) Polymorphic types

...

Bool Int Float Char

1 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 ... kind kind kind Maybe :: * -> * Either :: * -> * -> * (,,) :: * -> * -> * -> * kind Int :: * Maybe Int :: * Either Int Char :: * (,,) Int Int Float :: *

Int Maybe Int Either Int Char (,,) Int Int Float

Float IO Int State Int Char

Int -> Bool [] Int (,) Int Char

References : [B1] Ch.7, [B5] Ch.29, [H1] Ch.4 Type systems

TAPL [5] Ch.23 TAPL [5] Ch.29

corresponding to System F corresponding to System Fω

a Polymorphic types t a

(Quantified types) Maybe a

Int Maybe Int

TAPL [5] Ch.30

corresponding to λω

Type constructors (Type operators)

References : [B5] Ch.23, 29, 30 References References

Books

[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-languages

Documents

[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 References

Search

[S1] Hoogle https://www.haskell.org/hoogle

Specifications

[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 References

Furthermore 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