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Functional Programming Lecture 6 - Algebraic Data Types

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Algebraic Data Types

A powerful mechanism for definingyour own data types.

All other types in Haskell can be defined using algebraic data types.

We will start with simplest cases and work toward the most complicated.

Enumerated Types

Example:

data Colour = Red | Green | Blue

data is a keyword - defines an algebraic data type.

Colour is the type name.

Red, Green, Blue are constructors.

Red :: Colour, Green ::Colour, etc.

Red:: Colour Green :: Colour

The type name and constructors must begin

with an upper case letter.

Variable must begin with lower case letter.

Pattern Matching

You can pattern match on this enumerated type.

Example:

value :: Colour -> Int

value Red = 15

value Green = 42

value Blue = 5

In general, define equations by

function-name pattern = expression

pattern is a constructor.

data Day = Mon|Tues ..

weekend :: Day -> Bool

weekend d = (d==Sat) || (d==Sun)

Deriving

Often when you define a new data type, you also need to define a function that shows it.

(show -- convert to string, for printing)

Example

showColour :: Colour -> String

showColour Red = “Red”

showColour Blue = “Blue”

…...

showInt :: Int -> String

showInt 5 => “5”

deriving Show

data Colour = Red | Blue | Green deriving Show

deriving Show tells the compiler automatically to define

show :: Colour -> String

Example:

let c = Blue

in “The answer is” ++ c ++ “\n” wrong!

let c = Blue

in “The answer is” ++ (show c) ++ “\n”

today :: Colour -> String

today c = “the sky is “++ (show c) ++ “.”

Example: The Bool Type

data Bool = False | True deriving Show

This is a built-in declaration in Haskell.

False, True are the boolean constants -

constructors of type Bool.

Data Fields

Example:

data Thing = Item Int String deriving Show

A value of type Thing consists of a constructor Item (indicating what it is), a data field containing an Int, and another field containing a String.

u :: Int w :: String

Item u w :: Thing

Examples

x, y :: Thing

x = Item 3 “small”

y = Item (2+3) (str++”\n”)

show x => “Item 3 small”

Polymorphic Data Fields

Allow type a (where a is a type variable) instead of a specific type like Int.

Entire data type has type parameter a.

data Object a = Obj Int a

x = Obj 15 True :: Object Bool

y = Obj 27 13 :: Object Int

z = Obj 39 “Hello” :: Object String

compare :: (Eq a) => Object a -> Object a -> Bool

compare (Obj i x) (Obj j y) = (i ==j) && (x == y)

Examples

compare (Obj 27 13) (Obj 27 6) =>

compare (Obj 27 13) (Obj 27 “hello”) =>

compare1 :: Object a -> Object a -> Bool

compare1 (Obj i _) (Obj j _) = (i ==j)

Example: Tuple Types

data Pairs a b = Pair a b

data Triples a b c = Triple a b c

data Tuple4s a b c d = Tuple4 a b c d

…….

The builtin tuple types behave essentially like

this, except they have a special syntax with

(, , ,)

So,

Pair 2 3 :: Pairs Int Int

(2,3) :: (Int,Int)

Constructors and Union Types

There could be several different cases

each case has its own constructor

and can have different fields.

Example

data PhoneNumber

= LocalPhone Int

| BritishPhone Int Int

| InternationalPhone Int Int Int

Localphone 3304969 :: PhoneNumber

BritishPhone 141 3304969 :: PhoneNumber

InternationalPhone 44 141 3304969 :: PhoneNumber

Union Types

data Course = FP String String Room

| DSA String [String] [Room]

FP “Calder” “Haskell” lect2 ::Course

DSA “Miller” [“Ada”,”Java”] [lec4,lec5] ::Course

Computations that might fail

Sometimes a computation may succeed or fail

but the program should deal with failure,

instead of crashing.

Example

average :: [Double] -> Double

average xs = sum xs / fromInt (length xs)

average [4,12] => 8.0

average [ ] => divide by 0 error - crash!

Also, head []

The Maybe type

data Maybe a = Just a | Nothing

Represents result of a computation that maybe gives

you a value of type a, but could give you nothing.

Normal values of a Nothing

Just 0 Nothing

Just 1 Just 2 Just 3 Just 4

Just 5 Just 6 …..

Maybe Int

The Maybe type

average :: [Double] -> Maybe Double

average [ ] = Nothing

average xs = Just (sum xs / length xs)

head :: [a] -> Maybe a

head [] = Nothing

head (x:xs) = Just x

f xs ys = x : ys

where Just x = head xs

What is type of f?

[a] -> [a] -> [a] Why?

Recursive Types

A type T which has some data fields of type T.

Example

data T = T0 String | T1 Int T

s ::String x :: Int t :: T

TO s :: T T1 x t :: T

x, y, z :: T

x = T0 “aardvark”

y = T1 31 (T0 “cat”)

z = T1 17 (T1 86 (T0 “dog”))

Integer Lists

data List = Empty | Cons Int List

All equivalent representations!

Cons 3 (Cons 4 (Cons 1 Empty))

3:(4:(1:Empty))

[3,4,1]

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