 • Core program element is an expression
• Every valid expression has a type (determined at compile-time)
• Every valid expression reduces to a value (computed at run-time)

Basic values & operators

• Int, Bool, Char, Double
• +, -, ==, /=

Execution / Function Calls

• Just substitute equals by equals

Producing Collections

• Pack data into tuples & lists

Consuming Collections

• Unpack data via pattern-matching

Next: Creating and Using New Data Types

1. type Synonyms: Naming existing types

2. data types: Creating new types

Type Synonyms

Synonyms are just names (“aliases”) for existing types

• think typedef in C

A type to represent Circle

A tuple (x, y, r) is a circle with center at (x, y) and radius r

type Circle = (Double, Double, Double)

A type to represent Cuboid

A tuple (length, depth, height) is a cuboid

type Cuboid = (Double, Double, Double) Using Type Synonyms

We can now use synonyms by creating values of the given types

circ0 :: Circle
circ0 = (0, 0, 100)  -- ^ circle at "origin" with radius 100

cub0 :: Cuboid
cub0 = (10, 20, 30)  -- ^ cuboid with length=10, depth=20, height=30

And we can write functions over synonyms too

area :: Circle -> Double
area (x, y, r) = pi * r * r

volume :: Cuboid -> Double
volume (l, d, h) = l * d * h

We should get this behavior

>>> area circ0
31415.926535897932

>>> volume cub0
6000

QUIZ

Suppose we have the definitions

type Circle = (Double, Double, Double)
type Cuboid = (Double, Double, Double)

circ0 :: Circle
circ0 = (0, 0, 100)  -- ^ circle at "origin" with radius 100

cub0 :: Cuboid
cub0 = (10, 20, 30)  -- ^ cuboid with length=10, depth=20, height=30

area :: Circle -> Double
area (x, y, r) = pi * r * r

volume :: Cuboid -> Double
volume (l, d, h) = l * d * h

What is the result of

>>> volume circ0

A. 0

B. Type error

Beware!

Type Synonyms

• Do not create new types

• Just name existing types

And hence, synonyms

• Do not prevent confusing different values

Creating New Data Types

We can avoid mixing up by creating new data types

-- | A new type `CircleT` with constructor `MkCircle`
data CircleT = MkCircle Double Double Double

-- | A new type `CuboidT` with constructor `MkCuboid`
data CuboidT = MkCuboid Double Double Double

Constructors are the only way to create values

• MkCircle creates CircleT

• MkCuboid creates CuboidT

QUIZ

Suppose we create a new type with a data definition

-- | A new type `CircleT` with constructor `MkCircle`
data CircleT = MkCircle Double Double Double

What is the type of the MkCircle constructor?

A. MkCircle :: CircleT

B. MkCircle :: Double -> CircleT

C. MkCircle :: Double -> Double -> CircleT

D. MkCircle :: Double -> Double -> Double -> CircleT

E. MkCircle :: (Double, Double, Double) -> CircleT

Constructing Data

Constructors let us build values of the new type

circ1 :: CircleT
circ1 = MkCircle 0 0 100  -- ^ circle at "origin" w/ radius 100

cub1 :: Cuboid
cub1 = MkCuboid 10 20 30  -- ^ cuboid w/ len=10, dep=20, ht=30

QUIZ

Suppose we have the definitions

data CuboidT = MkCuboid Double Double Double

type Cuboid  = (Double, Double, Double)

volume :: Cuboid -> Double
volume (l, d, h) = l * d * h

What is the result of

>>> volume (MkCuboid 10 20 30)

A. 6000

B. Type error

Deconstructing Data

Constructors let us build values of new type … but how to use those values?

How can we implement a function

volume :: Cuboid -> Double
volume c = ???

such that

>>> volume (MkCuboid 10 20 30)
6000

Deconstructing Data by Pattern Matching

Haskell lets us deconstruct data via pattern-matching

volume :: Cuboid -> Double
volume c = case c of
MkCuboid l d h -> l * d * h

case e of Ctor x y z -> e1 is read as as

IF - e evaluates to a value that matches the pattern Ctor vx vy vz

THEN - evaluate e1 after naming x := vx, y := vy, z := vz

Pattern matching on Function Inputs

Very common to do matching on function inputs

volume :: Cuboid -> Double
volume c = case c of
MkCuboid l d h -> l * d * h

area :: Circle -> Double
area a  = case a of
MkCircle x y r -> pi * r * r

So Haskell allows a nicer syntax: patterns in the arguments

volume :: Cuboid -> Double
volume (MkCuboid l d h) = l * d * h

area :: Circle -> Double
area (MkCircle x y r) = pi * r * r

Nice syntax plus the compiler saves us from mixing up values!

But … what if we need to mix up values?

Suppose I need to represent a list of shapes

• Some Circles
• Some Cuboids

What is the problem with shapes as defined below?

shapes = [circ1, cub1]

Where we have defined

circ1 :: CircleT
circ1 = MkCircle 0 0 100  -- ^ circle at "origin" with radius 100

cub1 :: Cuboid
cub1 = MkCuboid 10 20 30  -- ^ cuboid with length=10, depth=20, height=30

Solution???

QUIZ: Variant (aka Union) Types

Lets create a single type that can represent both kinds of shapes!

data Shape
= MkCircle Double Double Double   -- ^ Circle at x, y with radius r
| MkCuboid Double Double Double   -- ^ Cuboid with length, depth, height

What is the type of MkCircle 0 0 100 ?

A. Shape

B. Circle

C. (Double, Double, Double)

Each Data Constructor of Shape has a different type

When we define a data type like the below

data Shape
= MkCircle  Double Double Double   -- ^ Circle at x, y with radius r
| MkCuboid  Double Double Double   -- ^ Cuboid with length, depth, height

We get multiple constructors for Shape

MkCircle :: Double -> Double -> Double -> Shape
MkCuboid :: Double -> Double -> Double -> Shape

Now we can create collections of Shape

Now we can define

circ2 :: Shape
circ2 = MkCircle 0 0 100  -- ^ circle at "origin" with radius 100

cub2 :: Shape
cub2 = MkCuboid 10 20 30  -- ^ cuboid with length=10, depth=20, height=30

and then define collections of Shapes

shapes :: [Shape]
shapes = [circ1, cub1]

EXERCISE

Lets define a type for 2D shapes

data Shape2D
= MkRect Double Double -- ^ 'MkRect w h' is a rectangle with width 'w', height 'h'
| MkCirc Double        -- ^ 'MkCirc r' is a circle with radius 'r'
| MkPoly [Vertex]      -- ^ 'MkPoly [v1,...,vn]' is a polygon with vertices at 'v1...vn'

type Vertex = (Double, Double)

Write a function to compute the area of a Shape2D

area2D :: Shape2D -> Double
area2D s = ???

HINT

You may want to use this helper that computes the area of a triangle at v1, v2, v3

areaTriangle :: Vertex -> Vertex -> Vertex -> Double
areaTriangle v1 v2 v3 = sqrt (s * (s - s1) * (s - s2) * (s - s3))
where
s  = (s1 + s2 + s3) / 2
s1 = distance v1 v2
s2 = distance v2 v3
s3 = distance v3 v1

distance :: Vertex -> Vertex -> Double
distance (x1, y1) (x2, y2) = sqrt ((x2 - x1) ** 2 + (y2 - y1) ** 2)

Polymorphic Data Structures

Next, lets see polymorphic data types

which contain many kinds of values.

Recap: Data Types

Recall that Haskell allows you to create brand new data types

data Shape
= MkRect  Double Double
| MkPoly [(Double, Double)]

QUIZ

What is the type of MkRect ?

data Shape
= MkRect  Double Double
| MkPoly [(Double, Double)]

a. Shape

b. Double

c. Double -> Double -> Shape

d. (Double, Double) -> Shape

e. [(Double, Double)] -> Shape

Tagged Boxes

Values of this type are either two doubles tagged with Rectangle

>>> :type (Rectangle 4.5 1.2)
(Rectangle 4.5 1.2) :: Shape

or a list of pairs of Double values tagged with Polygon

ghci> :type (Polygon [(1, 1), (2, 2), (3, 3)])
(Polygon [(1, 1), (2, 2), (3, 3)]) :: Shape

Recursive Data Types

We can define datatypes recursively too

data IntList
= INil                -- ^ empty list
| ICons Int IntList   -- ^ list with "hd" Int and "tl" IntList
deriving (Show)

(Ignore the bit about deriving for now.)

QUIZ

data IntList
= INil                -- ^ empty list
| ICons Int IntList   -- ^ list with "hd" Int and "tl" IntList
deriving (Show)

What is the type of ICons ?

A. Int -> IntList -> List

B. IntList

C. Int -> IntList -> IntList

D. Int -> List -> IntList

E. IntList -> IntList

Constructing IntList

Can only build IntList via constructors.

>>> :type INil
INil:: IntList

>>> :type ICons
ICons :: Int -> IntList -> IntList

EXERCISE

Write down a representation of type IntList of the list of three numbers 1, 2 and 3.

list_1_2_3 :: IntList
list_1_2_3 = ???

Hint Recursion means boxes within boxes

Trees: Multiple Recursive Occurrences

We can represent Int trees like

data IntTree
= ILeaf Int              -- ^ single "leaf" w/ an Int
| INode IntTree IntTree  -- ^ internal "node" w/ 2 sub-trees
deriving (Show)

A leaf is a box containing an Int tagged ILeaf e.g.

>>> it1  = ILeaf 1
>>> it2  = ILeaf 2

A node is a box containing two sub-trees tagged INode e.g.

>>> itt   = INode (ILeaf 1) (ILeaf 2)
>>> itt'  = INode itt itt
>>> INode itt' itt'
INode (INode (ILeaf 1) (ILeaf 2)) (INode (ILeaf 1) (ILeaf 2))

Multiple Branching Factors

e.g. 2-3 trees

data Int23T
= ILeaf0
| INode2 Int Int23T Int23T
| INode3 Int Int23T Int23T Int23T
deriving (Show)

An example value of type Int23T would be

i23t :: Int23T
i23t = INode3 0 t t t
where t = INode2 1 ILeaf0 ILeaf0

which looks like

Parameterized Types

We can define CharList or DoubleList - versions of IntList for Char and Double as

data CharList
= CNil
| CCons Char CharList
deriving (Show)

data DoubleList
= DNil
| DCons Char DoubleList
deriving (Show)

Don’t Repeat Yourself!

Don’t repeat definitions - Instead reuse the list structure across all types!

Find abstract data patterns by

• identifying the different parts and
• refactor those into parameters

A Refactored List

Here are the three types: What is common? What is different?

data IList = INil | ICons Int    IList

data CList = CNil | CCons Char   CList

data DList = DNil | DCons Double DList

Common: Nil/Cons structure

Different: type of each “head” element

Refactored using Type Parameter

data List a = Nil | Cons a  (List a)

Recover original types as instances of List

type IntList    = List Int
type CharList   = List Char
type DoubleList = List Double

Polymorphic Data has Polymorphic Constructors

Look at the types of the constructors

>>> :type Nil
Nil :: List a

That is, the Empty tag is a value of any kind of list, and

>>> :type Cons
Cons :: a -> List a -> List a

Cons takes an a and a List a and returns a List a.

cList :: List Char     -- list where 'a' = 'Char'
cList = Cons 'a' (Cons 'b' (Cons 'c' Nil))

iList :: List Int      -- list where 'a' = 'Int'
iList = Cons 1 (Cons 2 (Cons 3 Nil))

dList :: List Double   -- list where 'a' = 'Double'
dList = Cons 1.1 (Cons 2.2 (Cons 3.3 Nil))

Polymorphic Function over Polymorphic Data

Lets write the list length function

len :: List a -> Int
len Nil         = 0
len (Cons x xs) = 1 + len xs

len doesn’t care about the actual values in the list - only “counts” the number of Cons constructors

Hence len :: List a -> Int

• we can call len on any kind of list.
>>> len [1.1, 2.2, 3.3, 4.4]    -- a := Double
4

>>> len "mmm donuts!"           -- a := Char
11

>>> len [, [1,2], [1,2,3]]   -- a := ???
3

Built-in Lists?

This is exactly how Haskell’s “built-in” lists are defined:

data [a]    = [] | (:) a [a]

data List a = Nil | Cons a (List a)
• Nil is called []
• Cons is called :

Many list manipulating functions e.g. in [Data.List] are polymorphic - Can be reused across all kinds of lists.

(++) :: [a] -> [a] -> [a]
tail :: [a] -> [a]

Generalizing Other Data Types

Polymorphic trees

data Tree a
= Leaf a
| Node (Tree a) (Tree a)
deriving (Show)

Polymorphic 2-3 trees

data Tree23 a
= Leaf0
| Node2 (Tree23 a) (Tree23 a)
| Node3 (Tree23 a) (Tree23 a) (Tree23 a)
deriving (Show)

Kinds

List a corresponds to lists of values of type a.

If a is the type parameter, then what is List?

A type-constructor that - takes as input a type a - returns as output the type List a

But wait, if List is a type-constructor then what is its “type”?

• A kind is the “type” of a type.
>>> :kind Int
Int :: *
>>> :kind Char
Char :: *
>>> :kind Bool
Bool :: *

Thus, List is a function from any “type” to any other “type”, and so

>>> :kind List
List :: * -> *

QUIZ

What is the kind of ->? That, is what does GHCi say if we type

>>> :kind (->)

A. *

B. * -> *

C. * -> * -> *

We will not dwell too much on this now.

As you might imagine, they allow for all sorts of abstractions over data.