## Concept:

# Type class

## Headline

An abstraction mechanism for bounded polymorphism

## Illustration

Type classes are not to be confused with OO classes. In fact, type classes may be somewhat compared with OO interfaces. Type classes have been popularized by Haskell. Similar constructs exist in a few other languages. Type classes capture operations that may be defined for many types. The operations can be defined differently for each type, i.e., for each instance of a type class.

All subsequent illustrations leverage Haskell. Let us consider the following datatypes of bits and bitstreams which represent unsigned binary numbers. We are going to enrich these datatypes with some functionality eventually, with the help of type classes:

```
-- A bit can be zero or one
data Bit = Zero | One
-- Bit streams of any length
newtype Bits = Bits { getBits :: [Bit] }
```

Thus, the binary number "101" would be represented as follows:

```
Bits [One,Zero,One]
```

Now suppose that we want to define some standard operations for bits and bitstreams: equality, total order, unparsing to text, parsing from text, and possibly others. Let us begin with unparsing (conversion) to text. To this end, we should implement Haskell's type-class-polymorphic function *show* so that it produces text like this:

```
> show (Bits [One,Zero,One])
"101"
```

Here is the type class *Show* which declares indeed the polymorphic *show* function:

```
class Show a
where
show :: a -> String
```

In reality, the type class has not just one member, *show*, as shown, but we omit the discussion of the other members here for brevity. The type class is parameterized in a type *a* for the actual type for which to implement the members. Here are the type-class instances for bits and bit streams:

```
-- Show bits
instance Show Bit
where
show Zero = "0"
show One = "1"
-- Show bit streams
instance Show Bits
where
show = concat . map show . getBits
```

Thus, the instance fills the position of the type parameter with an actual type such as *Bit* and *Bits*. Also, the member function *show* is actually defined, while assuming the specific type. We show a bit as either "0" or "1". We show a bit stream by showing all the individual bits and concatenating the results.

The inverse of *show* is *read*. There is also a corresponding type class *Read*, which we skip here for brevity. Let us consider equality instead. There is again a type class which captures the potential of equality for many types:

```
class Eq a
where
(==) :: a -> a -> Bool
```

The member "(==)" is the infix operation for testing two bit streams to be equal. Arguably, bit streams are equal, if they are of the same length and they agree on each other bit by bit. In fact, the following definition is a bit more general in that it also trims away preceding zero bits:

```
-- Test bits for equality
instance Eq Bit
where
Zero == Zero = True
Zero == One = False
One == One = True
One == Zero = False
-- Test bit streams for equality
instance Eq Bits
where
x == y = length x' == length y'
&& and (map (uncurry (==)) (zip x' y'))
where
x' = trim (getBits x)
y' = trim (getBits y)
trim [] = []
trim z@(One: ) = z
trim (Zero:z) = trim z
```

For instance:

```
-- Test bit streams for equality
> let b101 = read "101" :: Bits
> let b0101 = read "0101" :: Bits
> let b1101 = read "1101" :: Bits
> b101 == b0101
True
> b101 == b1101
False
```

Actually, bit streams are (unsigned) binary numbers. Thus, we should also instantiate the corresponding type classes for number types. Operations on number types are grouped in multiple type classes. The type class *Num* deals with addition, subtraction, multiplication, and a few other operations, but notably no division:

```
class (Eq a, Show a) => Num a
where
(+) :: a -> a -> a
(*) :: a -> a -> a
(-) :: a -> a -> a
negate :: a -> a
abs :: a -> a
signum :: a -> a
fromInteger :: Integer -> a
```

We would like to instantiate the *Num* type class for bit streams. There are different ways of doing this. For instance, we could define addition by bitwise addition, right at the level of bit streams, or we could instead resort to existing number types. For simplicity, we do indeed conversions from and to *Integer*, in fact, any *integral* type:

```
-- Convert bits to an integer
bits2integral :: Integral a => Bits -> a
bits2integral = foldl f 0 . getBits
where
f a b = a * 2 + (bit2int b)
bit2int Zero = 0
bit2int One = 1
-- Convert a (non-negative) integral to bits
integral2bits :: Integral a => a -> Bits
integral2bits i | i < 0 = error "Bits are unsigned"
integral2bits i = Bits (f [] i)
where
f xs 0 = xs
f xs i = f (x:xs) (i `div` 2)
where
x = if odd i then One else Zero
```

On these grounds, we can trivially instantiate the *Num* type class for *Bits* by simply reusing the existing instance for Integer through systematic conversions.

```
-- Bits as a Num type
instance Num Bits
where
x + y = integral2bits z'
where
x' = bits2integral x
y' = bits2integral y
z' = x' + y'
x * y = integral2bits z'
where
x' = bits2integral x
y' = bits2integral y
z' = x' * y'
x - y = integral2bits z'
where
x' = bits2integral x
y' = bits2integral y
z' = x' - y'
abs = id
signum = integral2bits
. signum
. bits2integral
fromInteger = integral2bits
```

The examples given so far are concerned with predefined type classes. However, type classes can also be declared by programmers in their projects. Let's assume that we may need to convert data from different formats into ``Int*s. Here is a corresponding type class with a few instances:*

```
class ToInt a
where
toInt :: a -> Maybe Int
instance ToInt Int
where
toInt = Just
instance ToInt Float
where
toInt = Just . round
instance ToInt String
where
toInt s =
case reads s of
[(i, "")] -> Just i
_ -> Nothing
```

The conversion can be illustrated like this:

```
*Main> toInt "5"
Just 5
*Main> toInt "foo"
Nothing
*Main> toInt (5::Int)
Just 5
*Main> toInt (5.5::Float)
Just 6
```

In Haskell, type-class parameters are not limited to types, but, in fact, type classes may be parameterized in type constructors. Consider the following type class which models different notions of size for container types:

```
-- Notions of size for container types
class Size f
where
-- Number of constructors
consSize :: f a -> Int
-- Number of elements
elemSize :: f a -> Int
```

Here is a straightforward instance for lists:

```
instance Size []
where
consSize = (+1) . length
elemSize = length
```

Let's also consider sizes for rose trees:

```
-- Node-labeled rose trees
data NLTree a = NLTree a [NLTree a]
deriving (Eq, Show, Read)
instance Size NLTree
where
consSize (NLTree _ ts) =
1
+ consSize ts
+ sum (map consSize ts)
elemSize (NLTree _ ts) =
1
+ sum (map elemSize ts)
-- Leaf-labeled rose trees
data LLTree a = Leaf a | Fork [LLTree a]
deriving (Eq, Show, Read)
instance Size LLTree
where
consSize (Leaf _) = 1
consSize (Fork ts) =
consSize ts
+ sum (map consSize ts)
elemSize (Leaf _) = 1
elemSize (Fork ts) =
sum (map elemSize ts)
```

A few illustrations are due:

```
*Main> let list = [1,2,3]
*Main> let nltree = NLTree 1 [NLTree 2 [], NLTree 3 []]
*Main> let lltree = Fork [Leaf 1, Fork [Leaf 2, Leaf 3]]
*Main> consSize list
4
*Main> elemSize list
3
*Main> consSize nltree
8
*Main> elemSize nltree
3
*Main> consSize lltree
9
*Main> elemSize lltree
3
```

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