Exercise set 4 - from PIH

Exercise 1

Define an action adder :: IO () that reads a given number of integers from the keyboard, one per line, and displays their sum. For example:

> adder
How many numbers? 5 1
3
5
7
9
The total is 25

Exercise 2

Redefine adder using the function sequence :: [IO a] -> IO [a] that performs a list of actions and returns a list of the resulting values.

Exercuse 3

Define an instance of the Functor class for the following type of binary trees that have data in their nodes:

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

Exercise 4

There may be more than one way to make a parameterised type into an applicative functor. For example, the library Control.Applicative provides an alternative ¡®zippy¡¯ instance for lists, in which the function pure makes an infinite list of copies of its argument, and the operator <*> applies each argument function to the corresponding argument value at the same position. Complete the following declarations that implement this idea:

newtype ZipList a = Z [a]
  deriving Show

instance Functor ZipList where
-- fmap :: (a -> b) -> ZipList a -> ZipList b
   fmap g (Z xs) = ...

instance Applicative ZipList where
-- pure :: a -> ZipList a
   pure x = ...
-- <*> :: ZipList (a -> b) -> ZipList a -> ZipList b
   (Z gs) <$> (Z xs) = ...

The ZipList wrapper around the list type is required because each type can only have at most one instance declaration for a given class.

Exercise 4

Define an instance of the Functor class for the following type of binary trees that have data in their nodes:

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

Exercise 5

Given the following type of expressions

data Expr a = Var a | Val Int | Add (Expr a) (Expr a)
    deriving Show

that contain variables of some type a, show how to make this type into in- stances of the Functor, Applicative and Monad classes. With the aid of an example, explain what the >>= operator for this type does.