5. Algebraic Datatypes in Haskell

1. Define the type list with elements of type Integer:

data IntList = ...

2. Define a function which computes the sum of elements of such a list:

isum :: IntList -> Integer

3. Define type polymorfic type List a encoding lists with elements of type a:

data List a = 

4. Define a function which converts IntList lists to List Integer lists:

to_poly_list :: IntList -> List Integer

5. Define a function which displays lists. What type will this function have?

show_list = ...

Add the tree datatype definition from the lecture:

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

6. Implement the function flatten:

flatten :: Tree a -> List a

7. Define list concatenation over type List a:

app :: (List a) -> (List a) -> (List a)

8. Define the function tmap which is the Tree a correspondent to map::(a→b) → [a] → [b].

9. Define the function tzipWith:

tzipWith :: (a -> b -> c) -> (Tree a) -> (Tree b) -> (Tree c)

10. Define the function tfoldr:

tfoldr :: (a -> b -> b) -> b -> Tree a -> b

Before implementing it, consider what should tfoldr (+) 0 do, and how should this value be computed.

11. Implement the flattening function using tfoldr:

tflatten = ...

Sometimes, instead of doing pattern matching in a function definition, e.g.:

f [] = "Empty"
f (x:y:xs) = "At least two"
f _ = "Something else"

it is useful to do this in the body of the function, or in another expression. This can be done using the case instruction. Example:

f l = 
   case l of
     [] -> "Empty"
     (x:y:xs) -> "At least two"
     _ -> "Something else"

12. Consider the following definition of natural numbers extended with the value Infinity:

data Extended = Infinity | Value Integer

Define a function which computes the sum of two Extended values:

extSum :: Extended -> Extended -> Extended
extSum v v' =

(Question: can you implement it with a single case expression?)

13. Define a function which computes the equality of two Extended values:

equal :: Extended -> Extended -> Bool

The polymorphic datatype:

data Maybe a = Nothing | Just a

which is part of the Haskell default library is useful for error handling. For instance, if a computation fails, a function may return Nothing. If the computation returns a result x of type a, then the function will return a value Just x.

14. Implement the function lhead which behaves like head but returns Nothing if the argument of the function is the empty list:

lhead :: List a -> Maybe a

15. Implement the function ltail