Here are some exercises designed to help you practice programming with higher-order functions and test data.

Please prepare yourself for these exercises by printing them out, and also printing out the code samples and documentation that is referred to by links.

If you do not have time to do all these exercises, don't worry. The exercises are intended to provide enough work to keep the most experienced students busy. If you do all exercises marked with an (*) you have probably understood this week's material.

Some of the exercises below involve writing QuickCheck properties. The parts of those exercie that require you to run QuickCheck are obviously not so suited for doing in the group exercises. These parts you can do yourself at home, either before the exercise session (as preparation), or after.

Good luck!

1 (*). Exercises from chapter 9-10 of The Craft of Functional Programming

• (*) Define the length function using map and sum.
• (*) What does map (+1) (map (+1) xs) do? Can you conclude anything in general about properties of map f (map g xs), where f and g are arbitrary functions?
• Give the type of, and define the function iter so that

  iter n f x = f (f (... (f x)))
  

  where f occurs n times on the right-hand side of the equation. For instance, we should have

  iter 3 f x = f (f (f x))
  

  and iter 0 f x should return x.
• What is the type and effect of the following function?

  \n -> iter n succ
  

  succ is the successor function, which increases a value by one:

  Prelude> succ 33
  34
  

• (*) How would you define the sum of the squares of the natural numbers 1 to n using map and foldr?
• How does the function

  mystery xs = foldr (++) [] (map sing xs)
    where
      sing x = [x]
  

  behave?
• (*) If id is the polymorphic identity function, defined by id x = x, explain the behavior of the expressions

  (id . f)        (f . id)        (id f)
  

  If f is of type Int -> Bool, at what instance of its most general type a -> a is id used in each case?
• Define a function composeList which composes a list of functions into a single function. You should give the type of composeList, and explain why the function has this type. What is the effect of your function on an empty list of functions?
• (*) Define the function

  flip :: (a -> b -> c) -> (b -> a -> c)
  

  which reverses the order in which its function argument takes its arguments.

  The following example shows the effect of flip:

  Prelude> flip div 3 100
  33
  

Do not forget to write suitable QuickCheck properties checking that your definitions behave as expected.

(To QuickCheck properties that take function arguments, make sure to import Text.Show.Functions!)

2 (*). List Comprehensions and Higher-Order Functions

Can you rewrite the following list comprehensions using the higher-order functions map and filter? You might need the function concat too.

1. [ x+1 | x <- xs ]
2. [ x+y | x <- xs, y <-ys ]
3. [ x+2 | x <- xs, x > 3 ]
4. [ x+3 | (x,_) <- xys ]
5. [ x+4 | (x,y) <- xys, x+y < 5 ]
6. [ x+5 | Just x <- mxs ]

Can you it the other way around? I.e. rewrite the following expressions as list comprehensions.

1. map (+3) xs
2. filter (>7) xs
3. concat (map (\x -> map (\y -> (x,y)) ys) xs)
4. filter (>3) (map (\(x,y) -> x+y) xys)

Do not forget to check your answers using QuickCheck!

3 (*). Generating Lists

Sometimes we want to generate lists of a certain length.

A. Write a generator

    listOfLength :: Integer -> Gen a -> Gen [a]

such that listOf n g generates a list of n elements, where each element is generated by g. What property would you write to test that your generator behaves as it should?

B. Now use listOf to write a generator that generates pairs of lists of the same, random, length.

C.Take a look at the standard Haskell functions zip and unzip:

zip :: [a] -> [b] -> [(a,b)]
unzip :: [(a,b)] -> ([a],[b])

Write down two properties for these; one that says that zip is the inverse of unzip, and one that unzip is the inverse of zip. Note that unzip is not always the inverse of zip, so you need a condition! Could you make use of the generator you just defined?

Hint: make a datatype

  data TwoSameLengthLists a = SameLength [a] [a]

and use your generator for part B. to make a generator for the above type. Then, make the above type an instance of Arbitrary. Finally, you can use it to write your property.

Another possibility is to use the QuickCheck function forAll, which we might see later on in the course.

4. Generating Ordered Lists

Write a generator that generates random lists of integers that are sorted (ordered). You are not allowed to make use of the function sort! (And not of the generator orderedList from the lecture either...)

Hint: First generate a list of random, positive integers. Then, generate a first, random number in the list. Then, produce a list where the differences between each consecutive pair of elements is given by the list of positive integers.

Make sure to check that your generator generates ordered lists by writing a suitable property!