Exercises for Week 3: Lists and List Comprehensions
Here are some exercises designed to help you practice programming with lists and list comprehensions.
You may need the following useful standard functions:
or :: [Bool] -> Bool
Trueif any element of its argument list is
and :: [Bool] -> Bool
Trueif every element of its argument list is
nub :: Eq a => [a] -> [a]
- which removes duplicate elements from a list.
nubis defined in the standard library module
Data.List: you must write
at the beginning of any Haskell program that uses it.import Data.List
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.
0 (*). Defining Functions over Lists(Based on Thompson's book, Chapter 7)
A. The prelude defines a function
take which is
used to take a given number of elements from a list. For example,
A possible implementation oftake 5 "Programming in Haskell is fun!" = "Progr"
Use this definition oftake :: Int -> [a] -> [a] take n _ | n <= 0 =  take _  =  take n (x:xs) = x : take (n-1) xs
takeas a guide to implement the prelude functions
B. How would you define a function
zip3 which zips
together three lists? Try to write a recursive definition and also
one which uses
zip instead; what are the advantages and
disadvantages of the two definitions?
A permutation of a list is another list with the same elements, but in a possibly different order. For example, [1,2,1] is a permutation of [2,1,1], but not of [1,2,2]. Write a function
that returnsisPermutation :: Eq a => [a] -> [a] -> Bool
Trueif its arguments are permutations of each other.
Express suitable properties of the
function in the context of permutations.
2 (*). Sorting
Just as keeping a room tidy can make it easier to find things, so keeping information organised inside a computer can make tasks easier to accomplish. For that reason, lists are often kept sorted, with the least element first, and the greatest element last. In this exercise, we develop functions to take an unsorted list and convert it into a sorted one.
First of all, we will need to be able to check that lists are sorted. Define a function
which returns True if its argument is a sorted list. (The extra type restrictionsorted :: Ord a => [a] -> Bool
Ord a =>is there because it only works for types that have an ordering, i.e. we can use the operations <, >, >=, etc.)
sorted [1,2,3,2,4]> False
The sorting method we will use is called insertion sort — imagine sorting a collection of magazines by working through an unsorted pile and building a sorted pile, taking each magazine in turn from the unsorted pile, and inserting it into the right place in the sorted one. Clearly, when there are no magazines left in the unsorted pile, then we will have a pile containing all the magazines in the correct order. Each pile will be represented in our program by a list.
The first step is to define a function which inserts an element into a sorted list, in the correct position so that the result is still sorted. We call it
(yes,insert' :: Ord a => a -> [a] -> [a]
insertis a standard function too) and for example,
insert' 2 [1,3,4][1,2,3,4]
insert' and test it, by QuickChecking the following property:
prop_insert :: Integer -> [Integer] -> Property prop_insert x xs = sorted xs ==> sorted (insert' x xs)
Now use insert' to define
isort :: Ord a => [a] -> [a]
which sorts any list into order, using the insertion sort method. For example,
Write and test a QuickCheck property of isort that only holds if isort is a correct sorting function. (In particular, make sure that your property fails for this incorrect definition of isort: isort xs = ).
3. Pascal's TrianglePascal's triangle is a triangle of numbers
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 .............computed as follows:
- The first row just contains a 1.
- The following rows are computed by adding together adjacent numbers in the row above, and adding a 1 at the beginning and at the end.
Define a function
so thatpascal :: Int -> [Int]
pascaln computes the nth row of Pascal's triangle.
4. Erastosthenes' sieveEratosthenes' sieve is an ancient method for finding prime numbers. Start by writing out all the numbers from 2 to (say) 100. The first number (2) is prime. Now cross out all multiples of 2. The first remaining number (3) is also prime. Cross out all multiples of 3. The first remaining number (5) is also prime... and so on. When no numbers remain, you have found all the prime numbers in the range you started with.
Define a function
so thatcrossOut :: Int -> [Int] -> [Int]
crossOutm ns removes all multiples of m from ns. Try to not implement
crossOutrecursively, but use a list comprehension instead!
Now define a (recursive!) function
which applies Eratosthenes' sieve to the list of numbers it is given, and returns a list of all the prime numbers that it found. This is a recursive function with a list as its argument, so you must see to it that the list gets smaller in each recursive call. Take an empty argument list as your base case.sieve :: [Int] -> [Int]
Use sieve to construct the list of primes from 2 to 100.
5. Number GamesIn these examples we'll investigate the properties of prime numbers in the range 2 to 100. Define functions
- To test whether n is a prime (in the range 2 to 100).
- To test whether n is a sum of two primes (in the range 2 to 100).
6 (*). Occurrences in ListsDefine the following functions, and state their (polymorphic) types:
occursIn x xs, which returns
xis an element of
allOccurIn xs ys, which returns
Trueif all of the elements of
xsare also elements of
sameElements xs ys, which returns
yshave exactly the same elements.
numOccurrences x xs, which returns the number of times
In some ways, lists are like sets: both are collections of elements. But the order of elements in a list matters, while it does not matter in a set, and the number of occurrences in a list matters, while it does not matter in a set.
The concept of a bag is something between a list and a set: the number of occurrences matters, but the order of elements does not. One way to represent a bag is a list of pairs of values and the number of times the value occurs: for example
Define a function[("a",1), ("b",2)]
bagto convert a list into a bag. For example,
should bebag "hello"
7 Elements and PositionsElements which occur in lists do so at a particular position. For example, 'l' occurs in "hello" at positions 3 and 4. Define functions
positions xs, which converts a list into a list of pairs of elements and their positions. Hint: Make use of the standard function
firstPosition x xs, which returns the first position at which x occurs in xs.
remove1st x xs, which removes the first occurrence of x from xs. For example,
remove1st 'l' "hello" == "helo"
remove n x xs, which removes the first n occurrences of x from xs.
8 (*). List ComprehensionsExperiment with the function
and see what it does.pairs :: [a] -> [b] -> [(a,b)] pairs xs ys = [(x,y) | x<-xs, y<-ys]
A Pythagorean triad is a triple of integers (a,b,c) such that a2 + b2 = c2. Find all Pythagorean triads with a≤b≤c≤100.