fp/test1

Test 1

Marking table

The exercises are defined so that it is hard to get a first-class mark.

Mark	Cut-off
1st	28 marks and above
upper second	24-27 marks
lower second	20-23 marks
third	16-19 marks
fail	0-15 marks

All questions have equal weight, with eight marks each, with a total of 40 marks.

Preparation

• The test must be completed on JupyterLab.

• Run git pull on JupyterLab to make sure you have the latest version of the course repository.

• We have defined some helper functions for your in the Types.hs file. You are encouraged to make use of them in your solutions.

• Do not modify either the file Types.hs or the file Test1-Template.hs.

• Copy the file Test1-Template.hs to a new file called Test1.hs and write your solutions in Test1.hs.

  Don't change the header of this file, including the module declaration, and, moreover, don't change the type signature of any of the given functions for you to complete.

  If you do make changes, then we will not be able to mark your submission and hence it will receive zero marks!

• Solve the exercises below in the file Test1.hs.

Submission procedure

• If your submission doesn't compile or fails to pass the presubmit script on JupyterLab, it will get zero marks.
• Run the presubmit script provided to you on your submission from Jupyter by running ./presubmit.sh Test1 in the terminal (in the same folder as your submission).
• This will check that your submission is in the correct format.
• If it is, submit it on Canvas.
• Otherwise fix and repeat the presubmission procedure.

Plagiarism

Plagiarism will not be tolerated. Copying and contract cheating have led to full loss of marks, and even module or degree failure, in the past.

You will need to sign a declaration on Canvas, before submission, that you understand the rules and are abiding by them, in order for your submission to qualify.

Background material

• Each question has some Background Material, an Implementation Task and possibly some Examples.
• Read this material first, then implement the requested function.
• The corresponding type appears in the file Test1-Template.hs (to be copied by you).
• Replace the default function implementation of undefined with your own function.

More Rules

• This is an open book test.
• You may consult your own notes, the course materials, any of the recommended books or Hoogle.
• Feel free to write helper functions whenever convenient.
• All the exercises may be solved without importing additional modules. Do not import any modules, as it may interfere with the marking.

Submission Deadline

• The official submission deadline is 1pm.
• If you are provided extra time by the Welfare office then your submission deadline is 1:30pm or 2:00pm.
• Submissions close 10 minutes after your deadline, and late submissions have 5% penalty

Question 1 — Even majority (8 marks)

Task Write the following function evenMajority, that takes a list of integers, and tells whether more than half of them are even.

evenMajority :: [Int] -> Bool
evenMajority ns = undefined

Question 2 — 5-smooth numbers (8 marks)

A 5-smooth number is an integer which has no prime factor larger than 5. For an integer N, we define S(N) as the set of 5-smooth numbers less than or equal to N. For example S(20) = {1,2,3,4,5,6,8,9,10,12,15,16,18,20}

Task Define the following function get5SmoothNumbers, that gives a list of all 5-smooth numbers less than or equal to a given number.

get5SmoothNumbers :: Int -> [Int]
get5SmoothNumbers k = undefined

Examples:

*Test1> get5SmoothNumbers 25
[1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25]

*Test1> get5SmoothNumbers 50
[1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,27,30,32,36,40,45,48,50]

Question 3 — Train stops (8 marks)

Consider the following type of train stops on the West Midlands Railway line, from Redditch to Birmingham New Street.

data TrainStop = BirminghamNewStreet
               | FiveWays
               | University
               | SellyOak
               | Bournville
               | KingsNorton
               | Northfield
               | Longbridge
               | BarntGreen
               | Alvechurch
               | Redditch
               deriving (Eq, Show)

We define the function theStopAfter on this type, which encodes the information of which stop comes immediately after which stop.

theStopAfter :: TrainStop -> TrainStop
theStopAfter Redditch            = Alvechurch
theStopAfter Alvechurch          = BarntGreen
theStopAfter BarntGreen          = Longbridge
theStopAfter Longbridge          = Northfield
theStopAfter Northfield          = KingsNorton
theStopAfter KingsNorton         = Bournville
theStopAfter Bournville          = SellyOak
theStopAfter SellyOak            = University
theStopAfter University          = FiveWays
theStopAfter FiveWays            = BirminghamNewStreet
theStopAfter BirminghamNewStreet = undefined

Note that the function is undefined on BirminghamNewStreet because that is the last possible stop on this line. You should ensure that this function is never called on BirminghamNewStreet, because the program will crash if you do that.

Task Define the following function comesBefore, using the given theStopAfter function, such that comesBefore s1 s2 is True if and only if s1 is a stop preceding stop s2.

comesBefore :: TrainStop -> TrainStop -> Bool
comesBefore s1 s2 = undefined

Some examples:

*Test1> comesBefore University BirminghamNewStreet
True
*Test1> comesBefore Bournville FiveWays
True
*Test1> comesBefore BirminghamNewStreet University
False
*Test1> comesBefore University KingsNorton
False
*Test1> comesBefore University University
False

Question 4 — Repeated applications of a function (8 marks)

Task Write a function countApplications

countApplications :: (a -> a) -> (a -> Bool) -> a -> Int
countApplications f p x = undefined

that takes

1. a function f :: a -> a,
2. a termination condition p :: a -> Bool, and
3. an input x :: a,

and counts the number of times that the function f must be repeatedly applied to x until the output satisfies the condition p.

Here is an example of how to use this function:

*Test1> countApplications (\n -> n `div` 2) odd 8
3
*Test1> countApplications (\n -> n `div` 2) odd 10
1

We will only test your implementation of countApplications on functions that do terminate with respect to the termination condition.

Question 5 — Higher order functions (8 marks)

Task Write a function f of the following type

f :: (a -> a -> r) -> ((a -> r) -> a) -> r
f g h = undefined

The function should terminate for all terminating inputs. Your solution should not use recursion or undefined.