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Exercises for Week 2: Recursion and Datatypes

Here are some exercises designed to help you practice writing and reasoning about recursive definitions and datatypes.

Other helpful exercises can be found in Chapters 3 and 4 of the book.

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.

Good luck!

1. (*) The Maximum Function

Complete the following function definition:

        -- maxi x y returns the maximum of x and y

(We call this function maxi, because there is a standard function max which does the same thing. Of course, you should not use it.)

Begin by writing a type signature for maxi, and a left hand side "equal to undefined". Before you write the definition, think of at least one property that you will use for testing your code.

You will need to consider two cases: what are they? Write the left hand sides, and make sure Hugs accepts them.

Complete your definition and test it with QuickCheck.

2. Sum of squares.

Define a function which computes the sum of the squares of the numbers from 1 to n.

        -- sumsq n returns 1*1 + 2*2 + ... + n*n

Hint: use recursion to compute sumsq n from sumsq (n-1); do not use a formula (such as the one below) to compute this without recursion.

Some say that sumsq n is equal to n (n+1) (2n + 1) / 6. Use QuickCheck to investigate whether this holds.

3. (*) The Towers of Hanoi

The Towers of Hanoi is an ancient puzzle, consisting of a collection of rings of different sizes, and three posts mounted on a base. At the beginning all the rings are on the left-most post as shown, and the goal is to move them all to the rightmost post, by moving one ring at a time from one post to another. But, at no time may a larger ring be placed on top of a smaller one!

Can you find a strategy for solving the puzzle based on recursion? That is, if you already know how to move n-1 rings from one post to another, can you find a way to move n rings?

If you try out your strategy, you will quickly discover that quite a large number of moves are needed to solve the puzzle with, say, five rings. Can you define a Haskell function

hanoi n

which computes the number of moves needed to move n rings from one post to another using your strategy? How many moves would be needed to solve the puzzle with ten rings? Legend has it that the original version of the puzzle has 32 rings, and is being solved at this very moment by Bhuddist monks in a monastery. When the puzzle is complete, the world will end.

More difficult (only do this if you want a challenge): Now suppose we add a fourth post to the puzzle. Can you think of a strategy which makes use of the fourth post? Define a Haskell function to compute the number of moves your new strategy needs to solve the puzzle. How many moves are now required to solve the puzzle with ten rings? How much closer would the end of the world be if the monks added a fourth post to their puzzle?

Can you explain this behaviour? Hint: what kind of recursion do your two strategies use?

4. Fibonacci Numbers

The Fibonacci numbers are defined by

so the sequence of values begins 1, 1, 2, 3, 5, 8...

Write a recursive function

    -- fib n computes the nth Fibonacci number

based on the mathematical definition above. Use it to compute the 10th, 15th, 20th, 25th, and 30th Fibonacci number. What do you notice? Can you explain this behaviour?

More Difficult: There is a faster way to compute Fibonacci numbers. Suppose we define a function fibAux which satisfies the property

    fibAux i (fib n) (fib (n+1)) == fib (n+i)

Notice that this is not a definition, it is a property!

From this property can you see how we might redefine fib by using fibAux? (Hint: try setting n to 0 in the property).

It is possible to derive a recursive definition of fibAux from this property just using algebra: the two cases we want are

    fibAux 0 a b = ...

    fibAux i a b | i>0 = ...

See if you can use the property to figure out what the right hand sides should be.

Use QuickCheck to test whether the new definition of fib (in terms of fibAux) satisfies the property above.

Use the new version of fib to compute the 10th, 15th, 20th, 25th, and 30th Fibonacci number. What do you notice?

Calculate fibAux 4 1 1 by hand. If you have programmed before, then observe the way the values of i, a, and b change in successive recursive calls. Does it remind you of anything?

5. Factors.

A prime number p has only two factors, 1 and p itself. A composite number has more than two factors. Define a function

        smallestFactor n

which returns the smallest factor of n larger than one. For example,

        smallestFactor 14 == 2
        smallestFactor 15 == 3

Before you program smallestFactor, write at least two QuickCheck properties that it should satisfy. You will need functions for integer division and remainder: investigate the standard functions div and mod for this purpose.

Hint: write smallestFactor using an auxiliary function nextFactor k n which returns the smallest factor of n larger than k. You can define smallestFactor using nextFactor, and nextFactor by recursion. Write QuickCheck properties of nextFactor before defining it also.

Now define

numFactors n

which computes the number of factors of n in the range 1..n.

 

For the following exercises, you can either use built-in Haskell lists, or (if you want to) the type List we used in the lecture. These types are basically the same, only the syntax is slightly different. For more about the syntax of built-in Haskell lists, read the beginning of chapter 7 of the book.

6. (*) Multiplying list elements

Define a function

    multiply :: Num a => [a] -> a

which multiplies together all the elements of a list. (Think: what should its value be for the empty list?). For example

Main> multiply [1,2,3,4,5]
120

(This is actually a standard function, called product).

You may do this for the List datatype we saw in the lecture:

data List a = Empty | Add a (List a)
So, your function has the type:
multiply :: List Integer -> Integer

7. Avoiding Duplicates

In many situations, lists should not contain duplicate elements. For example, a pack of cards should not contain the same card twice. Define a function

    duplicates :: Eq a => [a] -> Bool

which returns True if its argument contains duplicate elements.

Main> duplicates [1,2,3,4,5]
False
Main> duplicates [1,2,3,2]
True

Hint: the standard function elem, which tests whether an element occurs in a list, is helpful here.

One way to ensure a list contains no duplicates is to start with a list that might contain duplicate elements, and remove them. Define a function

    removeDuplicates :: Eq a => [a] -> [a]

which returns a list containing the same elements as its argument, but without duplicates. Test it using the following property:

    prop_duplicatesRemoved :: [Integer] -> Bool
    prop_duplicatesRemoved xs = not (duplicates (removeDuplicates xs))

Does this property guarantee that removeDuplicates behaves correctly? If not, what is missing?

(removeDuplicates is actually a standard function, called nub).

8. Testing

Take the definition of rankBeats from the lecture, and sabotage it by changing one of the False results to True, or vice versa. Thorough testing of rankBeats ought to reveal the error. Your task is to define one or more properties of rankBeats, which are True for the correct definition, but which enable you to reliably detect the error in the sabotaged definition using QuickCheck.

(Note: Because QuickCheck chooses test data at random, it is always possible that no error is found among the first 100 test cases. If you expect an error, and none is found, just keep testing the same property. If it really is False, then QuickCheck should eventually find a test case that fails).

9. (*) Defining Types

Define a data type Month to represent months, and a function

    daysInMonth :: Month -> Integer -> Integer

which computes the number of days in a month, given also the year. (You can ignore leap centuries and the like: just assume that every fourth year is a leap year).

Define a data type Date, containing a year, month, and day, and a function

    validDate :: Date ->Bool

that returns True if the day in the date lies between 1 and the number of days in the month.