This document contains the first hand-in assignment ("lab") for the course. It's a small one just to get the ball rolling. See the course home page for the deadline (it is soon!).
We suggest that you find a lab partner and work together in pairs. (When you are done, please submit your solution using the Fire system.)
Please note that lab assignment 2 and onwards must be submitted by groups of exactly two people.
If you leave one group to join another group in Fire, save the join code before leaving the first group, in case you need to go back to it later.
Lab assignment 1: the power function
In this lab assignment, you will implement the well-known "power" function in two different new ways. The power function takes two arguments n and k and computes nk. Your implementation only has to work for non-negative k.
You have possibly seen one implementation of this function in the lecture. If not here it is:
You will implement two more ways in this lab assignment.power :: Integer -> Integer -> Integer power n k | k < 0 = error "power: negative argument" power n 0 = 1 power n k = n * power n (k-1)
Part 1
In order to calculatepower
n k,
for a given n and k, how many computing
"steps" are being used?
Hint:
power
n0
takes 1 step.power
n1
takes 1 step, and then usespower
n0
.power
n2
takes 1 step, and then usespower
n1
.power
n3
takes 1 step, and then usespower
n2
.- And so forth.
Part 2
A different way of computing the power function is to use the standard Haskell functionproduct
, which calculates the product (multiplication) of
all elements in a list.
To calculate power
n k, first construct
a list with k elements, all being n,
and then use product
.
Implement this idea as a Haskell function power1
.
Hint:
You have to come up with a way of producing a list with k
elements, all being equal to n.
Use a list comprehension, or use the standard
Haskell function replicate
.
If you use replicate
, you might want to
use the function fromInteger
too!
Use Hoogle to find out more about
standard functions (and also to search for standard functions by their type).
Part 3
A different approach to calculating the power function uses fewer computing steps.We use the fact that, if k is even, we can calculate nk as follows:
nk = (n2)k/2 (k is even)In other words:
nk = (n * n)k/2 (k is even)So, instead of recursively using the case for k-1, we use the (much smaller) case for k/2.
If k is not even, we simply go one step down in order to arrive at an even k, just as in the original definition of power:
nk = n * (nk-1) (k is not even)
To sum up, to calculate power
n k:
- If k is even, we use (n * n)k/2
- If k is odd, we use n*(nk-1)
power2
.
Hints:
- Do not forget to add a base case (what do you do when k=0?)
- You need to find out when numbers are even or odd. Use the standard Haskell
functions
even
and/orodd
. - To divide integer numbers, use the function
div
(and not the function/
, which is used to divide floating point and rational numbers)
Part 4
We would like the three functionspower
, power1
,
and power2
to calculate the same thing.
It is probably a good idea to test this!
A. Come up with a number of test cases (inputs you will test your functions on). Argue why you have chosen these test cases. (Think about for what inputs the functions are defined, and for what inputs the functions are not defined.)
B. Implement a function prop_powers
which given n
and k checks that
power
n k,
power1
n k, and
power2
n k
all give the same answer.
C. Write all the test cases you suggested in part A as a Haskell function that performs all test cases. It is probably a good idea to use the function that you defined in part B.
Hint: You can use a list comprehension to combine all possible cases
you would like to test for n and k. Use the standard Haskell function
and
to
combine the results. If you are not familiar with list
comprehensions, you do not have to use these.
D.
Try running quickcheck on the function prop_powers
.
It will probably fail. If so, define a new version, prop_powers'
,
for which quickcheck succeeds.
The new version should of course still test the intended property!
Submission Guidelines
See the course home page for lab deadlines.Write your answers in one file, called Lab1.hs. For each part, use Haskell comments to indicate what part of the file contains the answer to that part. For answers in natural language, use English; write your answers also in Haskell comments.
Before you submit your code, Clean It Up! Remember, submitting clean code is Really Important, and simply the polite thing to do. After you feel you are done, spend some time on cleaning your code; make it simpler, remove unneccessary things, etc. We will reject your solution if it is not clean. Clean code:
- Does not have long lines (< 78 characters)
- Has a consistent layout
- Has type signatures for all top-level functions
- Has good comments
- Has no junk (junk is unused code, commented code, unneccessary comments)
- Has no overly complicated function definitions
- Does not contain any repetitive code (copy-and-paste programming)
Good Luck!