Programming Languages  Homework Assignment 3


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Programming Languages
Homework Assignment 3

In this assignment, you will earn total 100 points. Here are some general instructions.
1. Read the problem descriptions and requirements carefully! There may be significant
penalties for not fulfilling the requirements.
2. Some problems ask you to explain the working of your function with the given input.
Your explanation must be consistent with your definition of the function. Your work
will be graded not only on the correctness of your answer, but also on the consistency
and clarity with which you express it.
3. This homework set is an individual homework, not a team-based effort. Discussion of
the concept is encouraged, but actual write-up of the solutions must be done individually and your final product – the code as well as the comments and explanations –
should never be shared.
4. Submit electronically exactly one file named YourLastName-YourFirstName-hw3.hs,
and nothing else, on
5. Make sure that the Haskell script (the .hs file) you submit compiles without any error
when compiled using the Glasgow Haskell Compilation System (ghc or ghci) version
7 and above1
If your program does not compile, you will receive very few points (more likely zero)
for this assignment. To avoid receiving zero for the entire assignment, if you cannot
complete defining a function correctly without compile error, you can set the function
definition undefined, see the skeleton code provided.
6. Remember to put the head comment in your file, including your name, UIN, and
acknowledgements of any help received in doing this assignment.
1Version 7 is installed in the departmental servers ( and, and
version 8 is what you will get if you install the Haskell system in your computer.
Below, the exercise problems are from the Haskell Textbook: “Programming in Haskell, 2nd
Ed.”, by Graham Hutton. Some problems are modified (with additional requirements) by
the instructor. Please read corresponding textbook chapters and the problem statements
carefully, paying attention to the requirements. There may be significant penalties for not
fulfilling such requirements.
Keep the name and type of each function exactly the same as given in the problem statement
and the skeleton code. Also, do not remove or modify the test list or the main function. If
you remove or modify those, then you will be penalized for that.
Problem 1. (5 points) Put your full name, UIN, and acknowledgements of any help received
in the head comment in your .hs file for this assignment.
Problem 2. (10 points) Chapter 8, Exercise 1. (The function definition for mult is given
in appendix A.) Carefully study the recursive type of natural numbers in Section 8.4. Using
the definitions of the recursive data type Nat, mult and add, show how 2 × 2 = 4 proceeds,
in the same way as showing how 2 + 1 = 3 proceeds given in Section 8.4 (page 97).
Problem 3. (25 points) [Make sure you read Chapter 16 before attempting this problem.]
Chapter 16. Exercise 6, page 247. This problem has two parts. Given the following data
data Tree = Leaf Int | Node Tree Tree
1. (5+5 = 10) Given a tree, function leaves counts the number of leaves in the tree, and
function nodes the number of internal nodes in the tree. Define leaves and nodes.
The function types are as follows.
leaves :: Tree – Int
nodes :: Tree – Int
2. (Base case 5 points + inductive case 10 points) Prove the following property by induction on trees.
leaves t = nodes t + 1
Problem 4. (60 points) Chapter 8, Exercise 9. Study Section 8.7 carefully before attempting this problem.
Have fun!

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