Introduction to Functional Programming – Exercises Week 6: "Recursive Data Types" | TDA555 / DIT440, LP1 2016 |

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Introduction to Functional Programming – Exercises Week 6: "Recursive Data Types" | TDA555 / DIT440, LP1 2016 |

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Here are some exercises designed to help you practice programming with data
structures and higher-order functions.
## 1 (*). Exercises from The Craft of Functional Programming

**Expressions**
## 2 (*). Exercises on type Expr from the Lecture

(This exercise consists of writing QuickCheck properties, and checking them
using QuickCheck. Define the properties in the groups exercises, but you can
leave the QuickChecking of them until a later time, if you do not have access to
a computer in your group room.)
## 3 (*). File Systems

A *file* either contains data or is a directory. A *directory*
contains other files (which may themselves be directories) along
with a name for each one.
## 4. Exercises on Propositional Logic

A *proposition* is a boolean formula of one of the following
forms:
## 5. Approximating 0-solutions of functions

Define a function:

Please prepare yourself for these exercises by printing them out, and also printing out the code samples and documentation that is referred to by links.

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!

Take a look at the following datatype:

data Expr = Num Int | Add Expr Expr | Mul Expr Expr

A. Define a function size :: Expr -> Int that counts the number of operators in an expression.

B. Add Sub (subtraction) and Div (integer division) to the datatype Expr and modify the functions eval, show, and size accordingly. What do you in eval when you divide by 0?

C. Implement one version of **eval** with the result
type Maybe Int. (You return Nothing when division by 0 occurs somewhere in the expression.)

**Integer Trees**

Take a look at the following datatype:

data Tree = Leaf Int | Split Tree Tree

D. Implement a function collapse :: Tree -> [Int] that returns all integers in a given tree as a list, in the same order as they occurred in the tree.

E. Implement a function mirror :: Tree -> Tree that mirrors a tree (i.e. swaps the left and right subtrees everywhere).

F. State a QuickCheck property that expresses what happens when a tree is mirrored twice.

G. State a QuickCheck property that expresses the relationship between mirror, collapse and reverse.

(In order to be able to test your properties, you have to
make **Tree** an instance of Arbitrary.)

Take a look at the datatype **Expr** for expressions with variables
from the lecture: ExprVar.hs.

In the lecture, there was a function for differentiating expressions, called
**diff** (Swedish: "derivat").

diff :: Expr -> Name -> Expr diff (Num n) x = Num 0 diff (Add a b) x = Add (diff a x) (diff b x) diff (Mul a b) x = Add (Mul a (diff b x)) (Mul b (diff a x)) diff (Var y) x | x == y = Num 1 | otherwise = Num 0

**A.** Define a property that checks that, for each expression e, the derivative of e
to x does not contain more variables than e.

Does the opposite hold also?

The result of the **diff** function often contains expressions that can be
enormously simplified.

**B.**(*) Define a function **simplify** that, given an expression e,
creates an expression that is equivalent to e, but simplified. Examples of
simplifications you could do are:

We have made a small start for you in the file ExprVar.hs

**Hint:** This exercise is more open-ended.

You can try to define **simplify** recursively. You will however
notice that it is difficult to accomplish many simplifications.

(A different, but more ambitious,
approach is to design a new type, that represents a *normal form* for
expressions, for example polynomials. Your **simplify** function could simply
transform an expression into a polynomial, and back into expressions again. How
would you model polynomials over multiple variables as a type in Haskell?)

Make sure that the following property holds for your **simplify
**
function:
For each expression e, evaluating it in an environment generates the
same result before and after simplifying:

prop_SimplifyCorrect e (Env env) = eval env e == eval env (simplify e)

**C.** Define a property:

prop_SimplifyNoJunk :: Expr -> Boolthat checks that the result of simplification does not "leave any junk". In other words, the result of simplification should for example not have subexpressions of the form: And so forth. What is allowed as "junk" and what is not of course depends on your simplification function, and what you expect from it.

This property boils down to defining a function **noJunk:: Expr -> Bool**. We
have already made a small start for you in ExprVar.hs.

**D.** Define a property:

prop_SimplifyDiff :: Expr -> Boolthat checks that differentiating an expression and then simplifying it should have the same result as simplifying it first, then deriving, and then simplifying.

Do you expect it to hold? Does it actually hold for your simplification function?

**A.** Design a data type to represent the
contents of a directory. Ignore the contents of files: you are
just trying to represent file names and the way they are
organised into directories here.

**B.** Define a function to search for a given file name in a
directory. You should return a path leading to a file with the
given name. Thus if your directory contains a, b, and c, and b is
a directory containing x and y, then searching for x should
produce b/x.

- a variable name (a string)
- p & q
*(and)* - p | q
*(or)* - ~p
*(not)*

where p and q are propositions. For example, **p | ~p** is a proposition.

**A.** Design a data type Proposition to
represent propositions.

**B.** Define a function

vars :: Proposition -> [String]

which returns a list of the variables in a proposition. Make sure each variable appears only once in the list you return.

Suppose you are given a list of variable names and their values, of type Bool, for example, [("p",True),("q",False)]. Define a function

truthValue :: Proposition -> [(String,Bool)] -> Bool

which determines whether the proposition is true when the variables have the values given.

**C.** Define a function

tautology :: Proposition -> Bool

which returns true if the proposition holds for all values of the variables appearing in it.

Congratulations! You have implemented a simple theorem prover.

solve0 :: (Double -> Double) -> (Double,Double) -> Double

The idea is that **solve f (x0,x1)** finds a value x in between x0 and x1
such that f x == 0.0.

Examples:

If needed, you may assume the following reasonable assumptions: So, you may produce an error when this does not hold:Main>solve0 (+1) (-3,3) -1.0Main>solve0 cos (2,5) 4.71238898031879Main>solve0 (\x -> x ^ 2 - 10) (1,10) 3.16227766017255Main>solve0 (\x -> x ^ 3) (1,1) Program error: no solution!

Your program has to approximate this value. For example, the answer to the first example above might be -0.999999999941792 instead of -1.0, and that is OK.Main>solve0 (\x -> x ^ 2) (1,10) Program error: bad interval!

One way to do this is to use the "halving" method: Calculate x that lies somewhere in between x0 and x1. Investigate f x. Then recursively solve f on either the interval (x0,x) or (x,x1).

There are other methods too!