# Functional Programming Coursework 2 - Simplify

## Introduction

In this coursework assignment, you will implement a simplifier for symbolic arithmetic expressions involving addition, multiplication and integer exponentiation (xn where n is a non-negative integer).

## Downloading the skeleton code

- Download the skeleton code zip file by going to this link.
- Extract the contents of the zip to wherever you want to save it • How to extract zip files on Windows • How to extract zip files on Ubuntu • How to extract zip files on Mac

## Compiling and running tests

If your code passes all the tests, you get all the marks! It’s therefore essential that you can compile and run the tests. To do this:

- In VSCode (in WSL mode if you’re on Windows), open the folder containing the skeleton code
- Open an integrated terminal in the VSCode window, either by using the menu bar (”Terminal -> New Terminal”) or by using the shortcut (Ctrl + Shift + ‘).
- Run cabal update (this will update your package list for Haskell libraries)
- Run cabal build (this will build your project and the tests, which may take a little while the first time)
- Run cabal test (this will run the tests) When you run this on the skeleton code all the tests should fail, because nothing has been implemented yet. Every time you make a change, run cabal test to test it! You can also load your code in GHCi/interactive mode using cabal repl Write all your code in Simplify.hs, but don’t change the name or type of the functions and don’t change any other files. You are free to write any helper functions you like, as long as you write them in Simplify.hs.

## Calculating your mark

Your mark will be based on the percentage of tasks you implement correctly. This is in contrast to the first coursework, where your mark was based on the percentage of passing tests. Instead, to get the marks for a task, you must pass every test for that task. This means all tasks are worth an equal number of marks, even if they have a different number of tests.

## Submitting your coursework

Submit your work via Blackboard. You should submit only Simplify.hs. Do not rename Simplify.hs. Do not submit any other files. Do not submit a zip file. You are responsible for early submission, therefore submit at least an hour before the deadline to ensure there are no upload problems. The university systems will automatically apply quite severe penalties if the coursework is late by even a second.

## The Expr Type

Expr is a data type which represents symbolic arithmetic expressions. It is defined in Expr.hs as:

```
data Expr
= Op BinOp Expr Expr
| NumLit Int
| ExpX Int
deriving (Eq, Show)
data BinOp = AddOp | MulOp
deriving (Eq, Show)
```

It can represent:

- Integers, e.g. – NumLit 3 represents 3
- Addition, e.g. – Op AddOp (NumLit 2) (NumLit 3) represents 2 + 3
- Multiplication, e.g. – Op MulOp (NumLit 2) (NumLit 3) represents 2 ∗ 3
- Non-negative integer powers of a single variable x, e.g. – ExpX 2 represents x2 – ExpX 9 represents x9 – ExpX 0 represents x0 – ExpX (−2) does not cause a type error, but we consider it to be invalid because it contains a negative power. The tests will not generate any negative powers in an Expr and your functions should not introduce any.
- Combinations of these terms, e.g. – Op MulOp (NumLit 1) (Op AddOp (ExpX 4) (NumLit 3)) represents 1 ∗ (x4 + 3) Expr.hs also exports two functions.

### eval

` `eval :: Int → Expr → Int``

eval evaluates an expression using the given a value for x, e.g.
`eval 2 (ExpX 3) ≡ 8`

### prettyExpr

``prettyExpr :: Expr → String``

prettyExpr takes an Expr and shows it in a prettier format than the default Show instance, e.g.

```
prettyExpr (NumLit 4) ≡ "4"
prettyExpr (ExpX 2) ≡ "x^2"
prettyExpr (ExpX 1) ≡ "x"
prettyExpr (Op AddOp (NumLit 1) (ExpX 2)) ≡ "1 + x^2"
```

This function isn’t needed for the tasks or the tests, but you may find it useful while writing/debugging your code.

### ‘Junk’

In the following tasks you’ll have to make sure you don’t introduce ‘junk’ into your expressions. Junk is defined to be: multiplication by one or zero, addition of zero, addition or multiplication of numbers, multiplication of two xn terms (e.g. x2 ∗ x3 is junk because it should be simplified to x5 ), or x0 .

## Task 1

Define a function add :: Expr → Expr → Expr, which adds 2 expressions together without introducing any ‘junk’, e.g. • adding 1 and x4 should result in 1 + x4 • adding 0 and x4 should result in x4 • adding 1 and 2 should result in 3

## Task 2

Define a function mul :: Expr → Expr → Expr, which multiplies 2 expressions together without introducing any ‘junk’, e.g. • multiplying 2 and x4 should result in 2 ∗ x4 • multiplying 0 and x4 should result in 0 • multiplying 1 and x4 should result in x4

## Task 3

Define a function addAll :: [Expr] → Expr, which adds a list of expressions together into a single expression without introducing any ‘junk’. Hint: You might want to reuse your add function.

## Task 4

Define a function mulAll :: [Expr] → Expr, which multiplies a list of expressions together into a single expression without introducing any ‘junk’. Hint: You might want to reuse your mul function.

## The Poly Type

Poly.hs exports a type Poly for representing polynomials (expressions like x3 − 3x + 7). Watch this short video for an introduction to Poly.hs. (Note: In the video Poly has a unicode Show instance, but our Poly.hs has a Show instance that’s closer to what the data actually looks like. To print a Poly as pretty unicode, use prettyPoly instead.) To represent a polynomial with the Poly data type, you need a list of the polynomial’s coefficients, most significant power first. For example, x3 + 2x − 1 corresponds to the list [1, 0, 2, −1]. The Poly module also exports a function listToPoly, which converts this list of coefficients into a Poly. Note that the Poly type is an instance of the Num typeclass. This means that all the functions that Num defines can be used on Poly. Use Hoogle to find out what these functions are. Note: If you are using Windows, it seems that printing Unicode characters does not work out of the box and you will get an exception when trying to prettyPoly a polynomial. To fix this you need to type chcp 65001 in your terminal before running the code.

## Task 5

Define the function:
`exprToPoly :: Expr → Poly`

which converts an expression into a polynomial.
Here it is important to think recursively: how can you convert a bigger expression by converting its subexpressions and then combining them in a suitable way?
Hint: Since Poly is an instance of Num, you can use + and × to add and multiply two polynomials, respectively.

## Task 6

Define the the function:
`polyToExpr :: Poly → Expr`

which converts a polynomial into an expression.
You can convert a polynomial to a list of its coefficients using the function polyToList :: Poly → [Int], which
Poly.hs exports.

## Task 7

Define a function which simplifies an expression by converting it to a polynomial and back again:
`simplify :: Expr → Expr`

Hint: Your previous functions can do the hard work here. Reuse them!