COMP3141 Assignment 1 Tortoise Graphics solved

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In this assignment, you will implement various extensions to a simple graphics drawing
language, the Tortoise Graphics Language, embedded in Haskell. It is intended to give
you experience writing Haskell programs, including functional programming idioms, as
well as experience programming to algebraic specifications expressed as QuickCheck

These specifications will introduce you to concepts such as monoids, the
distinction between syntax and semantics, and various notions of composition.

Provided Code

The provided code consists of a number of modules:
Main.hs contains a main function to save an example image to ‘tortoise.png’. It also
includes a series of example graphics of increasing complexity. Initially, only the
first example will work.

Tests.hs QuickCheck specifications for all functions you must implement, as well as
any support code to run the tests, depending on your environment.
TestSupport.hs contains support code such as Arbitrary instances and alternative
test data generation strategies for use when you are testing this assignment, and
for when we are marking it.

Tortoise.hs contains the definitions for the syntax and semantics of the Tortoise
Graphics Language.
TortoiseGraphics.hs contains a graphical backend (using the rasterific library) for
the Tortoise Graphics Language, to actually visualise your graphics.

TortoiseCombinators.hs contains stubs for the additional functions you are required
to implement.
Note: The only file you can submit is ‘TortoiseCombinators.hs’, so make sure your
submission compiles with the original versions of all other files.

The Tortoise Graphics Language

Simon the Tortoise has decided to take up line drawing as his latest hobby. Unfortunately
for him, he lacks any artistic inspiration and is unable to dream up even the simplest
picture. He is, however, very good at understanding Haskell programs1

To help Simon
out, we’ve come up with a little language of drawing commands, and defined it as a
Haskell data type (in ‘Tortoise.hs’):
data Instructions = Move Distance Instructions
| Turn Angle Instructions
| SetStyle LineStyle Instructions
| SetColour Colour Instructions
| PenDown Instructions
| PenUp Instructions
| Stop

type LineWidth = Int
data LineStyle = Solid LineWidth | Dashed LineWidth | Dotted LineWidth
type Angle = Integer — Degrees
1Not unlike many other people named Simon.

type Distance = Integer — Pixels
type Point = (Integer, Integer) — (x, y)
data Colour = Colour { redC, greenC, blueC, alphaC :: Int }
The data type Instructions is the syntax of our Tortoise Graphics Language. With this
language, we can do all the artistic thinking, and encode our pictures as Instructions
for Simon to follow. When Simon moves, if the pen is down, he will also draw a line
from his starting point to his ending point.

We define a Picture to be a series of Lines, drawn in order. A Line consists of a line
style, a colour, a start point, and an end point2
data Line = Line LineStyle Colour Point Point
type Picture = [Line]

Thus, Simon’s job is simple: Given an initial state (consisting of a starting position,
angle, colour etc.), follow the instructions to produce a Picture and a final state:
tortoise :: Instructions -> (TortoiseState -> (Picture, TortoiseState))

This tortoise function defines the semantics, or meaning, of our Tortoise Graphics
Language. It maps syntax (Instructions) to a domain of state transformers that
return a Picture. In computer science literature, the expression tortoise i would
usually be written as JiK, but we shall stick to our Haskell-based notation.

We also
define two utility functions that give the picture (resp. final state) produced for a given
set of instructions if we start from the default start state:
tortoisePic :: Instructions -> Picture
finalState :: Instructions -> TortoiseState
With the basic Tortoise Graphics Language defined, we can now (in ‘Main.hs’) produce
syntax for a simple square:

square :: Distance -> Instructions
square s = Move s $ Turn 90
$ Move s $ Turn 90
$ Move s $ Turn 90
$ Move s $ Turn 90
$ Stop — The dollar operator allows us to avoid nested parens.
You can use the provided drawPicture function (from ‘TortoiseGraphics.hs‘) to produce an image from a Picture, and writePng to save it to disk:
main = do

writePng “tortoise.png” (drawPicture (tortoisePic (square 100)))
We have defined the basic language, but large drawings are very cumbersome to write
directly. To remedy this, you must define a set of so-called combinators, functions
that act on Instructions to let build bigger drawings out of smaller ones. Each
of these combinators has been specified in ‘Test.hs’. You must implement them in

2Note that we have defined equality on Lines to treat a line A − B and a line B − A as equal.

Sequential Composition (4 marks)

The first combinator you must implement is a way to take two sets of Instructions
and combine them into one, one after another:
andThen :: Instructions -> Instructions -> Instructions
The specification for this function is given in the form of QuickCheck properties in
‘Test.hs’. As it is a composition operator, we expect it to be associative, and have a
left and right identity with Stop:
Stop ‘andThen‘ i = i (andThen left id)

i ‘andThen‘ Stop = i (andThen right id)
i1 ‘andThen‘ (i2 ‘andThen‘ i3) = (i1 ‘andThen‘ i2) ‘andThen‘ i3 (andThen assoc)
Algebraically, the above properties mean that the triple (Instructions, andThen, Stop)
form a monoid.

Semantically, this combinator corresponds to the notion of sequential composition — that
is, doing one thing after another. We define sequential composition of state transformers
by running the first state transformer with the given state, then running the second with
the output state of the first, and concatenating their outputs3

comp :: (a -> (Picture, b)) -> (b -> (Picture, c))
-> (a -> (Picture, c))
comp f g a = let (p , b) = f a
(p’, c) = g b
in (p ++ p’, c)

Then, our correctness property for andThen can succinctly state the relationship between
syntactic composition and semantic composition:
(andThen compose)
tortoise (i1 ‘andThen‘ i2) start = comp (tortoise i1) (tortoise i2) start
Marking Criteria

Marks Description
1 Left identity property passed
1 Right identity property passed
1 Associativity property passed
1 Semantic composition property passed
4 Total
3We use a more general type than necessary for comp. This helps the type system to aid us to get
the implementation correct. In practice, all the type variables a, b and c will be instantiated to

Bounded Looping (4 marks)

Our next combinator is for bounded looping, that is, repeating the same set of instructions
a fixed number of times:
loop :: Int -> Instructions -> Instructions
The expression loop 0 i should be equivalent to Stop, as should loop n i for any negative
n. Any positive n should produce the composition of n copies of i. For example, loop 3 i
should be equivalent to i ‘andThen‘ i ‘andThen‘ i.

To define what we expect this combinator to do semantically, we replicate n times the
state transformer for i, and then use the higher-order function foldr to compose them
all together:
(loop compose)
tortoise (loop n i) start = foldr comp nop (replicate n (tortoise i)) start

Here nop is the identity state transformer, that does not change the state and returns
an empty picture (i.e. tortoise Stop). To get a better sense of how this works, it may
be instructive to examine the following equational proof of the above property for the
case where n = 3:

foldr comp nop (replicate 3 (tortoise i)) start
= foldr comp nop (tortoise i : tortoise i : tortoise i : []) start
= (tortoise i ‘comp‘ tortoise i ‘comp‘ tortoise i ‘comp‘ nop) start
= (tortoise (i ‘andThen‘ i) ‘comp‘ tortoise i ‘comp‘ nop) start
= (tortoise (i ‘andThen‘ i ‘andThen‘ i) ‘comp‘ nop) start
= tortoise (i ‘andThen‘ i ‘andThen‘ i ‘andThen‘ Stop) start
= tortoise (i ‘andThen‘ i ‘andThen‘ i) start
= tortoise (loop 3 i) start

This combinator should allow you to try some interesting pictures! For example, the
circlograph and squareograph examples in ‘Main.hs0 are classic examples of Tortoise
graphics. We encourage you to try to express yourself creatively and come up with other
pretty pictures, and share them with your friends.
Marking Criteria
Marks Description
2 Passes for positive n
1 Passes for negative n
1 Passes for zero n
4 Total

Invisibility (4 marks)

The next combinator you must implement is invisibly:
invisibly :: Instructions -> Instructions
As the name suggests, this function takes in some Instructions and produces a new
set of Instructions that has the same effect on the tortoise state when starting from
the initial state start, but does not produce any lines in the picture.
tortoise (invisibly i) start = ([], finalState i) (invisibly sems)

The PenUp and PenDown constructors govern whether or not to draw lines. While the pen
is up, no lines are drawn, and while the pen is down, the Move constructor will draw a
line as well as move the tortoise. The complication when implementing this combinator
is that the given Instructions may contain these pen-controlling constructors.

may find it helpful to start by only handling the cases with simple Move and Turn
constructors, and gradually adding more constructors while keeping tests passing.
To assist you in testing your program, you may use the provided newtypes that restrict
test data generation to some subset of constructors, MoveTurnOnly and NoPenControl.

These types have different Arbitrary instances and therefore will produce different test
data. If you wish to run tests using these subsets, simply change the property definition
from, for example:
prop_invisibly_sems i = …
prop_invisibly_sems (MoveTurnOnly i) = …

Make sure you remove these changes and test with the full set of constructors after you
have implemented it, however! You are only allowed to submit ‘TurtleCombinators.hs’,
so any changes you make to ‘Test.hs’ will not be submitted.

Hint: You may find it helpful to define a separate helper function that takes additional
arguments, and define invisibly in terms of that helper function. It is also useful to
note that the pen starts down in the initial state start. You should only need to make
one pass through the given Instructions.
Marking Criteria
Marks Description
1 Passes with just Move, Turn
1 Additionally passes with SetStyle, SetColour
2 Additionally passes with PenUp, PenDown (i.e. everything)
4 Total

Retracing Backwards (4 marks)

The next combinator you are to implement is called retrace:
retrace :: Instructions -> Instructions
If a set of instructions i goes from state start to state σ and produces picture p, then the
instructions retrace i will go from state σ to start and produce the picture reverse p
— that is, the same lines, but in reverse order.
(retrace sems)

tortoise (retrace i) (finalState i) = (reverse (tortoisePic i), start)
Like the previous combinator, you may find it helpful to define retrace with a separate
helper function that takes additional arguments. Also like before, you may wish to
start by retracing simple Instructions such as those containing just Move and Turn
constructors, before moving on to more complicated ones involving style, colour and
pen state.

You may find it useful to make use of previously-defined combinators such
as andThen in your implementation, but be warned: andThen is O(n) time complexity.
Using it here can easily make your retrace function O(n
), which is unacceptably slow
for many real images.

To remedy this, you will need to introduce an accumulator. To
illustrate this concept, here is a slow, O(n
) implementation of reverse on lists:
reverse :: [a] -> [a]
reverse [] = []
reverse (x:xs) = reverse xs ++ x

And here is a linear-time version using an accumulator parameter a:
reverse :: [a] -> [a]
reverse l = go l []
where go [] a = a
go (x:xs) a = go xs (x:a)

Note that we avoid calling the O(n) time (++) function on each list element in the
accumulator version, and thus reduce the overall complexity from quadratic to linear.
One mark is on offer here for writing a linear-time version of retrace. You will also
find some examples (such as the flowers circlograph example) that do not perform in
reasonable time given a quadratic-time retrace.

If you have a working, but slow version
of retrace already implemented, I would recomment developing the faster version guided
by an additional QuickCheck property such as:
prop_retrace_same i = slowRetrace i == retrace i
Marking Criteria
Marks Description
1 Passes with just Move, Turn
1 Additionally passes with SetStyle, SetColour
1 Additionally passes with PenUp, PenDown (i.e. everything)
1 O(n) complexity; no quadratic blowup (or worse).
4 Total

Overlaying Images (4 marks)

The final combinator you must implement is called overlay. It takes in a list of
Instructions and produces a single set of Instructions:
overlay :: [Instructions] -> Instructions

If instructions i1, i2 and i3 produce images p1, p2 and p3 respectively, then the expression
overlay [i1, i2, i3] returns instructions that produce the combined picture where p1 is
drawn, then p2, then p3, such that p3 appears “on top” of p2, which is in turn “on top”
of p1. The overlay function should ensure that the tortoise returns to the initial state
start after drawing any images. If the provided list is empty, the resultant instructions
should draw nothing.

finalState (overlay is) = start (overlay state)
tortoisePic (overlay is) = concatMap tortoisePic is (overlay pic)
Here we use the higher-order function concatMap to express what we mean about overlaid
pictures. Here is a proof sketch for the second property, for our three Instructions i1,
i2 and i3:

concatMap tortoisePic [i1, i2, i3]
= concat (map tortoisePic [i1, i2, i3])
= concat [tortoisePic i1, tortoisePic i2, tortoisePic i3])
= tortoisePic i1 ++ tortoisePic i2 ++ tortoisePic i3
= tortoisePic (overlay [i1, i2, i3])
Tip: You can use the combinators you have previously defined such as invisibly and
retrace to implement overlay very succinctly.

Once you have implemented overlay, you can now generate all of the example images
in ‘Main.hs’, including the complex circlographograph example.
Marking Criteria
Marks Description
1 Empty list works
1 State is preserved
2 Images are concatenated
4 Total

Compiling and Building

This project has a number of dependencies, specifically the rasterific graphics library,
the JuicyPixels image library, the QuickCheck testing library and the test framework
called tasty. For CSE machines, we have already a configured a package database
on the course account that should build the assignment without difficulty using the
standard Haskell build tool cabal.

For students using their own computer, we instead
recommend the alternative build tool stack, available from the Haskell Stack website
at We have provided a stack configuration file that
fixes the versions of each of our dependencies to known-working ones.

If you use stack to
set up your toolchain, you should have minimal compatibility difficulties regardless of the
platform you are using. If you are using versions of GHC or Haskell build tools supplied
by your Linux distribution, these are commonly out of date or incorrectly configured. We
cannot provide support for these distributions.

Detailed, assignment-specific instructions for each build tool are presented below.
On CSE Machines
Enter a COMP3141 subshell by typing 3141 into a CSE terminal:
$ 3141
newclass starting new subshell for class COMP3141…
From there, if you navigate to the directory containing the assignment code, you can
build the assignment by typing:
$ cabal build

To run the program from ‘Main.hs’, which saves an image tortoise.png, type:
$ ./dist/build/Tortoise/Tortoise
To run the program from ‘Tests.hs’, which contains all the QuickCheck properties,
$ ./dist/build/TortoiseTests/TortoiseTests
To start a ghci session for the Tortoise program, type:
$ cabal repl Tortoise

Similarly, cabal repl can be used with TortoiseTests. Lastly, if for whatever reason
you want to remove all build artefacts, type:
$ cabal clean
For stack users
Firstly, ensure that GHC has been setup for this project by typing, in the directory that
contains the assignment code:
$ stack setup

If stack reports that it has already set up GHC, you should be able to build the assignment with:
$ stack build
This build command will, on first run, download and build the library dependencies
as well, so be sure to have an internet connection active. To run the program from
‘Main.hs’, which saves an image tortoise.png, type:
$ stack exec Tortoise

To run the program from ‘Tests.hs’, which contains all the QuickCheck properties,
$ stack exec TortoiseTests
To start a ghci session for the Tortoise program, type:
$ stack repl Tortoise:exe:Tortoise
Similarly, stack repl can be used with TortoiseTests:
$ stack repl Tortoise:exe:TortoiseTests

Lastly, if for whatever reason you want to remove all build artefacts, type:
$ stack clean

Marking and Testing

All marks for this assignment are awarded based on automatic marking scripts, which
are comprised of several QuickCheck properties (based on, but not exactly the same
as, the QuickCheck properties given in this assignment spec).

Marks are not awarded
subjectively, and are allocated according to the criteria presented in each section.
Barring exceptional circumstances, the marks awarded by the automatic marking
script are final. For this reason, please make sure that your submission compiles and
runs correctly on CSE machines.

We will use similar machines to mark your assignment.
A dry-run script that runs the tests provided in the assignment code will be provided.
When you submit the assignment, please make sure the script does not report any

Late Submissions

Unless otherwise stated if you wish to submit an assignment late, you may do so, but
a late penalty reducing the maximum available mark applies to every late assignment.
The maximum available mark is reduced by 10% if the assignment is one day late, by
25% if it is 2 days late and by 50% if it is 3 days late.

Assignments that are late 4 days
or more will be awarded zero marks. So if your assignment is worth 88% and you submit
it one day late you still get 88%, but if you submit it two days late you get 75%, three
days late 50%, and four days late zero.


Assignment extensions are only awarded for serious and unforeseeable events. Having
the flu for a few days, deleting your assignment by mistake, going on holiday, work
commitments, etc do not qualify. Therefore aim to complete your assignments well
before the due date in case of last minute illness, and make regular backups of your


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