- To sharpen your skills using higher-order functions
- To learn to program in a type-directed fashion
The DSL: Purpose and Design
The DSL is you will implement is based on the one presented in a paper by Conal Elliott, titled Functional Images. There is no requirement to read this paper, though it may be of interest to you. A summary of background information and the paper follows.
Consider the problem of programming a computer to draw a particular image. One approach is to carefully lay out a series of instructions to draw pixels to the screen. While this approach works and is commonly done, it is not without its flaws. For one, the code that draws a particular image may not immediately convey what the image is. As an anaology to connecting the dots on a piece of paper, the code focuses on the connections between individual dots, as opposed to the picture formed by the collective connections between all the dots. This disconnect can make writing image-drawing code difficult, as one must focus on how to draw a particular image, as opposed to explaining what the image is.
Another issue with this approach relates to the typical software engineering concerns of code maintainability and reusability. Say we have code that draws some particular image, and we later decide that the image should be shifted over to the right. From a high-level standpoint, this sounds quite simple: just shift what is already there to the right. However, this is not so simple with respect to using a series of instructions to draw an image. Every single instruction which writes a pixel needs modification for the shift to work correctly, which could mean a significant amount of code change. In fact, this sort of transformation may be so painful with this approach that we may design the code in such a way as to abstract away this particular transformation. Such abstraction would gracefully handle the shift to the right. There is, however, still a problem: what about other sorts of transformations? How do we handle, say, a rotation? Can our abstractions handle all possible kinds of transformations, or are their certain kinds of transformations which break the model?
Fundamentally, the problem with the approach of using a series of instructions for drawing is that the model is too low-level. Ideally, we'd like to work at a higher level of abstraction, one that focuses on images as opposed to the technical details behind drawing images.
Functional image synthesis is a technique that aims to be as close to real images as possible. Relative to the approach of specifying a series of instructions for drawing, almost everything in the functional images approach is abstracted over, allowing for a better reflection of reality. For example, with actual images, we could have Cartesian coordinate systems, polar coordinate systems, 2D or 3D spaces, discrete or continuous intervals, etc. Indeed, even the idea of what exists at a particular point in a particular space can differ based on the image type, as with images with transparent portions. The functional images approach can encompass all these real-world concerns into a single uniform library.
The key observation that for this level of abstraction is the following.
Consider for a moment a 2D canvas of colored pixels addressed by integral Cartesian coordinates.
In this type of environment, it is possible to represesent any image as a function from coordinate to color (
image: (X, Y) => Color).
That is, given some coordinate, the function returns the color of the pixel at that coordinate.
This idea can be further abstracted to other contexts.
For example, let's extend this to work in a 3D coordinate plane.
In 3D, the function simply takes an additional
Z coordinate (
image: (X, Y, Z) => Color).
In fact, for any kind of coordinate, we have
image: Coordinate => Color.
This idea works even for animations, considering the time as an additional kind of coordinate (4D).
We can abstract this idea further.
Consider drawing specifically black-and-white images.
In this context, every color is simply a measure of how close to black (or equivalently white) it is.
As such, a color is most simply represented as a number in the interval
[low, high], where white and black are represented with
With this in mind, images are functions from coordinates to intervals (
image: Coordinate => Interval).
Abstracting the color viewpoint and the black-and-white viewpoint together yields that an image is a function from a coordinate to something that can be drawn or otherwise represented (
image: Coordinate => Drawable).
The observation that images are representable as generic functions affords us a great deal of flexibility, as long as we have higher-order functions at our disposal. For example, consider the problem of shifting an image to the right by some number of pixels in a 2D Cartesian space. This can very naturally be expressed like so (using Scala-like notation):
def shiftRight(byAmount: Int)(image: (Int, Int) => Drawable): (Int, Int) => Drawable = (x, y) => image(x - byAmount, y)(One may notice that this may appear invalid as now negative coordinates could be passed to
image. However, there is no reason why negative coordinates are fundamentally invalid with this technique - with the provided context, there is not even enough information to determine if the image is of finite size.) The above code returns a function that, if given a coordinate, will return whatever the original function returned at
byAmountpixels to the left. For example, if we were shifting to the right by 5 pixels, then coordinate
(5, 0)will now return whatever was originally at
5 - 5). It should be clear that this is highly modular:
shiftRightis given one image and an amount to shift by, and returns a new image based on the old without any knowledge of what the particular image is. Not only is the code short, it straightforwardly addresses what it means to shift an image to the right.
As discussed in lecture, there are a number of valid ways to look at types. For example, two common ways to look at types are:
- A way to explain to the compiler/interpreter what sort of operation should occur
- A way to statically check for certain kinds of programmatic errors
For example, say we have the following two functions available:
def foo(a: A): B = ... def bar(b: B): C = ...Say we've determined that we need a new function
baz, which takes something of type
Aand returns something of type
def baz(a: A): CGiven the functions available, an implementation which typechecks is immediately apparent:
def baz(a: A): C = bar(foo(a))Additionally, given the provided types and type information, it's immediately apparent that the following would fail to typecheck (i.e., cannot possibly be a valid implementation):
def baz(a: A): C = foo(bar(a))Note how little information we're using here: not a thing has been stated about what
bazdoes. The only information used is type information. Even so, this is sufficient to eliminate candidate programs.
With the above example, we're taking the error-checking capabilities of types one step further to actively eliminate programs which cannot possibly exist. This sort of technique can be used to help guide the implementation when only the most basic of information is available. In the real world, this can be used to eliminate the vast majority of otherwise syntactically-valid implementations. Specifically with this assignment, you'll likely find that in many contexts very few functions are applicable based on the needed input and output types. If you're stuck, types can be exploited in this way to help narrow the range of possible implementations.
What to Implement
Download the template code here.
In this file, you will find prose descriptions of a variety of functions.
Your goal is to translate these descriptions into implementations.
To help this process forward, we have provided a series of
main that utilize all the functions
that you must provide. The resulting images are here.
The vast majority of this assignment's difficulty is expected to be in writing something that compiles.
Once the types are correct, the actual implementation should be much simpler - a direct consequence of the fact that the types have heavily constrained what sort of implementations are possible.
To this end, we recommend first writing signatures for everything missing, and then using
??? as the implementation for said methods.
This way, you get to something that compiles as quickly as possible (which will substantially improve your grade), and then you can fill in the
??? bits with actual code as you go.
For this assignment, all methods must be defined in a curried fashion, and you should try to use currying where possible.
Specifically, for full credit, you must use currying to implement
diffM as curried applications of the various
lift methods (e.g.,
lift1(...) _ returns a curried
There are additional positions as well which currying should prove useful.
For more information on currying, you may wish to revisit
12_currying.scala in the tutorial.
For higher-order functions and generics, which are used extensively in this assignment, you may wish to revisit
13_generics.scala, respectively, in the tutorial.
Bonus: Additional Library Usage
It's a shame to implement a library without getting a chance to try it out. To this end, we encourage you to experiment with the library and use it to draw something interesting. Images of higher complexity will be awarded more points. As a suggestion, fractals like the Mandelbrot set form significantly complex patterns.
You may find that the library routines you have already written are insufficient or otherwise not expressive enough to draw your particular drawing. Feel free to write additional routines as you see fit. However, take care not to modify the existing routines or any of the provided examples; we will test based on our intended definitions, so any mismatches will be flagged as incorrect.
If you do attempt the bonus, please
turnin an additional
README.txt file stating (in plain English) what you're drawing and which function represents this image.
Be sure to
If you aren't attempting the bonus:
turnin assign2@cs162 fi.scala...will suffice. If you are attemping the bonus, then you should use:
turnin assign2@cs162 fi.scala README.txt...instead.