blog-static/content/blog/00_cs325_languages_hw1.md

title	date	draft	tags
A Language for an Assignment - Homework 1	2019-12-27T23:27:09-08:00	true	Haskell Python Algorithms

On a rainy Oregon day, I was walking between classes with a group of friends. We were discussing the various ways to obfuscate solutions to the weekly homework assignments in our Algorithms course: replace every if with a ternary expression, use single variable names, put everything on one line. I said:

The {{< sidenote "right" "chad-note" "chad" >}} This is in reference to a meme, Virgin vs Chad. A "chad" characteristic is masculine or "alpha" to the point of absurdity. {{< /sidenote >}} move would be to make your own, different language for every homework assignment.

It was required of us to use {{< sidenote "left" "python-note" "Python" >}} A friend suggested making a Haskell program that generates Python-based interpreters for languages. While that would be truly absurd, I'll leave this challenge for another day. {{< /sidenote >}} for our solutions, so that was the first limitation on this challenge. Someone suggested to write the languages in Haskell, since that's what we used in our Programming Languages class. So the final goal ended up:

• For each of the 10 homework assignments in CS325 - Analysis of Algorithms,
• Create a Haskell program that translates a language into,
• A valid Python program that works (nearly) out of the box and passes all the test cases.

It may not be worth it to create a whole {{< sidenote "right" "general-purpose-note" "general-purpose" >}} A general purpose language is one that's designed to be used in vairous domains. For instance, C++ is a general-purpose language because it can be used for embedded systems, GUI programs, and pretty much anything else. This is in contrast to a domain-specific language, such as Game Maker Language, which is aimed at a much narrower set of uses. {{< /sidenote >}} language for each problem, but nowhere in the challenge did we say that it had to be general-purpose. In fact, some interesting design thinking can go into designing a domain-specific language for a particular assignment. So let's jump right into it, and make a language for the the first homework assignment.

Homework 1

There are two problems in Homework 1. Here they are, verbatim:

{{< codelines "text" "cs325-langs/hws/hw1.txt" 32 38 >}}

And the second:

{{< codelines "text" "cs325-langs/hws/hw1.txt" 47 68 >}}

We want to make a language specifically for these two tasks (one of which is split into many tasks). What common things can we isolate? I see two:

First, all the problems deal with lists. This may seem like a trivial observation, but these two problems are the only thing we use our language for. We have list access, {{< sidenote "right" "filterting-note" "list filtering" >}} Quickselect is a variation on quicksort, which itself finds all the "lesser" and "greater" elements in the input array. {{< /sidenote >}} and list creation. That should serve as a good base!

If you squint a little bit, all the problems are recursive with the same base case. Consider the first few lines of search, implemented naively:

def search(xs, k):
    if xs == []:
        return false

How about sorted? Take a look:

def sorted(xs):
    if xs == []:
        return []

I'm sure you see the picture. But it will take some real mental gymnastics to twist the rest of the problems into this shape. What about qselect, for instance? There's two cases for what it may return:

• None or equivalent if the index is out of bounds (we give it 4 an a list [1, 2]).
• A number if qselect worked.

The test cases never provide a concrete example of what should be returned from qselect in the first case, so we'll interpret it like {{< sidenote "right" "undefined-note" "undefined behavior" >}} For a quick sidenote about undefined behavior, check out how C++ optimizes the Collatz Conjecture function. Clang doesn't know whether or not the function will terminate (whether the Collatz Conjecture function terminates is an unsolved problem), but functions that don't terminate are undefined behavior. There's only one other way the function returns, and that's with "1". Thus, clang optimzes the entire function to a single "return 1" call. {{< /sidenote >}} in C++: we can do whatever we want. So, let's allow it to return [] in the None case. This makes this base case valid:

def qselect(xs, k):
    if xs == []:
        return []

"Oh yeah, now it's all coming together." With one more observation (which will come from a piece I haven't yet shown you!), we'll be able to generalize this base case.

The observation is this section in the assignment:

{{< codelines "text" "cs325-langs/hws/hw1.txt" 83 98 >}}

The real key is the part about "returning the [] where x should be inserted". It so happens that when the list given to the function is empty, the number should be inserted precisely into that list. Thus:

def _search(xs, k):
    if xs == []:
        return xs

The same works for qselect:

def qselect(xs, k):
    if xs == []:
        return xs

And for sorted, too:

def sorted(xs):
    if xs == []:
        return xs

There are some functions that are exceptions, though:

def insert(xs, k):
    # We can't return early here!
    # If we do, we'll never insert anything.

Also:

def search(xs, k):
    # We have to return true or false, never
    # an empty list.

So, whenever we don't return a list, we don't want to add a special case. We arrive at the following common base case: whenever a function returns a list, if its first argument is the empty list, the first argument is immediately returned.

We've largely exhasuted the conclusiosn we can draw from these problems. Let's get to designing a language.

A Silly Language

Let's start by visualizing our goals. Without base cases, the solution to _search would be something like this:

{{< codelines "text" "cs325-langs/sols/hw1.lang" 11 14 >}}

Here we have an if-expression. It has to have an else, and evaluates to the value of the chosen branch. That is, if true then 0 else 1 evaluates to 0, while if false then 0 else 1 evaluates to 1. Otherwise, we follow the binary tree search algorithm faithfully.

Using this definition of _search, we can define search pretty easily:

{{< codelines "text" "cs325-langs/sols/hw1.lang" 17 17 >}}

Let's use Haskell's (++) operator for concatentation. This will help us understand when the user is operating on lists, and when they're not. With this, sorted becomes:

{{< codelines "text" "cs325-langs/sols/hw1.lang" 16 16 >}}

Let's go for qselect now. We'll introduce a very silly language feature for this problem: {{< sidenote "right" "selector-note" "list selectors" >}} You've probably never heard of list selectors, and for a good reason: this is a terrible language feature. I'll go in more detail later, but I wanted to make this clear right away. {{< /sidenote >}}. We observe that qselect aims to partition the list into other lists. We thus add the following pieces of syntax:

~xs -> {
    pivot <- xs[rand]!
    left <- xs[#0 <= pivot]
    ...
} -> ...

There are three new things here.

1. The actual "list selector": ~xs -> { .. } -> .... Between the curly braces are branches which select parts of the list and assign them to new variables. Thus, pivot <- xs[rand]! assigns the element at a random index to the variable pivot. the ! at the end means "after taking this out of xs, delete it from xs". The syntax {{< sidenote "right" "curly-note" "starts with "~"" >}} An observant reader will note that there's no need for the "xs" after the "~". The idea was to add a special case syntax to reference the "selected list", but I ended up not bothering. So in fact, this part of the syntax is useless. {{< /sidenote >}} to make it easier to parse.
2. The rand list access syntax. xs[rand] is a special case that picks a random element from xs.
3. The xs[#0 <= pivot] syntax. This is another special case that selects all elements from xs that match the given predicate (where #0 is replaced with each element in xs).

The big part of qselect is to not evaluate right unless you have to. So, we shouldn't eagerly evaluate the list selector. We also don't want something like right[|right|-1] to evaluate right twice. So we settle on {{< sidenote "right" "lazy-note" "lazy evaluation" >}} Lazy evaluation means only evaluating an expression when we need to. Thus, although we might encounter the expression for right, we only evaluate it when the time comes. Lazy evaluation, at least the way that Haskell has it, is more specific: an expression is evaluated only once, or not at all. {{</ sidenote >}}. Ah, but the ! marker introduces {{< sidenote "left" "side-effect-note" "side effects" >}} A side effect is a term frequently used when talking about functional programming. Evaluating the expression xs[rand]! doesn't just get a random element, it also changes something else. In this case, that something else is the xs list. {{< /sidenote >}}. So we can't just evaluate these things all willy-nilly. So, let's make it so that each expression in the selector list requires the ones above it. Thus, left will require pivot, and right will require left and pivot. So, lazily evaluated, ordered expressions. The whole qselect becomes:

{{< codelines "text" "cs325-langs/sols/hw1.lang" 1 9 >}}

We've now figured out all the language constructs. Let's start working on some implementation!

Data Definitions

Let's start with defining the AST and other data types for our language:

{{< codelines "Haskell" "cs325-langs/src/LanguageOne.hs" 14 52 >}}

The PossibleType class will be used when we figure out if a function returns a list or not, for our base case insertion rule. The Selector type will hold a single line in the list selector we defined earlier, and the SelectorMarker will indicate if the user added the ! "remove from list" marker at the end. To represent the various operators in our language, we create the Op data type. Note that unlike Python, ++ (list concatenation) and + (addition) are different operators in our language.

We then define valid expressions. Obviously, a variable (like xs), an integer literal (like 1) and a list literal (like []) are allowed. We also put in our selector, which consists of the expression on the left, the list of selector branches ([Selector]) and the expression of "what to actually do with the new variables". We also add if-expressions (like we discussed), and function calls. Lastly, we add binary operators like (x+y), the length operator (|xs|), and the list access operator (xs[0]). We also make #0 a part of the expression syntax, even though it's only allowed inside a list access.

Of course, we wouldn't want to write our language using Haskell. We want to actually write a text file, like hw1.lang, and then have our program translate that to Python. The first step to that is parsing: we need to turn our language text into the Expr structure we have.

Parsing

We'll be using Parsec for parsing. Parsec is a parsing library based on {{< sidenote "right" "monad-note" "monadic" >}} Haskell is a language with more monad tutorials than programmers. For this reason, I will resist the temptation to explain what monads are. If you don't know what they are, don't worry, there are plenty of other resources. {{< /sidenote >}} parser combinators.