blog-static/content/blog/01_cs325_languages_hw2.md

title	date	tags
A Language for an Assignment - Homework 2	2019-12-30T20:05:10-08:00	Haskell Python Algorithms

After the madness of the [language for homework 1]({{< relref "00_cs325_languages_hw1.md" >}}), the solution to the second homework offers a moment of respite. Let's get right into the problems, shall we?

Homework 2

Besides some free-response questions, the homework contains two problems. The first:

{{< codelines "text" "cs325-langs/hws/hw2.txt" 29 34 >}}

And the second:

{{< codelines "text" "cs325-langs/hws/hw2.txt" 36 44 >}}

At first glance, it's not obvious why these problems are good for us. However, there's one key observation: num_inversions can be implemented using a slightly-modified mergesort. The trick is to maintain a counter of inversions in every recursive call to mergesort, updating it every time we take an element from the {{< sidenote "right" "right-note" "right list" >}} If this nomeclature is not clear to you, recall that mergesort divides a list into two smaller lists. The "right list" refers to the second of the two, because if you visualize the original list as a rectangle, and cut it in half (vertically, down the middle), then the second list (from the left) is on the right. {{< /sidenote >}} while there are still elements in the {{< sidenote "left" "left-note" "left list" >}} Why this is the case is left as an exercise to the reader. {{< /sidenote >}}. When we return from the call, we add up the number of inversions from running num_inversions on the smaller lists, and the number of inversions that we counted as I described. We then return both the total number of inversions and the sorted list.

So, we either perform the standard mergesort, or we perform mergesort with additional steps added on. The additional steps can be divided into three general categories:

1. Initialization: We create / set some initial state. This state doesn't depend on the lists or anything else.
2. Effect: Each time that an element is moved from one of the two smaller lists into the output list, we may change the state in some way (create an effect).
3. Combination: The final state, and the results of the two sub-problem states, are combined into the output of the function.

This is all very abstract. In the concrete case of inversions, these steps are as follows:

1. Initializtion: The initial state, which is just the counter, is set to 0.
2. Effect: Each time an element is moved, if it comes from the right list, the number of inversions is updated.
3. Combination: We update the state, simply adding the left and right inversion counts.

We can make a language out of this!

A Language

Again, let's start by visualizing what the solution will look like. How about this:

{{< rawblock "cs325-langs/sols/hw2.lang" >}}

We divide the code into the same three steps that we described above. The first section is the initial state. Since it doesn't depend on anything, we expect it to be some kind of literal, like an integer. Next, we have the effect section, which has access to variables such as "STATE" (to access the current state) and "LEFT" (to access the left list), or "L" to access the "name" of the left list. We use an if-statement to check if the origin of the element that was popped (held in the "SOURCE" variable) is the right list (denoted by "R"). If it is, we increment the counter (state) by the proper amount. In the combine step, we simply increment the state by the counters from the left and right solutions, stored in "LSTATE" and "RSTATE". That's it!

Implementation

The implementation is not tricky at all. We don't need to use monads like we did last time, and nor do we have to perform any fancy Python nested function declarations.

To keep with the Python convention of lowercase variables, we'll translate the uppercase "global" variables to lowercase. We'll do it like so:

{{< codelines "Haskell" "cs325-langs/src/LanguageTwo.hs" 167 176 >}}

Note that we translated "L" and "R" to integer literals. We'll indicate the source of each element with an integer, since there's no real point to representing it with a string or a variable. We'll need to be aware of this when we implement the actual, generic mergesort code. Let's do that now:

{{< codelines "Haskell" "cs325-langs/src/LanguageTwo.hs" 101 161 >}}

This is probably the ugliest part of this assignment: we handwrote a Python AST in Haskell that implements mergesort with our augmentations. Note that this is a function, which takes a Py.PyExpr (the initial state expression), and two lists of Py.PyStmt, which are the "effect" and "combination" code, respectively. We simply splice them into our regular mergesort function. The translation is otherwise pretty trivial, so there's no real reason to show it here.

The Output

What's the output of our solution to num_inversions? Take a look for yourself:

def prog(xs):
    if len(xs)<2:
        return (0, xs)
    leng = len(xs)//2
    left = xs[:(leng)]
    right = xs[(leng):]
    (ls,left) = prog(left)
    (rs,right) = prog(right)
    left.reverse()
    right.reverse()
    state = 0
    source = 0
    total = []
    while (left!=[])and(right!=[]):
        if left[-1]<=right[-1]:
            total.append(left.pop())
            source = 1
        else:
            total.append(right.pop())
            source = 2
        if source==2:
            state = state+len(left)
    state = state+ls+rs
    left.reverse()
    right.reverse()
    return (state, total+left+right)

Honestly, that's pretty clean. As clean as left.reverse() to allow for \(O(1)\) pop is. What's really clean, however, is the implementation of mergesort in our language. It goes as follows:

state 0;
effect {}
combine {}

To implement mergesort in our language, which describes mergesort variants, all we have to do is not specify any additional behavior. Cool, huh?

That's the end of this post. If you liked this one (and the previous one!), keep an eye out for more!