---
title: A Language for an Assignment - Homework 3
date: 2020-01-02T22:17:43-08:00
tags: ["Haskell", "Python", "Algorithms"]
---

It rained in Sunriver on New Year's Eve, and it continued to rain
for the next couple of days. So, instead of going skiing as planned,
to the dismay of my family and friends, I spent the majority of
those days working on the third language for homework 3. It
was quite the language, too - the homework has three problems, each of
which has a solution independent of the others. I invite you
to join me in my descent into madness as we construct another language.

### Homework 3
Let's take a look at the three homework problems. The first two are
related, but are solved using a different technique:

{{< codelines "text" "cs325-langs/hws/hw3.txt" 18 30 >}}

This problem requires us to find the `k` numbers closest to some
query (which I will call `n`) from a list `xs`. The list isn't sorted, and the
problem must run in linear time. Sorting the list would require
the standard
{{< sidenote "right" "n-note" "\(O(n\log n)\) time." >}}
The \(n\) in this expression is not the same as the query <code>n</code>,
but rather the length of the list. In fact, I have not yet assigned
the length of the input <code>xs</code> to any variable. If we say that
\(m\) is a number that denotes that length, the proper expression
for the complexity is \(O(m \log m)\).
{{< /sidenote >}} Thus, we have to take another route, which should
already be familiar: quickselect. Using quickselect, we can find the `k`th
closest number, and then collect all the numbers that are closer than the `kth`
closest number. So, we need a language that:

* Supports quickselect (and thus, list partitioning and recursion).
* Supports iteration, {{< sidenote "left" "iteration-note" "multiple times." >}}
Why would we need to iterate multiple times? Note that we could have a list
of numbers that are all the same, <code>[1,1,1,1,1]</code>. Then, we'll need
to know how many of the numbers <em>equally close</em> as the <code>k</code>th
element we need to include, which will require another pass through the list.
{{< /sidenote >}}

That's a good start. Let's take a look at the second problem:

{{< codelines "text" "cs325-langs/hws/hw3.txt" 33 47 >}}

This problem really is easier. We have to find the position of _the_ closest
element, and then try expand towards either the left or right, depending on
which end is better. This expansion will take several steps, and will
likely require a way to "look" at a given part of the list. So let's add two more
rules. We need a language that also:

* Supports looping control flow, such as `while`.
* {{< sidenote "right" "view-note" "Allows for a \"view\" into the list" >}}
We could, of course, simply use list indexing. But then, we'd just be making
a simple imperative language, and that's boring. So let's play around
with our design a little, and experimentally add such a "list view" component.
{{< /sidenote >}}
(like an abstraction over indexing).

This is shaping up to be a fun language. Let's take a look at the last problem:
{{< codelines "text" "cs325-langs/hws/hw3.txt" 50 64 >}}

This problem requires more iterations of a list. We have several
{{< sidenote "right" "cursor-note" "\"cursors\"" >}}
I always make the language before I write the post, since a lot of
design decisions change mid-implementation. I realize now that
"cursors" would've been a better name for this language feature,
but alas, it is too late.
{{< /sidenote >}} looking into the list, and depending if the values
at each of the cursors add up, we do or do not add a new tuple to a list. So,
two more requirements:

* The "cursors" must be able to interact.
* The language can represent {{< sidenote "left" "tuple-note" "tuples." >}}
We could, of course, hack some other way to return a list of tuples, but
it turns out tuples are pretty simple to implement, and help make for nicer
programming in our language.
{{< /sidenote >}}

I think we've gathered what we want from the homework. Let's move on to the
language!

### A Language
As is now usual, let's envision a solution to the problems in our language. There
are actually quite a lot of functions to look at, so let's see them one by one.
First, let's look at `qselect`.

{{< codelines "text" "cs325-langs/sols/hw3.lang" 1 19 >}}

After the early return, the first interesting part of the language is the
use of what I have decided to call a __list traverser__. The list
traverser is a __generalization of a list index__. Whenever we use a list
index variable, we generally use the following operations:

* __Initialize__: we set the list index to some initial value, such as 0.
* __Step__: If we're walking the list from left to right, we increment the index.
If we're walking the list from right to left, we decrement the index.
* __Validity Check__: We check if the index is still valid (that is, we haven't
gone past the edge of the list).
* __Access__: Get the element the cursor is pointing to.

A {{< sidenote "right" "cpp-note" "traverser declaration" >}}
A fun fact is that we've just rediscovered C++
<a href="http://www.cplusplus.com/reference/iterator/">iterators</a>. C++
containers and their iterators provide us with the operations I described:

We can initialize an iterator like <code>auto it = list.begin()</code>. We
can step the iterator using <code>it++</code>. We can check its validity
using <code>it != list.end()</code>, and access what it's pointing to using
<code>*it</code>. While C++ uses templates and inheritance for this,
we define a language feature specifically for lists.

{{< /sidenote >}} describes these operations. The declartion for the `bisector`
traverser creates a "cursor" over the list `xs`, that goes between the 0th
and last elements of `xs`. The declaration for the `pivot` traverser creates
a "cursor" over the list `xs` that jumps around random locations in the list.

The next interesting part of the language is a __traverser macro__. This thing,
that looks like a function call (but isn't), performs an operation on the
cursor. For instance, `pop!` removes the element at the cursor from the list,
whereas `bisect!` categorizes the remaining elements in the cursor's list
into two lists, using a boolean-returning lambda (written in Java syntax).

Note that this implementation of `qselect` takes a function `c`, which it
uses to judge the actual value of the number. This is because our `qselect`
won't be finding _the_ smallest number, but the number with the smallest difference
with `n`. `n` will be factored in via the function.

Next up, let's take a look at the function that uses `qselect`, `closestUnsorted`:

{{< codelines "text" "cs325-langs/sols/hw3.lang" 21 46 >}}

Like we discussed, it finds the `k`th closest element (calling it `min`),
and counts how many elements that are __equal__ need to be included,
by setting the number to `k` at first, and subtracting 1 for every number
it encounters that's closer than `min`. Notice that we use the `valid!` and
`step!` macros, which implement the operations we described above. Notice
that the user doesn't deal with adding and subtracting numbers, and doing
comparisons. All they have to do is ask "am I still good to iterate?"

Next, let's take a look at `closestSorted`, which will require more
traverser macros.

{{< codelines "text" "cs325-langs/sols/hw3.lang" 48 70 >}}

The first new macro is `canstep!`. This macro just verifies that
the traverser can make another step. We need this for the "reverse" iterator,
which indicates the lower bound of the range of numbers we want to return,
because `subset!` (which itself is just Python's slice, like `xs[a:b]`), uses an inclusive bottom
index, and thus, we can't afford to step it before knowing that we can, and that
it's a better choice after the step.

Similarly, we have the `at!(t, i)` macro, which looks at the
traverser `t`, with offset `i`.

We have two loops. The first loop runs as long as we can expand the range in both
directions, and picks the better direction at each iteration. The second loop
runs as long as we still want more numbers, but have already hit the edge
of the list on the left or on the right.

Finally, let's look at the solution to `xyz`:

{{< codelines "text" "cs325-langs/sols/hw3.lang" 72 95 >}}

I won't go in depth, but notice that the expression in the `span` part
of the `traverser` declaration can access another traverser. We treat
as a feature the fact that this expression isn't immediately evaluated at the place
of the traverser declaration. Rather, every time that a comparison for a traverser
operation is performed, this expression is re-evaluated. This allows us to put
dynamic bounds on traversers `y` and `z`, one of which must not exceed the other.

Note also a new keyword that was just used: `sorted`. This is a harmless little
language feature that automatically calls `.sort()` on the first argument of
the function.

This is more than enough to work with. Let's move on to the implementation.

#### Implementation
Again, let's not go too far into the details of implementing the language from scratch.
Instead, let's take a look into specific parts of the language that deserve attention.

##### Revenge of the State Monad
Our previous language was, indeed, a respite from complexity. Translation was
straightforward, and the resulting expressions and statements were plugged straight
into a handwritten AST. We cannot get away with this here; the language is powerful
enough to implement three list-based problems, which comes at the cost of increased 
complexity.

We need, once again, to generate temporary variables. We also need to keep track of
which variables are traversers, and the properties of these traversers, throughout
each function of the language. We thus fall back to using `Control.Monad.State`:

{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 198 198 >}}

There's one part of the state tuple that we haven't yet explained: the list of
statements.

##### Generating Statements
Recall that our translation function for expressions in the first homework had the type:

```Haskell
translateExpr :: Expr -> Translator ([Py.PyStmt], Py.PyExpr)
```

We then had to use `do`-notation, and explicitly concatenate lists
of emitted statements. In this language, I took an alternative route: I made
the statements part of the state. They are thus implicitly generated and
stored in the monad, and expression generators don't have to worry about
concatenating them. When the program is ready to use the generated statements
(say, when an `if`-statement needs to use the statements emitted by the condition
expression), we retrieve them from the monad:

{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 228 234 >}}

I should note, for transparency, that there's a bug in my use of this function.
When I compile `if`-statements, I accidentally place statements generated by
the condition into the body of the `if`. This bug doesn't manifest
in the solutions to the homework problems, and so I decided not to spend any more
time on fixing it.

##### Validating Traverser Declarations
We declare two separate types that hold traverser data. The first is a kind of "draft"
type, `TraverserData`:

{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 184 190 >}}

This record holds all possible configurations of a traverser
that occur as the program is iterating through the various `key: value` pairs in
the declaration. For instance, at the very beginning of processing a traverser declaration,
our program will use a "default" `TraverserData`, with all fields set to `Nothing` or
their default value. This value will then be modified by the first key/value pair,
changing, for instance, the list that the traverser operates on. This new modified
`TraverserData` will then be modified by the next key/value pair, and so on. Doing
this with every key/value pair (called an option in the below snippet)
is effectively a foldl operation.

{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 378 387 >}}

The data may not have all the required fields until the very end, and its type
reflects that: `Maybe String` here, `Maybe TraverserBounds` there. We don't
want to deal with unwrapping the `Maybe a` values every time we use the traverser,
especially if we've done so before. So, we define a `ValidTraverserData` record
that does not have `Maybe` arguments, and thus, has all the required data. At the
end of a traverser declaration, we attempt to translate a `TraverserData` into
a `ValidTraverserData`, invoking `fail` if we can't, and storing the `ValidTraverserData`
into the state otherwise:

{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 408 420 >}}

Then, every time we retrieve a traverser from the state,
define a lookup monadic operation like this:

{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 240 244 >}}

##### Compiling Macros
I didn't call them macros for no reason. Clearly, we don't want to generate
code that
{{< sidenote "right" "increment-note" "calls functions only to increment an index." >}}
In fact, there's no easy way to do this at all. Python's integers (if we choose to
represent our traversers using integers), are immutable. Furthermore, unlike C++,
where passing by reference allows a function to change its parameters "outside"
the call, Python offers no way to reassign a different value to a variable given
to a function.
<br><br>
For an example use of C++'s pass-by-reference mechanic, consider <code>std::swap</code>:
it's a function, but it modifies the two variables given to it. There's no
way to generically implement such a function in Python.
{{< /sidenote >}} We also can't allow arbitrary expressions to serve as traversers:
our translator keeps some context about which variables are traversers, what their
bounds are, and how they behave. Thus, __calls to traverser macros are very much macros__:
they operate on AST nodes, and __require__ that their first argument is a variable,
named like the traverser. We use the `requireTraverser` monadic operation
to get the traverser associated with the given variable name, and then perform
the operation as intended. The `at!(t)` operation is straightforward:

{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 317 319 >}}

The `at!(t,i)` is less so, since it deals with the intricacies of accessing
the list at either a positive of negative offset, depending on the direction
of the traverser. We implement a function to properly generate an expression for the offset:

{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 246 249 >}}

We then implement `at!(t,i)` as follows:

{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 320 323 >}}

The most complicated macro is `bisect!`. It must be able to step the traverser,
and also return a tuple of two lists that the bisection yields. We also
prefer that it didn't pollute the environment with extra variables. To
achieve this, we want `bisect!` to be a function call. We want this
function to implement the iteration and list construction.

`bisect!`, by definition, takes a lambda. This lambda, in our language, is declared
in the lexical scope in which `bisect!` is called. Thus, to guarantee correct translation,
we must do one of two things:

1. Translate 1-to-1, and create a lambda, passing it to a fixed `bisect` function declared
elsewhere.
2. Translate to a nested function declaration,
{{< sidenote "right" "inline-note" "inlining the lambda." >}}
Inlining, in this case, means replacing a call to a function with the function's body.
We do this to prevent the overhead of calling a function, which typically involves pushing
on a stack and other extraneous work. If our function is simple, like a simple
comparison, it doesn't make sense to spend the effort calling it.
{{< /sidenote >}}

Since I quite like the idea of inlining a lambda, let's settle for that. To do this,
we pull a fresh temporary variable and declare a function, into which we place
the traverser iteration code, as well as the body of the lambda, with the variable
substituted for the list access expression.
{{< sidenote "left" "nonlocal-note" "Here's the code:" >}}
Reading the lexical scope is one thing, but modifying it is another. To prevent
accidental changes to the variables outside a nested function, Python assumes
that variables assigned inside the function body are local to the function. Thus, to make
sure changing our variable (the traverser index) has an effect outside the function
(as it should) we must include the <code>nonlocal</code> keyword, telling
Python that we're not declaring a new, local variable, but mutating the old one.
{{< /sidenote >}}

{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 342 363 >}}

### The Output
Let's see what the compiler spits out:

```Python
from bisect import bisect
import random
def qselect(xs,k,c):
    if xs==[]:
        return 0
    bisector = 0
    pivot = random.randrange(len(xs))
    pivotE = xs.pop(pivot)
    def temp1():
        nonlocal bisector
        l = []
        r = []
        while bisector<len(xs):
            if c(xs[bisector])<c(pivotE):
                l.append(xs[bisector])
            else:
                r.append(xs[bisector])
            bisector = bisector+1
        return (l, r)
    (leftList,rightList) = temp1()
    if k>len(leftList)+1:
        return qselect(rightList, k-len(leftList)-1, c)
    elif k==len(leftList)+1:
        return pivotE
    else:
        return qselect(leftList, k, c)
def closestUnsorted(xs,k,n):
    min = qselect(list(xs), k, (lambda x: abs(x-n)))
    out = []
    countEqual = k
    iter = 0
    while iter<len(xs):
        if abs(xs[iter]-n)<abs(min-n):
            countEqual = countEqual-1
        iter = iter+1
        0
    iter = 0
    while iter<len(xs):
        if abs(xs[iter]-n)==abs(min-n) and countEqual>0:
            countEqual = countEqual-1
            out = out+[xs[iter]]
        elif abs(xs[iter]-n)<abs(min-n):
            out = out+[xs[iter]]
        iter = iter+1
        0
    return out
def closestSorted(xs,k,n):
    start = bisect(xs, n)
    counter = 0
    left = start
    right = start
    while counter!=k and left-1*1>=0 and right<len(xs):
        if abs(xs[left-1*1]-n)<abs(xs[right]-n):
            left = left-1
            0
        else:
            right = right+1
            0
        counter = counter+1
    while counter!=k and (left-1*1>=0 or right<len(xs)):
        if left-1*1>=0:
            left = left-1
            0
        else:
            right = right+1
            0
        counter = counter+1
    return xs[(left):(right)]
def xyz(xs,k):
    xs.sort()
    x = 0
    dest = []
    while x<len(xs):
        z = x+2
        y = x+1
        while y<z and z<len(xs):
            if xs[x]+xs[y]==xs[z]:
                dest = dest+[(xs[x], xs[y], xs[z])]
                z = z+1
                0
            elif xs[x]+xs[y]>xs[z]:
                z = z+1
                0
            else:
                y = y+1
                0
        x = x+1
        0
    return dest
```

Observe that the generated code just uses indices, `+`, `-`, and various comparison operators.
Our traverser is an example of a __zero cost abstraction__, a feature that, conceptually,
operates at a higher level, making us no longer worry about adding, subtracting, and
comparing numbers, while, in the final output, not damaging the performance of safety
of the code. Also observe the various `0` standalone statements. This is an issue
with the translator: traverser macros may not always yield an expression, but
the type of `translateExpr` and `translateStmt` effectively requires one. Thus,
when a macro doesn't generate anything useful, we give it the placeholder expression `0`.

That concludes this third post in the series. I hope to see you in the next one!