2020-01-03 21:09:15 -08:00
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---
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title: A Language for an Assignment - Homework 3
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date: 2020-01-02T22:17:43-08:00
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tags: ["Haskell", "Python", "Algorithms"]
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---
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It rained in Sunriver on New Year's Eve, and it continued to rain
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for the next couple of days. So, instead of going skiing as planned,
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to the dismay of my family and friends, I spent the majority of
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those days working on the third language for homework 3. It
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was quite the language, too - the homework has three problems, each of
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which has a solution independent of the others. I invite you
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to join me in my descent into madness as we construct another language.
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### Homework 3
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Let's take a look at the three homework problems. The first two are
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related, but are solved using a different technique:
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{{< codelines "text" "cs325-langs/hws/hw3.txt" 18 30 >}}
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This problem requires us to find the `k` numbers closest to some
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query (which I will call `n`) from a list `xs`. The list isn't sorted, and the
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problem must run in linear time. Sorting the list would require
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the standard
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{{< sidenote "right" "n-note" "\(O(n\log n)\) time." >}}
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The \(n\) in this expression is not the same as the query <code>n</code>,
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but rather the length of the list. In fact, I have not yet assigned
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the length of the input <code>xs</code> to any variable. If we say that
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\(m\) is a number that denotes that length, the proper expression
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for the complexity is \(O(m \log m)\).
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{{< /sidenote >}} Thus, we have to take another route, which should
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already be familiar: quickselect. Using quickselect, we can find the `k`th
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closest number, and then collect all the numbers that are closer than the `kth`
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closest number. So, we need a language that:
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* Supports quickselect (and thus, list partitioning and recursion).
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* Supports iteration, {{< sidenote "left" "iteration-note" "multiple times." >}}
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Why would we need to iterate multiple times? Note that we could have a list
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of numbers that are all the same, <code>[1,1,1,1,1]</code>. Then, we'll need
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to know how many of the numbers <em>equally close</em> as the <code>k</code>th
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element we need to include, which will require another pass through the list.
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{{< /sidenote >}}
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That's a good start. Let's take a look at the second problem:
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{{< codelines "text" "cs325-langs/hws/hw3.txt" 33 47 >}}
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This problem really is easier. We have to find the position of _the_ closest
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element, and then try expand towards either the left or right, depending on
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which end is better. This expansion will take several steps, and will
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likely require a way to "look" at a given part of the list. So let's add two more
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rules. We need a language that also:
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* Supports looping control flow, such as `while`.
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* {{< sidenote "right" "view-note" "Allows for a \"view\" into the list" >}}
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We could, of course, simply use list indexing. But then, we'd just be making
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a simple imperative language, and that's boring. So let's play around
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with our design a little, and experimentally add such a "list view" component.
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{{< /sidenote >}}
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(like an abstraction over indexing).
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This is shaping up to be a fun language. Let's take a look at the last problem:
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{{< codelines "text" "cs325-langs/hws/hw3.txt" 50 64 >}}
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This problem requires more iterations of a list. We have several
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{{< sidenote "right" "cursor-note" "\"cursors\"" >}}
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I always make the language before I write the post, since a lot of
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design decisions change mid-implementation. I realize now that
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"cursors" would've been a better name for this language feature,
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but alas, it is too late.
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{{< /sidenote >}} looking into the list, and depending if the values
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at each of the cursors add up, we do or do not add a new tuple to a list. So,
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two more requirements:
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* The "cursors" must be able to interact.
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* The language can represent {{< sidenote "left" "tuple-note" "tuples." >}}
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We could, of course, hack some other way to return a list of tuples, but
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it turns out tuples are pretty simple to implement, and help make for nicer
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programming in our language.
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{{< /sidenote >}}
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I think we've gathered what we want from the homework. Let's move on to the
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language!
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### A Language
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As is now usual, let's envision a solution to the problems in our language. There
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are actually quite a lot of functions to look at, so let's see them one by one.
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First, let's look at `qselect`.
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{{< codelines "text" "cs325-langs/sols/hw3.lang" 1 19 >}}
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After the early return, the first interesting part of the language is the
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use of what I have decided to call a __list traverser__. The list
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traverser is a __generalization of a list index__. Whenever we use a list
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index variable, we generally use the following operations:
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* __Initialize__: we set the list index to some initial value, such as 0.
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* __Step__: If we're walking the list from left to right, we increment the index.
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If we're walking the list from right to left, we decrement the index.
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* __Validity Check__: We check if the index is still valid (that is, we haven't
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gone past the edge of the list).
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* __Access__: Get the element the cursor is pointing to.
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A {{< sidenote "right" "cpp-note" "traverser declaration" >}}
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A fun fact is that we've just rediscovered C++
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<a href="http://www.cplusplus.com/reference/iterator/">iterators</a>. C++
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containers and their iterators provide us with the operations I described:
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We can initialize an iterator like <code>auto it = list.begin()</code>. We
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can step the iterator using <code>it++</code>. We can check its validity
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using <code>it != list.end()</code>, and access what it's pointing to using
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<code>*it</code>. While C++ uses templates and inheritance for this,
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we define a language feature specifically for lists.
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{{< /sidenote >}} describes these operations. The declartion for the `bisector`
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traverser creates a "cursor" over the list `xs`, that goes between the 0th
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and last elements of `xs`. The declaration for the `pivot` traverser creates
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a "cursor" over the list `xs` that jumps around random locations in the list.
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The next interesting part of the language is a __traverser macro__. This thing,
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that looks like a function call (but isn't), performs an operation on the
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cursor. For instance, `pop!` removes the element at the cursor from the list,
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whereas `bisect!` categorizes the remaining elements in the cursor's list
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into two lists, using a boolean-returning lambda (written in Java syntax).
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Note that this implementation of `qselect` takes a function `c`, which it
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uses to judge the actual value of the number. This is because our `qselect`
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won't be finding _the_ smallest number, but the number with the smallest difference
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with `n`. `n` will be factored in via the function.
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Next up, let's take a look at the function that uses `qselect`, `closestUnsorted`:
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{{< codelines "text" "cs325-langs/sols/hw3.lang" 21 46 >}}
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Like we discussed, it finds the `k`th closest element (calling it `min`),
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and counts how many elements that are __equal__ need to be included,
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by setting the number to `k` at first, and subtracting 1 for every number
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it encounters that's closer than `min`. Notice that we use the `valid!` and
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`step!` macros, which implement the operations we described above. Notice
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that the user doesn't deal with adding and subtracting numbers, and doing
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comparisons. All they have to do is ask "am I still good to iterate?"
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Next, let's take a look at `closestSorted`, which will require more
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traverser macros.
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{{< codelines "text" "cs325-langs/sols/hw3.lang" 48 70 >}}
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The first new macro is `canstep!`. This macro just verifies that
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the traverser can make another step. We need this for the "reverse" iterator,
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which indicates the lower bound of the range of numbers we want to return,
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because `subset!` (which itself is just Python's slice, like `xs[a:b]`), uses an inclusive bottom
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index, and thus, we can't afford to step it before knowing that we can, and that
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it's a better choice after the step.
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Similarly, we have the `at!(t, i)` macro, which looks at the
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traverser `t`, with offset `i`.
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We have two loops. The first loop runs as long as we can expand the range in both
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directions, and picks the better direction at each iteration. The second loop
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runs as long as we still want more numbers, but have already hit the edge
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of the list on the left or on the right.
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Finally, let's look at the solution to `xyz`:
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{{< codelines "text" "cs325-langs/sols/hw3.lang" 72 95 >}}
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I won't go in depth, but notice that the expression in the `span` part
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of the `traverser` declaration can access another traverser. We treat
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as a feature the fact that this expression isn't immediately evaluated at the place
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of the traverser declaration. Rather, every time that a comparison for a traverser
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operation is performed, this expression is re-evaluated. This allows us to put
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dynamic bounds on traversers `y` and `z`, one of which must not exceed the other.
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2020-01-03 23:47:36 -08:00
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Note also a new keyword that was just used: `sorted`. This is a harmless little
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language feature that automatically calls `.sort()` on the first argument of
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the function.
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2020-01-03 21:09:15 -08:00
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This is more than enough to work with. Let's move on to the implementation.
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#### Implementation
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Again, let's not go too far into the details of implementing the language from scratch.
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Instead, let's take a look into specific parts of the language that deserve attention.
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##### Revenge of the State Monad
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Our previous language was, indeed, a respite from complexity. Translation was
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straightforward, and the resulting expressions and statements were plugged straight
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into a handwritten AST. We cannot get away with this here; the language is powerful
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enough to implement three list-based problems, which comes at the cost of increased
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complexity.
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We need, once again, to generate temporary variables. We also need to keep track of
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which variables are traversers, and the properties of these traversers, throughout
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each function of the language. We thus fall back to using `Control.Monad.State`:
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{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 198 198 >}}
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There's one part of the state tuple that we haven't yet explained: the list of
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statements.
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##### Generating Statements
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Recall that our translation function for expressions in the first homework had the type:
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```Haskell
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translateExpr :: Expr -> Translator ([Py.PyStmt], Py.PyExpr)
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```
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We then had to use `do`-notation, and explicitly concatenate lists
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of emitted statements. In this language, I took an alternative route: I made
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the statements part of the state. They are thus implicitly generated and
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stored in the monad, and expression generators don't have to worry about
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concatenating them. When the program is ready to use the generated statements
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(say, when an `if`-statement needs to use the statements emitted by the condition
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expression), we retrieve them from the monad:
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{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 228 234 >}}
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I should note, for transparency, that there's a bug in my use of this function.
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When I compile `if`-statements, I accidentally place statements generated by
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the condition into the body of the `if`. This bug doesn't manifest
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in the solutions to the homework problems, and so I decided not to spend any more
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time on fixing it.
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##### Validating Traverser Declarations
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We declare two separate types that hold traverser data. The first is a kind of "draft"
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type, `TraverserData`:
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{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 184 190 >}}
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This record holds all possible configurations of a traverser
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that occur as the program is iterating through the various `key: value` pairs in
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the declaration. For instance, at the very beginning of processing a traverser declaration,
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our program will use a "default" `TraverserData`, with all fields set to `Nothing` or
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their default value. This value will then be modified by the first key/value pair,
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changing, for instance, the list that the traverser operates on. This new modified
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`TraverserData` will then be modified by the next key/value pair, and so on. Doing
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this with every key/value pair (called an option in the below snippet)
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is effectively a foldl operation.
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{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 378 387 >}}
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The data may not have all the required fields until the very end, and its type
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reflects that: `Maybe String` here, `Maybe TraverserBounds` there. We don't
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want to deal with unwrapping the `Maybe a` values every time we use the traverser,
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especially if we've done so before. So, we define a `ValidTraverserData` record
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that does not have `Maybe` arguments, and thus, has all the required data. At the
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end of a traverser declaration, we attempt to translate a `TraverserData` into
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a `ValidTraverserData`, invoking `fail` if we can't, and storing the `ValidTraverserData`
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into the state otherwise:
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{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 408 420 >}}
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Then, every time we retrieve a traverser from the state,
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define a lookup monadic operation like this:
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{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 240 244 >}}
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##### Compiling Macros
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I didn't call them macros for no reason. Clearly, we don't want to generate
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code that
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{{< sidenote "right" "increment-note" "calls functions only to increment an index." >}}
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In fact, there's no easy way to do this at all. Python's integers (if we choose to
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represent our traversers using integers), are immutable. Furthermore, unlike C++,
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where passing by reference allows a function to change its parameters "outside"
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the call, Python offers no way to reassign a different value to a variable given
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to a function.
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<br><br>
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For an example use of C++'s pass-by-reference mechanic, consider <code>std::swap</code>:
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it's a function, but it modifies the two variables given to it. There's no
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way to generically implement such a function in Python.
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{{< /sidenote >}} We also can't allow arbitrary expressions to serve as traversers:
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our translator keeps some context about which variables are traversers, what their
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bounds are, and how they behave. Thus, __calls to traverser macros are very much macros__:
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they operate on AST nodes, and __require__ that their first argument is a variable,
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named like the traverser. We use the `requireTraverser` monadic operation
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to get the traverser associated with the given variable name, and then perform
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the operation as intended. The `at!(t)` operation is straightforward:
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{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 317 319 >}}
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The `at!(t,i)` is less so, since it deals with the intricacies of accessing
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the list at either a positive of negative offset, depending on the direction
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of the traverser. We implement a function to properly generate an expression for the offset:
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{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 246 249 >}}
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We then implement `at!(t,i)` as follows:
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{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 320 323 >}}
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The most complicated macro is `bisect!`. It must be able to step the traverser,
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and also return a tuple of two lists that the bisection yields. We also
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prefer that it didn't pollute the environment with extra variables. To
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achieve this, we want `bisect!` to be a function call. We want this
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function to implement the iteration and list construction.
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`bisect!`, by definition, takes a lambda. This lambda, in our language, is declared
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in the lexical scope in which `bisect!` is called. Thus, to guarantee correct translation,
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we must do one of two things:
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1. Translate 1-to-1, and create a lambda, passing it to a fixed `bisect` function declared
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elsewhere.
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2020-01-03 23:47:36 -08:00
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2. Translate to a nested function declaration,
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{{< sidenote "right" "inline-note" "inlining the lambda." >}}
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Inlining, in this case, means replacing a call to a function with the function's body.
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We do this to prevent the overhead of calling a function, which typically involves pushing
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on a stack and other extraneous work. If our function is simple, like a simple
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comparison, it doesn't make sense to spend the effort calling it.
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{{< /sidenote >}}
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2020-01-03 21:09:15 -08:00
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Since I quite like the idea of inlining a lambda, let's settle for that. To do this,
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we pull a fresh temporary variable and declare a function, into which we place
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the traverser iteration code, as well as the body of the lambda, with the variable
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2020-01-03 23:47:36 -08:00
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substituted for the list access expression.
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{{< sidenote "left" "nonlocal-note" "Here's the code:" >}}
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Reading the lexical scope is one thing, but modifying it is another. To prevent
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accidental changes to the variables outside a nested function, Python assumes
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that variables assigned inside the function body are local to the function. Thus, to make
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sure changing our variable (the traverser index) has an effect outside the function
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(as it should) we must include the <code>nonlocal</code> keyword, telling
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Python that we're not declaring a new, local variable, but mutating the old one.
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{{< /sidenote >}}
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{{< codelines "Haskell" "cs325-langs/src/LanguageThree.hs" 342 363 >}}
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### The Output
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Let's see what the compiler spits out:
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```Python
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from bisect import bisect
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import random
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def qselect(xs,k,c):
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if xs==[]:
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return 0
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bisector = 0
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pivot = random.randrange(len(xs))
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pivotE = xs.pop(pivot)
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def temp1():
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nonlocal bisector
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l = []
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r = []
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while bisector<len(xs):
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if c(xs[bisector])<c(pivotE):
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l.append(xs[bisector])
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else:
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r.append(xs[bisector])
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bisector = bisector+1
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return (l, r)
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(leftList,rightList) = temp1()
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if k>len(leftList)+1:
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return qselect(rightList, k-len(leftList)-1, c)
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elif k==len(leftList)+1:
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return pivotE
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else:
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return qselect(leftList, k, c)
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def closestUnsorted(xs,k,n):
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min = qselect(list(xs), k, (lambda x: abs(x-n)))
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out = []
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countEqual = k
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iter = 0
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while iter<len(xs):
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if abs(xs[iter]-n)<abs(min-n):
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countEqual = countEqual-1
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iter = iter+1
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0
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iter = 0
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while iter<len(xs):
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if abs(xs[iter]-n)==abs(min-n) and countEqual>0:
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countEqual = countEqual-1
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out = out+[xs[iter]]
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elif abs(xs[iter]-n)<abs(min-n):
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out = out+[xs[iter]]
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iter = iter+1
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0
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return out
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def closestSorted(xs,k,n):
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start = bisect(xs, n)
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counter = 0
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left = start
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right = start
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while counter!=k and left-1*1>=0 and right<len(xs):
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if abs(xs[left-1*1]-n)<abs(xs[right]-n):
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left = left-1
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0
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else:
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right = right+1
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0
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counter = counter+1
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while counter!=k and (left-1*1>=0 or right<len(xs)):
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if left-1*1>=0:
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left = left-1
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0
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else:
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right = right+1
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0
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counter = counter+1
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return xs[(left):(right)]
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def xyz(xs,k):
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xs.sort()
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x = 0
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dest = []
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while x<len(xs):
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z = x+2
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y = x+1
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while y<z and z<len(xs):
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if xs[x]+xs[y]==xs[z]:
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dest = dest+[(xs[x], xs[y], xs[z])]
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z = z+1
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0
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elif xs[x]+xs[y]>xs[z]:
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z = z+1
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0
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else:
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y = y+1
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0
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x = x+1
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0
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return dest
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```
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Observe that the generated code just uses indices, `+`, `-`, and various comparison operators.
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Our traverser is an example of a __zero cost abstraction__, a feature that, conceptually,
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operates at a higher level, making us no longer worry about adding, subtracting, and
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comparing numbers, while, in the final output, not damaging the performance of safety
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of the code. Also observe the various `0` standalone statements. This is an issue
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with the translator: traverser macros may not always yield an expression, but
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the type of `translateExpr` and `translateStmt` effectively requires one. Thus,
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|
when a macro doesn't generate anything useful, we give it the placeholder expression `0`.
|
2020-01-03 21:09:15 -08:00
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2020-01-03 23:47:36 -08:00
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That concludes this third post in the series. I hope to see you in the next one!
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