Now that our language's type system is more fleshed out and pleasant to use, it's time to shift our focus to the ergonomics of the language itself. I've been mentioning `let/in` expressions and __lambda expressions__ for a while now. The former will let us create names for expressions that are limited to a certain scope (without having to create global variable bindings), while the latter will allow us to create functions without giving them any name at all.
Let's take a look at `let/in` expressions first, to make sure we're all on the same page about what it is we're trying to implement. Let's start with some rather basic examples, and then move on to more complex examples. The most basic use of a `let/in` expression is, in Haskell:
In the above example, we bind the variable `x` to the value `5`, and then refer to `x` twice in the expression after the `in`. The whole snippet is one expression, evaluating to what the `in` part evaluates to. Additionally, the variable `x` does not escape the expression -
programming style may be slightly twitching right now, the words <em>eta reduction</em> swirling in their mind. What do you know, <code>fold</code>-based <code>sum</code> is even one of the examples on the Wikipedia page! I assure you, I left the code as you see it deliberately, to demonstrate a principle.
{{</sidenote>}} the list to be summed. We will want this to be valid in our language, as well. We will soon see how this particular feature is related to lambda functions, and why I'm covering these two features in the same post.
The part that doesn't translate well is the whole deal with patterns in function arguments, as well as the notion of having more than one equation for a single function, as is the case with <code>safeTail</code>.
It's not that these things are <em>impossible</em> to translate; it's just that translating them may be worthy of a post in and of itself, and would only serve to bloat and complicate this part. What can be implemented with pattern arguments can just as well be implemented using regular case expressions; I dare say most "big" functional languages actually just convert from the former to the latter as part of the compillation process.
[let-polymorphism](https://en.wikipedia.org/wiki/Hindley%E2%80%93Milner_type_system#Let-polymorphism)? This is it: we're allowing polymorphic variable bindings, but only when they're bound in a `let/in` expression (or at the top level).
The principles demonstrated by the last two snippets mean that compiling `let/in` expressions, at least with the power we want to give them, will require the same kind of dependency analysis we had to go through when we implemented polymorphically typed functions. That is, we will need to analyze which functions calls which other functions, and typecheck the callees before the callers. We will continue to represent callee-caller relationships using a dependency graph, in which nodes represent functions, and an edge from one function node to another means that the former function calls the latter. Below is an image of one such graph:
{{<figuresrc="fig_graph.png"caption="Example dependency graph without `let/in` expressions.">}}
Since we want to typecheck callees first, we effectively want to traverse the graph in reverse
topological order. However, there's a slight issue: a topological order is only defined for acyclic graphs, and it is very possible for functions in our language to mutually call each other. To deal with this, we have to find groups of mutually recursive functions, and and treat them as a single unit, thereby eliminating cycles. In the above graph, there are two groups, as follows:
{{<figuresrc="fig_colored_ordered.png"caption="Previous depndency graph with mutually recursive groups highlighted.">}}
As seen in the second image, according to the reverse topological order of the given graph, we will typecheck the blue group containing three functions first, since the sole function in the orange group calls one of the blue functions.
Things are more complicated now that `let/in` expressions are able to introduce their own, polymorphic and recursive declarations. However, there is a single invariant we can establish: function definitions can only depend on functions defined at the same time as them. That is, for our purposes, functions declared in the global scope can only depend on other functions declared in the global scope, and functions declared in a `let/in` expression can only depend on other functions declared in that same expression. That's not to say that a function declared in a `let/in` block inside some function `f` can't call another globally declared function `g` - rather, we allow this, but treat the situation as though `f` depends on `g`. In contrast, it's not at all possible for a global function to depend on a local function, because bindings created in a `let/in` expression do not escape the expression itself. This invariant tells us that in the presence of nested function definitions, the situation looks like this:
{{<figuresrc="fig_subgraphs.png"caption="Previous depndency graph augmented with `let/in` subgraphs.">}}
In the above image, some of the original nodes in our graph now contain other, smaller graphs. Those subgraphs are the graphs created by function declarations in `let/in` expressions. Just like our top-level nodes, the nodes of these smaller graphs can depend on other nodes, and even form cycles. Within each subgraph, we will have to perform the same kind of cycle detection, resulting in something like this:
{{<figuresrc="fig_subgraphs_colored_all.png"caption="Augmented dependency graph with mutually recursive groups highlighted.">}}
When typechecking a function, we must be ready to perform dependency analysis at any point. What's more is that the free variable analysis we used to perform must now be extended to differentiate between free variables that refer to "nearby" definitions (i.e. within the same `let/in` expression), and "far away" definitions (i.e. outside of the `let/in` expression). And speaking of free variables...
What do we do about variables that are captured by a local definition? Consider the following snippet:
```Haskell {linenos=table}
addToAll n xs = map addSingle xs
where
addSingle x = n + x
```
In the code above, the variable `n`, bound on line 1, is used by `addSingle` on line 3. When a function refers to variables bound outside of itself (as `addSingle` does), it is said to be _capturing_ these variables, and the function is called a _closure_. Why does this matter? On the machine level, functions are represented as sequences of instructions, and there's a finite number of them (as there is finite space on the machine). But there is an infinite number of `addSingle` functions! When we write `addToAll 5 [1,2,3]`, `addSingle` becomes `5+x`. When, on the other hand, we write `addToAll 6 [1,2,3]`, `addSingle` becomes `6+x`. There are certain ways to work around this - we could, for instance, dynamically create machine code in memory, and then execute it (this is called [just-in-time compilation](https://en.wikipedia.org/wiki/Just-in-time_compilation)). This would end up with a collections of runtime-defined functions that can be represented as follows:
```Haskell {linenos=table}
-- Version of addSingle when n = 5
addSingle5 x = 5 + x
-- Version of addSingle when n = 6
addSingle6 x = 6 + x
-- ... and so on ...
```
But now, we end up creating several functions with almost identical bodies, with the exception of the free variables themselves. Wouldn't it be better to perform the well-known strategy of reducing code duplication by factoring out parameters, and leaving only instance of the repeated code? We would end up with:
```Haskell {linenos=table}
addToAll n xs = map (addSingle n) xs
addSingle n x = n + x
```
Observe that we no longer have the "infinite" number of functions - the infinitude of possible behaviors is created via currying. Also note that `addSingle`
Wait a moment, didn't we just talk about nested polymorphic definitions, and how they change our typechecking model? If we transform our program into a bunch of global definitions, we don't need to make adjustments to our typechecking. <br><br>
This is true, but why should we perform transformations on a malformed program? Typechecking before pulling functions to the global scope will help us save the work, and breaking down one dependency-searching problem (which is \(O(n^3)\) thanks to Warshall's) into smaller, independent problems may even lead to better performance. Furthermore, typechecking before program transformations will help us come up with more helpful error messages.
{{</sidenote>}} and can be transformed into a sequence of instructions just like any other global function. It has been pulled from its `where` (which, by the way, is pretty much equivalent to a `let/in`) to the top level.
This technique of replacing captured variables with arguments, and pulling closures into the global scope to aid compilation, is called [Lambda Lifting](https://en.wikipedia.org/wiki/Lambda_lifting). Its name is no coincidence - lambda functions need to undergo the same kind of transformation as our nested definitions (unlike nested definitions, though, lambda functions need to be named). This is why they are included in this post together with `let/in`!
