blog-static/content/blog/06_compiler_semantics.md

---
title: Compiling a Functional Language Using C++, Part 6 - Compilation
date: 2019-08-06T14:26:38-07:00
draft: true
tags: ["C and C++", "Functional Languages", "Compilers"]
---
In the previous post, we defined a machine for graph reduction,
called a G-machine. However, this machine is still not particularly
connected to __our__ language. In this post, we will give
meanings to programs in our language in the context of
this G-machine. We will define a __compilation scheme__,
which will be a set of rules that tell us how to
translate programs in our language into G-machine instructions.
To mirror _Implementing Functional Languages: a tutorial_, we'll
call this compilation scheme \\(\\mathcal{C}\\), and write it
as \\(\\mathcal{C} ⟦e⟧ = i\\), meaning "the expression \\(e\\)
compiles to the instructions \\(i\\)".

To follow our route from the typechecking, let's start
with compiling expressions that are numbers. It's pretty easy:
$$
\\mathcal{C} ⟦n⟧ = [\\text{PushInt} \\; n]
$$

Here, we compiled a number expression to a list of
instructions with only one element - PushInt.

Just like when we did typechecking, let's
move on to compiling function applications. As
we informally stated in the previous chapter, since
the thing we're applying has to be on top,
we want to compile it last:

$$
\\mathcal{C} ⟦e\_1 \\; e\_2⟧ = \\mathcal{C} ⟦e\_2⟧ ⧺ \\mathcal{C} ⟦e\_1⟧ ⧺ [\\text{MkApp}]
$$

Here, we used the \\(⧺\\) operator to represent the concatenation of two
lists. Otherwise, this should be pretty intutive - we first run the instructions
to create the parameter, then we run the instructions to create the function,
and finally, we combine them using MkApp.

It's variables that once again force us to adjust our strategy. If our
program is well-typed, we know our variable will be on the stack:
our definition of Unwind makes it so for functions, and we will
define our case expression compilation scheme to match. However,
we still need to know __where__ on the stack each variable is,
and this changes as the stack is modified.

To accommodate for this, we define an environment, \\(\\rho\\),
to be a partial function mapping variable names to thier
offsets on the stack. We write \\(\\rho = [x \\rightarrow n, y \\rightarrow m]\\)
to say "the environment \\(\\rho\\) maps variable \\(x\\) to stack offset \\(n\\),
and variable \\(y\\) to stack offset \\(m\\)". We also write \\(\\rho \\; x\\) to
say "look up \\(x\\) in \\(\\rho\\)", since \\(\\rho\\) is a function. Finally,
to help with the ever-changing stack, we define an augmented environment
\\(\\rho^{+n}\\), such that \\(\\rho^{+n} \\; x = \\rho \\; x + n\\). In words,
this basically means "\\(\\rho^{+n}\\) has all the variables from \\(\\rho\\),
but their addresses are incremented by \\(n\\)". We now pass \\(\\rho\\)
in to \\(\\mathcal{C}\\) together with the expression \\(e\\). Let's
rewrite our first two rules. For numbers:

$$
\\mathcal{C} ⟦n⟧ \\; \\rho = [\\text{PushInt} \\; n]
$$

For function application:
$$
\\mathcal{C} ⟦e\_1 \\; e\_2⟧ \\; \\rho = \\mathcal{C} ⟦e\_2⟧ \\; \\rho ⧺ \\mathcal{C} ⟦e\_1⟧ \\; \\rho^{+1} ⧺ [\\text{MkApp}]
$$

Notice how in that last rule, we passed in \\(\\rho^{+1}\\) when compiling the function's expression. This is because
the result of running the instructions for \\(e\_2\\) will have left on the stack the function's parameter. Whatever
was at the top of the stack (and thus, had index 0), is now the second element from the top (address 1). The
same is true for all other things that were on the stack. So, we increment the environment accordingly.

With the environment, the variable rule is simple:
$$
\\mathcal{C} ⟦x⟧ \\; \\rho = [\\text{Push} \\; (\\rho \\; x)]
$$

One more thing. If we run across a function name, we want to
use PushGlobal rather than Push. Defining \\(f\\) to be a name
of a global function, we capture this using the following rule:

$$
\\mathcal{C} ⟦f⟧ \\; \\rho = [\\text{PushGlobal} \\; f]
$$

Now it's time for us to compile case expressions, but there's a bit of
an issue - our case expressions branches don't map one-to-one with
the \\(t \\rightarrow i\_t\\) format of the Jump instruction.
This is because we allow for name patterns in the form \\(x\\),
which can possibly match more than one tag. Consider this
rather useless example:

```
data Bool = { True, False }
defn weird b = { case b of { b -> { False } } }
```

We only have one branch, but we have two tags that should
lead to it! Not only that, but variable patterns are
location-dependent: if a variable pattern comes
before a constructor pattern, then the constructor
pattern will never be reached. On the other hand,
if a constructor pattern comes before a variable
pattern, it will be tried before the varible pattern,
and thus is reachable.

We will ignore this problem for now - we will define our semantics
as though each case expression branch can match exactly one tag.
In our C++ code, we will write a conversion function that will
figure out which tag goes to which sequence of instructions.
Effectively, we'll be performing [desugaring](https://en.wikipedia.org/wiki/Syntactic_sugar).

Now, on to defining the compilation rules for case expressions.
It's helpful to define compiling a single branch of a case expression
separately. For a branch in the form \\(t \\; x\_1 \\; x\_2 \\; ... \\; x\_n \\rightarrow \text{body}\\),
we define a compilation scheme \\(\\mathcal{A}\\) as follows:

$$
\\begin{align}
\\mathcal{A} ⟦t \\; x\_1 \\; ... \\; x\_n \\rightarrow \text{body}⟧ \\; \\rho & =
t \\rightarrow [\\text{Split} \\; n] \\; ⧺ \\; \\mathcal{C}⟦\\text{body}⟧ \\; \\rho' \\; ⧺ \\; [\\text{Slide} \\; n] \\\\\\
\text{where} \\; \\rho' &= \\rho^{+n}[x\_1 \\rightarrow 0, ..., x\_n \\rightarrow n - 1]
\\end{align}
$$

First, we run Split - the node on the top of the stack is a packed constructor,
and we want access to its member variables, since they can be referenced by
the branch's body via \\(x\_i\\). For the same reason, we must make sure to include
\\(x\_1\\) through \\(x\_n\\) in our environment. Furthermore, since the split values now occupy the stack,
we have to offset our environment by \\(n\\) before adding bindings to our new variables.
Doing all these things gives us \\(\\rho'\\), which we use to compile the body, placing
the resulting instructions after Split. This leaves us with the desired graph on top of
the stack - the only thing left to do is to clean up the stack of the unpacked values,
which we do using Slide.

Notice that we didn't just create instructions - we created a mapping from the tag \\(t\\)
to the instructions that correspond to it.

Now, it's time for compiling the whole case expression. We first want
to construct the graph for the expression we want to perform case analysis on.
Next, we want to evaluate it (since we need a packed value, not a graph,
to read the tag). Finally, we perform a jump depending on the tag. This
is capture by the following rule:

$$
\\mathcal{C} ⟦\\text{case} \\; e \\; \\text{of} \\; \\text{alt}_1 ... \\text{alt}_n⟧ \\; \\rho =
\\mathcal{C} ⟦e⟧ \\; \\rho \\; ⧺ [\\text{Eval}, \\text{Jump} \\; [\\mathcal{A} ⟦\\text{alt}_1⟧ \; \\rho, ..., \\mathcal{A} ⟦\\text{alt}_n⟧ \; \\rho]]
$$

This works because \\(\\mathcal{A}\\) creates not only instructions,
but also a tag mapping. We simply populate our Jump instruction such mappings
resulting from compiling each branch.

You may have noticed that we didn't add rules for binary operators. Just like
with type checking, we treat them as function calls. However, rather that constructing
graphs when we have to instantiate those functions, we simply
evaluate the arguments and perform the relevant arithmetic operation using BinOp.
We will do a similar thing for constructors.

With that out of the way, we can get around to writing some code. We can envision
a method on the `ast` struct that takes an environment (just like our compilation
scheme takes the environment \\(\\rho\\\)). Rather than returning a vector
of instructions (which involves copying, unless we get some optimization kicking in),
we'll pass to it a reference to a vector. The method will then place the generated
instructions into the vector.