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Do the first round of revisions on part 3

This commit is contained in:
Danila Fedorin 2019-08-26 17:08:05 -07:00
parent 30c881ce3f
commit 619c346897
2 changed files with 116 additions and 52 deletions
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20
code/compiler/03/ast.cpp
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@ -28,28 +28,22 @@ type_ptr ast_binop::typecheck(type_mgr& mgr, const type_env& env) const {
type_ptr ftype = env.lookup(op_name(op));
if(!ftype) throw 0;
type_ptr place_a = mgr.new_type();
type_ptr place_b = mgr.new_type();
type_ptr place_c = mgr.new_type();
type_ptr arrow_one = type_ptr(new type_arr(place_b, place_c));
type_ptr arrow_two = type_ptr(new type_arr(place_a, arrow_one));
type_ptr return_type = mgr.new_type();
type_ptr arrow_one = type_ptr(new type_arr(rtype, return_type));
type_ptr arrow_two = type_ptr(new type_arr(ltype, arrow_one));
mgr.unify(arrow_two, ftype);
mgr.unify(place_a, ltype);
mgr.unify(place_b, rtype);
return place_c;
return return_type;
}
type_ptr ast_app::typecheck(type_mgr& mgr, const type_env& env) const {
type_ptr ltype = left->typecheck(mgr, env);
type_ptr rtype = right->typecheck(mgr, env);
type_ptr place_a = mgr.new_type();
type_ptr place_b = mgr.new_type();
type_ptr arrow = type_ptr(new type_arr(place_a, place_b));
type_ptr return_type = mgr.new_type();
type_ptr arrow = type_ptr(new type_arr(rtype, return_type));
mgr.unify(arrow, ltype);
mgr.unify(place_a, rtype);
return place_b;
return return_type;
}
type_ptr ast_case::typecheck(type_mgr& mgr, const type_env& env) const {

148
content/blog/03_compiler_trees.md
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@ -36,8 +36,8 @@ defn main = { 1 2 3 4 5 }
```
What is this? It's a sequence of applications, starting with `1 2`. Numbers
are not functions. Their type is `Int`, not `Int -> a`. I can't even think of a type `1`
would need to have for this program to be valid.
are not functions. Their type is `Int`, not `Int -> a`. I can't even think of a type numbers
would need to have for this program to be valid (though perhaps one could come up with one).
Before we give meaning to programs in our language, we'll need to toss away the ones
that don't make sense. To do so, we will use type checking. During the process of type
@ -47,20 +47,18 @@ to throw away blatantly incorrect programs.
### Basic Type Checking
You may have noticed in the very first post that I have chosen to avoid polymorphism.
This will significantly simplify our type checking algorithm. If a more robust
This will simplify our type checking algorithm. If a more robust
algorithm is desired, take a look at the
[Hindley-Milner type system](https://en.wikipedia.org/wiki/Hindley%E2%80%93Milner_type_system).
Personally, I enjoyed [this](http://dev.stephendiehl.com/fun/006_hindley_milner.html) section
of _Write You a Haskell_.
of _Write You a Haskell_, which I used to sanity check this very post.
Let's start with the types of constants - those are pretty obvious. The constant `3` is an integer,
and we shall mark it as such: `3 :: Int`. Variables seem like the natural
next step, but they're fairly different. Without outside knowledge, all we can do is say
that a variable has __some__ type. If we stick with Haskell's notation (used in polymorphic types),
we can say a variable has a type `a`, where `a` can be replaced with any other type. If we
know more information, like the fact that `x` was declared to be an integer, we can instead
say that. This tells us that throughout type checking we'll have to keep some kind
of record of names and their associated types.
next step, but they're fairly different. Without outside knowledge, we can't
tell what type a variable `x` is. If we know more information, like the fact that `x`
was declared to be an integer, we can instead say that. This tells us that throughout
type checking we'll have to keep some kind of record of names and their associated types.
Next, let's take a look at functions, which are admittedly more interesting
than the previous two examples. I'm not talking about the case of seeing something
@ -68,13 +66,16 @@ like a function name `f`. This is the same as the variable case - we don't even
it's a function unless there is context, and if there __is__ context, then that
context is probably the most useful information we have. I'm talking about
something like the application of a function to another value, in the form
`f x`. In this case, we know that `f :: a -> b`, a function from something
to something. However, we know even more. For this program to be correct,
the `a` in `f :: a -> b`, and the type of `x` (let's call it `c`), must
be compatible. In order to do that, we will use what can be considered
simplified [unification](https://en.wikipedia.org/wiki/Unification_(computer_science)).
Conceptually, what this means is that we will attempt to perform substitutions
in various equations in search of a solution.
`f x`. In this case, we know that for the program to be valid,
`f` must have the type `a -> b`, a function from something
to something. Furthermore, since `f` is being applied to `x`,
the type of `x` (let's call it `c`) must be the same as the type `a`.
Our rules are getting more complicated. In order to check that they hold, we will use
[unification](https://en.wikipedia.org/wiki/Unification_(computer_science)).
This means that we'll be creating various equations (such as "the type of `f` is equal to `a -> b`"),
and finding substitutions to solve these equations. If we're unable to
find a solution to our equations, the program is invalid and we throw it away.
#### Basic Examples
Let's try an example. We'll try to determine the type of the following
@ -87,7 +88,7 @@ foo 320 6
In out parse tree, this will be represented as `(foo 320) 6`, meaning
that the outermost application will be at the top. Let's assume
we know __nothing__ about `foo`.
we know only that `foo` is defined.
To figure out the type of the application, we will need to know the types
of the thing being applied, and the thing that serves as the argument.
@ -99,12 +100,13 @@ an argument, it has to be of type `a -> b`.
Let's proceed to the left child. It's another application, this time
of `foo` to `320`. Again, the right child is simple: the type of
`320` is `Int`. Again, we know that `foo` has to have a type
`c -> d` (we're using different variable names to avoid ambiguity).
`c -> d` (we're using different variable names since the types
of `foo` and `foo 320` are not the same).
Now, we need to combine the pieces of information that we have. Since
`foo :: c -> d`, we know that its first parameter __must__ be of
type `c`. We also know that its first parameter is of type `Int`.
The only way for both of these to be tree is for `c = Int`.
The only way for both of these to be true is for `c = Int`.
This also tells us that `foo :: Int -> d`. Finally,
since `foo` has now been applied to its first argument,
we know that the `foo 320 :: d`.
@ -116,7 +118,7 @@ is for `d = a -> b`. So, `foo 320` is a function from `a` to `b`.
Again, we can conclude the first parameter has to be of type
`a`. We also know that the first parameter is of type `Int`. Clearly,
this means that `a = Int`. After the application, we know
that the whole expression has some type `d`.
that the whole expression has some type `b`.
Let's revisit what we know about `foo`. Last time we checked in on it,
`foo` was of type `Int -> d`. But since we know that `d = a -> b`,
@ -156,7 +158,7 @@ $$
\\frac{A\_1 \\ldots A\_n} {B\_1 \\ldots B\_m}
$$
This reads, "given that the premises \\(A\_1\\) through \\(A\_n\\) are true,
it holds that the conclusions \\(B\_1\\) through \\(B\_n\\) are true".
it holds that the conclusions \\(B\_1\\) through \\(B\_m\\) are true".
For example, we can have the following inference rule:
$$
@ -197,18 +199,27 @@ $$
{e_1 \\; e_2 : \\tau\_2}
$$
This rule includes everything we've said before:
the thing being applied has to have a function type
(\\(\\tau\_1 \\rightarrow \\tau\_2\\)), and
the expression the function is applied to
has to have the same type \\(\\tau\_1\\) as the
left type of the function.
It's the variable rule that forces us to adjust our notation.
Our rules don't take into account the context that we've
already discussed. Let's fix that! It's convention
already discussed, and thus, we can't bring
in any outside information. Let's fix that! It's convention
to use the symbol \\(\\Gamma\\) for the context. We then
add notation to say, "using the context \\(\\Gamma\\),
we can deduce that \\(e\\) has type \\(\\tau\\)". We will
write this as \\(\\Gamma \\vdash e : \\tau\\).
But what __is__ our context? It's just a set of pairs
But what __is__ our context? We can think of it
as a mapping from variable names to their known types. We
can represent such a mapping using a set of pairs
in the form \\(x : \\tau\\), where \\(x\\) represents
a variable name. Each pair tells us that the variable
\\(x\\) is known to have a type \\(\\tau\\).
a variable name.
Since \\(\\Gamma\\) is just a regular set, we can
write \\(x : \\tau \\in \\Gamma\\), meaning that
@ -236,7 +247,7 @@ $$
In these three expressions, we've captured all of the rules
that we've seen so far. It's important to know
that These rules leave out the process
that these rules leave out the process
of unification altogether: we use unification to find
types that satisfy the rules.
@ -271,7 +282,7 @@ such as `Cons x xs` will introduce new type information, namely \\(\text{x} : \t
4. Finally, the conclusion is that the case expression, if all the premises are met, is of type \\(\\tau\_c\\).
For completeness, let's add rules for \\(\\text{matcht}(\\tau, p\_i) = b\_i\\). We'll need two: one for
the "basic" pattern, which always matches the value and binds it to the variable, and one
the "basic" pattern, which always matches the value and binds a variable to it, and one
for a constructor pattern, that matches a constructor and its parameters.
Let's define \\(v\\) to be a variable name in the context of a pattern. For the basic pattern:
@ -325,7 +336,7 @@ of the chain is a placeholder, it will set `var` to be a pointer
to that placeholder.
* `bind`, inspired by [this post](http://dev.stephendiehl.com/fun/006_hindley_milner.html),
will map a type variable of some name to a type. This function will also check if
the thing we're binding to is the same type variable and not do anything in that case,
we're binding a type variable to itself, and do nothing in that case,
since `a = a` is not a very useful equation to have.
To fit its original purpose, we also give the manager class the methods
@ -358,7 +369,7 @@ to accidentally compare `a` and `b` (and try to bind `a` to
From resolve, we will have `lvar` and `rvar` set to
something not NULL if `l` or `r` were type variables
that haven't been yet mapped (we defined `resolve` to behave this way).
that haven't yet been bound (we defined `resolve` to behave this way).
So, if one of the variables is not NULL, we try to bind it.
Next, `unify` checks if both types are either base types or
@ -378,12 +389,13 @@ need to implement our rules. Our rules
usually include three things: an environment
\\(\\Gamma\\), an expression \\(e\\),
and a type \\(\\tau\\). We will
represent this as a method on `ast`, which is
our representation of an expression \\(e\\). This
represent this as a method on `ast`, our struct
for an expression tree. This
method will take an environment and return
a type.
But first, how should we implement our environment?
##### Environment
How should we implement our environment?
Conceptually, an environment maps a name string
to a type. So naively, we can implement this simply
using an `std::map`. But observe
@ -429,7 +441,8 @@ creating new environments on the stack, and we'll
take care not to let a parent go out of scope
before the child.
At least, it's time to declare a new type checking method.
##### Typechecking Expressions
At last, it's time to declare a new type checking method.
We start with with a signature inside `ast`:
```
virtual type_ptr typecheck(type_mgr& mgr, const type_env& env) const;
@ -445,11 +458,68 @@ at the implementation now:
{{< codeblock "C++" "compiler/03/ast.cpp" >}}
{{< todo >}}Explain implementation, at least a little.{{< /todo >}}
The `typecheck` implementation for `ast_int`, `ast_lid`, and `ast_uid`
is quite intuitive. The type of a number is always `Int`, and varible
names we simply look up in the environment.
For `ast_binop` and `ast_app`, we check the types of the children first,
and store their resulting types. In the case of `binop`, we assume
that we have the type of the binary operator (such as `+`) in the
environment, and throw if it isn't. Then, we know that
the operator has to be a function of at least two arguments,
with the types of the left and right children of the application.
We also know its actual type, as it's in the environment. We unify
these two types, and then return.
In the case of `app`, we know that the left side (the thing
being applied), has to be a function from the type of the right
child. We also know its computed type. Once again, we unify the two,
and then return.
The type checking for `case` offloads most of the work onto the
`match` method on patterns. We start by computing the type
of the expression we're matching. Then, for each branch,
we create a new scope, and populate it with the new bindings
created by the pattern match (`match` does this). Once
we have a new environment, we typecheck the body
of the branch. Finally, we unify the type of the branch's
body with the previous body types (since branches of the
case expression have to have the same type).
Let's take a look at `match` now. The `match` method
for a variable pattern is very simple: it always
introduces a new variable bindings, and stops there.
The `match` method for a constructor pattern is more
interesting. First, it looks up the constructor
that the pattern is trying to match. This needs to exist,
so if we don't find a type for it, we throw. Next,
for each variable name in the pattern, we know
that there has to be a corresponding parameter in the
constructor. Because of this, we cast the constructor type
to an function type (throwing if it isn't one),
and create a new mapping from the variable name to the
left side (parameter) of the function type. We then
move on to examine the right side of the function
type, and the remaining variables of the pattern.
Once we have gone through the parameters,
what remains __should__ be the type of the thing
the constructor creates. Not only that, but
the remaining type should match the type of the value
the pattern is being matched against. To ensure
this, we unify the two types.
There's only one thing that can still go wrong.
The value __and__ the pattern can __both__ be
a partial application. For ease of implementation,
we will not allow this case. Thus, we make sure
the final type is a base type, and throw otherwise.
##### Typechecking Definitions
This looks good, but we're not done yet. We can type
check expressions, but our program ins't
made up of expressions. Rather, it's made up of
declarations. Further, we can't look at the declarations
definitions. Further, we can't look at the definitions
in isolation. Consider these two functions:
```
@ -503,11 +573,11 @@ virtual void typecheck_second(type_mgr& mgr, const type_env& env) const;
Furthermore, in the `definition_defn`, we will keep an
`std::vector` of `type_ptr`, in which the first element is the
type of the __last__ parameter, and so on. We switch around
the order of arguments because we build up the `a -> b -> ... -> x`
the order of arguments because we build up the `a -> b -> ...`
type signature from the right (`->` is right associative), and
thus we'll be creating the types right-to-left, too. We also
add a `type_ptr` field which holds the type for the function's
return value. We keep these two things in the `definition_defn` so
add a `type_ptr` field which holds the function's return type.
We keep these two things in the `definition_defn` so
that they persist between the two typechecking stages: we want to use
the types from the first stage to aid in checking the body in the second stage.