I think tokenizing and parsing are boring. The thing is, looking at syntax is a pretty shallow measure of how interesting a language is. It’s like the cover of a book. Every language has one, and it so happens that to make our “book”, we need to start with making the cover. But the content of the book is what matters, and that’s where we’ve arrived now. We must make decisions about our language, and give meaning to programs written in it. But before we can give our programs meaning, we need to make sense of the current domain of programs that we receive from our parser. Let’s consider a few wonderful examples.
defn main = { plus 320 6 }
defn plus x y = { x + y }
This is a valid program, as far as we’re concerned. But are all programs we get from the parser valid? See for yourself:
data Bool = { True, False }
defn main = { 3 + True }
Obviously, that’s not right. The parser accepts it  this matches our grammar. But giving meaning to this program is not easy, since we have no clear way of adding 3 and some data type. Similarly:
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 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 checking, we will collect information about various parts of our abstract syntax trees, classifying them by the types of values they create. Using this information, we’ll be able 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 simplify our type checking algorithm. If a more robust algorithm is desired, take a look at the HindleyMilner type system. Personally, I enjoyed this section 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, 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
like a function name f
. This is the same as the variable case  we don’t even know
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 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.
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 expression, and extract any other information from this expression that we might use later.
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 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.
The latter is easy: the type of 6
is Int
. Before we proceed
into the left child of the application, there’s one more
piece of information we can deduce: since foo 320
is applied to
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 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 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
.
We’ve done what we can from this innermost application; it’s time to return
to the outermost one. We now know that the left child is of type d
, and
that it also has to be of type a > b
. The only way for this to be true
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 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
,
and that a = Int
, we can now say that foo :: Int > Int > b
.
We haven’t found any issues with the expression, and we learned
some new information about the type of foo
. Awesome!
Let’s apply this to a simplified example from the beginning of this post. Let’s check the type of:
1 2
Once again, the application is what we see first. The right child
of the application, just like in the previous example, is Int
.
We also kno that since 1
is being applied as a function,
its type must be a > b
. However, we also know that the left
child, being a number, is also of type Int
! There’s no
way to combine a > b
with Int
, and thus, there is no solution
we can find for the type of 1 2
. This means our program is invalid.
We can toss it away, give an error, and exit.
Some Notation
If you go to the Wikipedia page on the HindleyMilner type system, you will see quite a lot of symbols and greek letters. This is a good thing, because the way that I presented to you the rules for figuring out types of expressions is very verbose. You have to read several paragraphs of text, and that’s only for the first three logical rules! If you’re anything like me, you want to be able to read just the important parts, and with some notation, I’ll be able to show you these important parts concisely, while continuing to explain the rules in detail in paragraphs of text.
Let’s start with inference rules. An inference rule is an expression in the form:
$\frac{A_1 \ldots A_n} {B_1 \ldots B_m}$This reads, “given that the premises $A_1$ through $A_n$ are true, it holds that the conclusions $B_1$ through $B_m$ are true”.
For example, we can have the following inference rule:
$\frac {\text{if it's cold, I wear a jacket} \quad \text{it's cold}} {\text{I wear a jacket}}$Since you wear a jacket when it’s cold, and it’s cold, we can conclude that you will wear a jacket.
When talking about type systems, it’s common to represent a type with $\tau$. The letter, which is the greek character “tau”, is used as a placeholder for some concrete type. It’s kind of like a template, to be filled in with an actual value. When we plug in an actual value into a rule containing $\tau$, we say we are instantiating it. Similarly, we will use $e$ to serve as a placeholder for an expression (matched by our $A_{add}$ grammar rule from part 2). Next, we have the typing relation, written as $e:\tau$. This says that “expression $e$ has the type $\tau$”.
Alright, this is enough to get us started with some typing rules. Let’s start with one for numbers. If we define $n$ to mean “any expression that is a just a number, like 3, 2, 6, etc.”, we can write the typing rule as follows:
$\frac{}{n : \text{Int}}$There’s nothing above the line 
there are no premises that are needed for a number to
have the type Int
.
Now, let’s move on to the rule for function application:
$\frac {e_1 : \tau_1 \rightarrow \tau_2 \quad e_2 : \tau_1} {e_1 \; e_2 : \tau_2}$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, 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? 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.
Since $\Gamma$ is just a regular set, we can write $x : \tau \in \Gamma$, meaning that in the current context, it is known that $x$ has the type $\tau$.
Let’s update our rules with this new addition.
The integer rule just needs a slight adjustment:
$\frac{}{\Gamma \vdash n : \text{Int}}$The same is true for the application rule:
$\frac {\Gamma \vdash e_1 : \tau_1 \rightarrow \tau_2 \quad \Gamma \vdash e_2 : \tau_1} {\Gamma \vdash e_1 \; e_2 : \tau_2}$And finally, we can represent the variable rule:
$\frac{x : \tau \in \Gamma}{\Gamma \vdash x : \tau}$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 of unification altogether: we use unification to find types that satisfy the rules.
Checking Case Expressions
So far, we’ve only checked types of numbers, applications, and variables. Our language has more than that, though!
Binary operators are by far the simplest to extend our language with;
We can simply say that (+)
is a function, Int > Int > Int
, and
x+y
is the same as (+) x y
. This way, we simply translate
operators into function application, and the same typing rules apply.
Next up, we have case expressions. This is one of the two places where we will introduce new variables into the context, and also a place where we will need several rules.
Let’s first take a look at the whole case expression rule:
$\frac {\Gamma \vdash e : \tau \quad \text{matcht}(\tau, p_i) = b_i \quad \Gamma,b_i \vdash e_i : \tau_c} {\Gamma \vdash \text{case} \; e \; \text{of} \; \{ (p_1,e_1) \ldots (p_n, e_n) \} : \tau_c }$This is a lot more complicated than the other rules we’ve seen, and we’ve used some notation that we haven’t seen before. Let’s take this step by step:

$e : \tau$, in this case, means that the expression between
case
andof
, is of type $\tau$. 
$\text{matcht}(\tau, p_i) = b_i$ means that the pattern $p_i$ can match a value of type
$\tau$, producing additional type pairs $b_i$. We need $b_i$ because a pattern
such as
Cons x xs
will introduce new type information, namely $\text{x} : \text{Int}$ and $\text{xs} : \text{List}$.  $\Gamma,b_i \vdash e_i : \tau_c$ means that each individual branch can be deduced to have the type $\tau_c$, using the previously existing context $\Gamma$, with the addition of $b_i$, the new type information.
 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 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:
$\frac {} {\text{matcht}(\tau, v) = \{v : \tau \}}$For the next rule, let’s define $c$ to be a constructor name. The rule for the constructor pattern, then, is:
$\frac {\Gamma \vdash c : \tau_1 \rightarrow ... \rightarrow \tau_n \rightarrow \tau} {\text{matcht}(\tau, c \; v_1 ... v_n) = \{ v_1 : \tau_1, ..., v_n : \tau_n \}}$This rule means that whenever we have a pattern in the form of a constructor applied to $n$ variable names, if the constructor takes $n$ arguments of types $\tau_1$ through $\tau_n$, then the each variable will have a corresponding type.
We didn’t include lambda expressions in our syntax, and thus we won’t need typing rules for them, so it actually seems like we’re done with the first draft of our type rules.
Implementation
Let’s work towards some code. Before we write anything down though, let’s get a definition of what a “type” is, in the context of our type checker. Let’s say a type is one of 3 things:
 A “base type”, like
Int
,Bool
, orList
.  A type that’s a function from one type to another.
 A placeholder / type variable (like the kind we used for type inference).
We represent a plceholder type with a unique string, such as “a”, or “b”, and this makes our placeholder type class very similar to the base type class.


As you can see, we also declared a type_mgr
, or type manager class.
This class will keep the state used for generating more placeholder
type names, as well as the information about which
placeholder type is mapped to what. We gave it 3 methods:

unify
, to perform unification. It will take two types and find values for placeholder variables such that they can equal. 
resolve
, to get to the “bottom” of a chain of equations. For instance, we have placeholdera
be mapped to a placeholderb
, an finally, the placeholderb
to be mapped toInt
.resolve
will return for usInt
, and, if the “bottom” of the chain is a placeholder, it will setvar
to be a pointer to that placeholder. 
bind
, inspired by this post, will map a type variable of some name to a type. This function will also check if we’re binding a type variable to itself, and do nothing in that case, sincea = a
is not a very useful equation to have.
To fit its original purpose, we also give the manager class the methods
new_type_name
, and two convenience methods to create placeholder types,
new_type
(in the form a
) and new_arrow_type
(in the form a>b
).
Let’s take a look at the implementation now:


Here, new_type_name
is actually pretty boring. My goal
was to generate type names like a
, then b
, eventually getting
to z
, and then move on to aa
. This provides is with an
endless stream of placeholder types.
Time for the interesting functions. resolve
keeps
trying dynamic_cast
to a type variable, and if that succeeds,
then either:
 It’s a type variable that’s already been set
to something, in which case we try resolve the thing it was
set to (
t = it>second
)  It’s a type variable that hasn’t been set to something.
We set
var
to it (the caller will use this info), and stop our resolution loop (break
).
In unify
, we start by calling resolve
 we don’t want
to accidentally compare a
and b
(and try to bind a
to
b
) when a
is already bound to something else (like Int
).
From resolve, we will have lvar
and rvar
set to
something not NULL if l
or r
were type variables
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
arrow types. If they’re base types, it compares their names,
and if they’re arrow types, it recursively unifies their children.
We return in all cases when unification succeeds, and then throw
an exception (currently 0) if all the cases fell thorugh, and thus,
unification failed.
Finally, bind
places the type we’re binding to into
the types
map, but not before it checks that the type
we’re binding is the same as the string we’re binding it to
(since, again, a=a
is not a useful equation).
We now have a unification algorithm, but we still
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
, our struct
for an expression tree. This
method will take an environment and return
a type.
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
that we only extend the environment in one case so far:
a case expression. In a case expression, we have the base
envrionment $\Gamma$, and for each branch,
we extend it with the bindings produced by
the pattern match. Each branch receives a modified
copy of the original environment, one that
doesn’t see the effects of the other branches.
Using our naive approach, we’d create a new std::map
for
each branch that’s a copy of the original environment,
and place into it the new pairs. But this means we’ll
need to copy a map for each branch of the pattern!
There’s a better way. We structure our environment like
a linked list. Each node in the linked list
contains an std::map
. When we encounter a new
scope (such as in a case branch), we create a new such node, and add all
variables introduced in this scope to that node’s map. We make
it point to our current environment. Then, we pass around the new node
as the environment.
When we look up a variable name, we first look in this node we created. If we don’t find the variable we’re looking for, we move on to the next node. The benefit of this is that we won’t be recreating a map for each branch, and just creating a node with the changes. Let’s implement exactly that. the header:


And the source file:


Nothing should seem too surprising. Of note is the fact
that we’re not using smart pointers for scope
,
and that the child we create during the call
would be invalid if the parent goes out of scope
/ is released. We’re gearing this towards
creating new environments on the stack, and we’ll
take care not to let a parent go out of scope
before the child.
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;
We also implement the $\text{matchp}$ function
as a method match
on pattern
with the following signature:
virtual void match(type_ptr t, type_mgr& mgr, type_env& env) const;
We declare this in every subclass of ast
. Let’s take a look
at the implementation now:


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 definitions. Further, we can’t look at the definitions in isolation. Consider these two functions:
defn double x = { x + x }
defn quadruple x = { double (double x) }
Assuming we have an environment containing x
when we typecheck the body
of double
, our algorithm will work out fine. But what about
quadruple
? It needs to know what double
is, or at least that it exists.
We could also envision two mutually recursive functions. Let’s
assume we have the functions eq
and if
in global scope. We can write
two functions, even
and odd
:
defn even x = { if (eq x 0) True (odd (x1)) }
defn odd x = { if (eq x 0) False (even (n1)) }
odd
needs to know about even
, and even
needs
to know about odd
. Thus, before we do any checking,
we need to populate a global environment with some
type for each function we declare. We will
use what we know about the function for our
initial declaration: if the function
takes two parameters, its type will be a > b > c
.
If it takes one parameter, its type will be a > b
.
What’s more, though, is that we need to make sure
that the function’s parameters are passed in the environment
when checking its body, and that these parameters’ types
are the same as the placeholder types in the function’s
“declaration”.
We’ll typecheck the program in two passes. First, we’ll go through each definition, and add any functions it declares to the global scope. Then, we will go through each definition again, and, if it’s a function, typecheck its body using the previously fleshed out global scope.
We’ll add two functions, typecheck_first
and typecheck_second
corresponding to
these two stages. Their signatures:
virtual void typecheck_first(type_mgr& mgr, type_env& env);
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 > ...
type signature from the right (>
is right associative), and
thus we’ll be creating the types righttoleft, too. We also
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.
Here’s the code for the implementation:


And finally, our updated main:


Notice that we manually add the types for our binary operators to the environment.
Let’s run our project on a few examples. On our two “bad” examples, we get the very eloquent error:
terminate called after throwing an instance of 'int'
[2] 9776 abort (core dumped) ./a.out < bad2.txt
That’s what we get for throwing 0.
So far, our program has thrown in 100% of cases. Let’s verify it actually accepts valid programs! We’ll try our very first example from today, as well as these two:
defn add x y = { x + y }
defn double x = { add x x }
defn main = { double 163 }
data List = { Nil, Cons Int List }
defn length l = {
case l of {
Nil > { 0 }
Cons x xs > { 1 + length xs }
}
}
All of our examples print the number of declarations in the program, which means they don’t throw 0. And so, we have typechecking! Before we look at how we will execute our source code, we will slow down and make quality of life improvements in our codebase in Part 4  Small Improvements.