blog-static/content/blog/11_compiler_polymorphic_data_types.md

title	date	draft	tags
Compiling a Functional Language Using C++, Part 11 - Polymorphic Data Types	2020-03-28T20:10:35-07:00	true	C and C++ Functional Languages Compilers

[In part 10]({{< relref "10_compiler_polymorphism.md" >}}), we managed to get our compiler to accept functions that were polymorphically typed. However, a piece of the puzzle is still missing: while our functions can handle values of different types, the same cannot be said for our data types. This means that we cannot construct data structures that can contain arbitrary types. While we can define and use a list of integers, if we want to also have a list of booleans, we must copy all of our constructors and define a new data type. Worse, not only do we have to duplicate the constructors, but also all the functions that operate on the list. As far as our compiler is concerned, a list of integers and a list of booleans are entirely different beasts, and cannot be operated on by the same code.

To make polymorphic data types possible, we must extend our language (and type system) a little. We will now allow for something like this:

data List a = { Nil, Cons a List }

In the above snippet, we are no longer declaring a single type, but a collection of related types, parameterized by a type a. Any type can take the place of a to get a list containing that type of element. Then, List Int is a type, as is List Bool and List (List Int). The constructors in the snippet also get polymorphic types:

{{< latex >}} \text{Nil} : \forall a ; . ; \text{List} ; a \ \text{Cons} : \forall a ; . ; a \rightarrow \text{List} ; a \rightarrow \text{List} ; a {{< /latex >}}

When you call Cons, the type of the resulting list varies with the type of element you pass in. The empty list Nil is a valid list of any type, since, well, it's empty.

Let's talk about List itself, now. I suggest that we ponder the following table:

\(\text{List}\)	\(\text{Cons}\)
\(\text{List}\) is not a type; it must be followed up with arguments, like \(\text{List} \; \text{Int}\).	\(\text{Cons}\) is not a list; it must be followed up with arguments, like \(\text{Cons} \; 3 \; \text{Nil}\).
\(\text{List} \; \text{Int}\) is in its simplest form.	\(\text{Cons} \; 3 \; \text{Nil}\) is in its simplest form.
\(\text{List} \; \text{Int}\) is a type.	\(\text{Cons} \; 3 \; \text{Nil}\) is a value of type \(\text{List} \; \text{Int}\).

I hope that the similarities are quite striking. I claim that List is quite similar to a constructor Cons, except that it occurs in a different context: whereas Cons is a way to create values, List is a way to create types. Indeed, while we call Cons a constructor, it's typical to call List a type constructor. We know that Cons is a function which assigns to values (like 3 and Nil) other values (like Cons 3 Nil, or [3] for short). In a similar manner, List can be thought of as a function that assigns to types (like Int) other types (like List Int). We can even claim that it has a type:

{{< latex >}} \text{List} : \text{Type} \rightarrow \text{Type} {{< /latex >}}

{{< sidenote "right" "dependent-types-note" "Unless we get really wacky," >}} When your type constructors take as input not only other types but also values such as 3, you've ventured into the territory of dependent types. This is a significant step up in complexity from what we'll be doing in this series. If you're interested, check out Idris (if you want to know about dependent types for functional programming) or Coq (to see how propositions and proofs can be encoded in a dependently typed language). {{< /sidenote >}} our type constructors will only take zero or more types as input, and produce a type as output. In this case, writing \(\text{Type}\) becomes quite repetitive, and we will adopt the convention of writing \(*\) instead. The types of such constructors are called kinds. Let's look at a few examples, just to make sure we're on the same page:

• The kind of \(\text{Bool}\) is \(*\): it does not accept any type arguments, and is a type in its own right.
• The kind of \(\text{List}\) is \(*\rightarrow *\): it takes one argument (the type of the things inside the list), and creates a type from it.
• If we define a pair as data Pair a b = { MkPair a b }, then its kind is \(* \rightarrow * \rightarrow *\), because it requires two parameters.

As one final observation, we note that effectively, all we're doing is tracking the arity of the constructor type.

Let's now enumerate all the possible forms that (mono)types can take in our system:

1. A type can be a placeholder, like \(a\), \(b\), etc.
2. A type can be a type constructor, applied to {{< sidenote "right" "zero-more-note" "zero ore more arguments," >}} It is convenient to treat regular types (like \text{Bool}) as type constructors of arity 0 (that is, type constructors with kind *). In effect, they take zero arguments and produce types (themselves). {{< /sidenote >}} such as \(\text{List} ; \text{Int}\) or \(\text{Bool}\).
3. A function from one type to another, like \(\text{List} \; a \rightarrow \text{Int}\).

Polytypes (type schemes) in our system can be all of the above, but may also include a "forall" quantifier at the front, generalizing the type (like \(\forall a \; . \; \text{List} \; a \rightarrow \text{Int}\)).

Let's start implementing all of this. Why don't we start with the change to the syntax of our language? We have complicated the situation quite a bit. Let's take a look at the old grammar for data type declarations (this is going back as far as [part 2]({{< relref "02_compiler_parsing.md" >}})). Here, \(L_D\) is the nonterminal for the things that go between the curly braces of a data type declaration, \(D\) is the nonterminal representing a single constructor definition, and \(L_U\) is a list of zero or more uppercase variable names:

{{< latex >}} \begin{aligned} L_D & \rightarrow D ; , ; L_D \ L_D & \rightarrow D \ D & \rightarrow \text{upperVar} ; L_U \ L_U & \rightarrow \text{upperVar} ; L_U \ L_U & \rightarrow \epsilon \end{aligned} {{< /latex >}}

This grammar was actually too simple even for our monomorphically typed language! Since functions are not represented using a single uppercase variable, it wasn't possible for us to define constructors that accept as arguments anything other than integers and user-defined data types. Now, we also need to modify this grammar to allow for constructor applications (which can be nested!) To do so, we will define a new nonterminal, \(Y\), for types:

{{< latex >}} \begin{aligned} Y & \rightarrow N ; ``\rightarrow" Y \ Y & \rightarrow N \end{aligned} {{< /latex >}}

We make it right-recursive (because the \(\rightarrow\) operator is right-associative). Next, we define a nonterminal for all types except those constructed with the arrow, \(N\).

{{< latex >}} \begin{aligned} N & \rightarrow \text{upperVar} ; L_Y \ N & \rightarrow \text{typeVar} \ N & \rightarrow ( Y ) \end{aligned} {{< /latex >}}

The first of the above rules allows a type to be a constructor applied to zero or more arguments (generated by \(L_Y\)). The second rule allows a type to be a placeholder type variable. Finally, the third rule allows for any type (including functions, again) to occur between parentheses. This is so that higher-order functions, like \((a \rightarrow b) \rightarrow a \rightarrow a \), can be represented.

Unfortunately, the definition of \(L_Y\) is not as straightforward as we imagine. We could define it as just a list of \(Y\) nonterminals, but this would make the grammar ambigous: something like List Maybe Int could be interpreted as "List, applied to types Maybe and Int", and "List, applied to type Maybe Int". To avoid this, we define a "type list element" \(Y'\), which does not take arguments:

{{< latex >}} \begin{aligned} Y' & \rightarrow \text{upperVar} \ Y' & \rightarrow \text{lowerVar} \ Y' & \rightarrow ( Y ) \end{aligned} {{< /latex >}}

We then make \(L_Y\) a list of \(Y'\):

{{< latex >}} \begin{aligned} L_Y & \rightarrow Y' ; L_Y \ L_Y & \rightarrow \epsilon \end{aligned} {{< /latex >}}

Finally, we update the rules for the data type declaration, as well as for a single constructor:

{{< latex >}} \begin{aligned} T & \rightarrow \text{data} ; \text{upperVar} ; L_T = { L_D } \ D & \rightarrow \text{upperVar} ; L_Y \ \end{aligned} {{< /latex >}}

Now that we have a grammar for all these things, we have to implement the corresponding data structures. We define a new family of structs, extending parsed_type, which represent types as they are received from the parser. These differ from regular types in that they do not require that the types they represent are valid; validating types requires two passes, which is a luxury we do not have when parsing. We can define them as follows:

{{< codeblock "C++" "compiler/11/parsed_type.hpp" >}}

We define the conversion function to_type, which requires a set of type variables quantified in the given type, and the environment in which to look up the arities of various type constructors. The implementation is as follows:

{{< codeblock "C++" "compiler/11/parsed_type.cpp" >}}

With this definition in hand, we can now update the grammar in our Bison file. First things first, we'll add the type parameters to the data type definition:

{{< codelines "plaintext" "compiler/11/parser.y" 127 130 >}}

Next, we add the new grammar rules we came up with:

{{< codelines "plaintext" "compiler/11/parser.y" 138 163 >}}

Finally, we define the types for these new rules at the top of the file:

{{< codelines "plaintext" "compiler/11/parser.y" 43 44 >}}

{{< todo >}} Nullary is not the right word. {{< /todo >}}