This term, I’m a TA for Oregon State University’s Programming Languages course.
The students in the course are tasked with using Haskell to implement a programming
language of their own design. One of the things they can do to gain points for the
project is implement type checking, rejecting
Whether or not the below example is ill-typed actually depends on your language.
Many languages (even those with a static type system, like C++ or Crystal)
have a notion of "truthy" and "falsy" values. These values can be used
in the condition of an if-expression, and will be equivalent to "true" or "false",
respectively. However, for simplicity, I will avoid including
truthy and falsy values into the languages in this post. For the same reason, I will avoid
which make expressions like
if "Hello" then 0 else 1
For instance, a student may have a function
typecheck, with the following
signature (in Haskell):
typecheck :: Expr -> Either TypeError ExprType
The function will return an error if something goes wrong, or, if everything goes well, the type of the given expression. So far, so good.
A student asked, however:
Now that I ran type checking on my program, surely I don’t need to include errors in my [note: I'm using "valuation function" here in the context of denotational semantics. In short, a valuation function takes an expression and assigns to it some representation of its meaning. For a language of arithmetic expression, the "meaning" of an expression is just a number (the result of simplifying the expression). For a language of booleans,
or, the "meaning" is a boolean for the same reason. Since an expression in the language can be ill-formed (like
list(5)in Python), the "meaning" (semantic domain) of a complicated language tends to include the possibility of errors. ] I should be able to make my function be of type
Expr -> Val, and not
Expr -> Maybe Val!
Unfortunately, this is not quite true. It is true that if the student’s type checking
function is correct, then there will be no way for a type error to occur during
the evaluation of an expression “validated” by said function. The issue is, though,
that the type system does not know about the expression’s type-correctness. Haskell
doesn’t know that an expression has been type checked; worse, since the function’s type
indicates that it accepts
Expr, it must handle invalid expressions to avoid being partial. In short, even if we know that the
expressions we give to a function are type safe, we have no way of enforcing this.
A potential solution offered in class was to separate the expressions into several
ArithExpr, and finally, a more general
Expr' that can
be constructed from the first two. Operations such as
will then only be applicable to boolean expressions:
data BoolExpr = BoolLit Bool | And BoolExpr BoolExpr | Or BoolExpr BoolExpr
It will be a type error to represent an expression such as
true or 5. Then,
Expr' may have a constructor such as
IfElse that only accepts a boolean
expression as the first argument:
data Expr' = IfElse BoolExpr Expr' Expr' | ...
All seems well. Now, it’s impossible to have a non-boolean condition, and thus,
this error has been eliminated from the evaluator. Maybe we can even have
our type checking function translate an unsafe, potentially incorrect
a more safe
typecheck :: Expr -> Either TypeError (Expr', ExprType)
However, we typically also want the branches of an if-expression to both have the same
if x then 3 else False may work sometimes, but not always, depending of the
x. How do we encode this? Do we have two constructors,
IfElseInt, with one in
BoolExpr and the other in
ArithExpr? What if we add strings?
We’ll be copying functionality back and forth, and our code will suffer. Wouldn’t it be
nice if we could somehow tag our expressions with the type they produce? Instead of
ArithExpr, we would be able to have
Expr BoolType and
which would share the
It’s not easy to do this in canonical Haskell, but it can be done in Idris!
Enter Dependent Types
Idris is a language with support for dependent types. Wikipedia gives the following definition for “dependent type”:
In computer science and logic, a dependent type is a type whose definition depends on a value.
This is exactly what we want. In Idris, we can define the possible set of types in our language:
Then, we can define a
SafeExpr type family, which is indexed by
I should probably note that the definition of
SafeExpr is that of
Generalized Algebraic Data Type,
or GADT for short. This is what allows each of our constructors to produce
values of a different type:
which we will discuss below:
The first line of the above snippet says, “
SafeExpr is a type constructor
that requires a value of type
ExprType”. For example, we can have
SafeExpr IntType, or
SafeExpr BoolType. Next, we have to define constructors
SafeExpr. One such constructor is
IntLiteral, which takes a value of
Int (which represents the value of the integer literal), and builds
a value of
SafeExpr IntType, that is, an expression that we know evaluates
to an integer.
The same is the case for
StringLiteral, only they build
values of type
SafeExpr BoolType and
SafeExpr StringType, respectively.
The more complicated case is that of
BinOperation. Put simply, it takes
a binary function of type
a->b->c (kind of), two
and combines the values of those expressions using the function to generate
a value of type
c. Since the whole expression returns
builds a value of type
That’s almost it. Except, what’s up with
repr? We need it because
is parameterized by a value of type
all values in the definition of
BinOperation. However, in a function
output have to be types, not values.
Thus, we define a function
repr which converts values such as
the actual type that
eval would yield when running our expression:
The power of dependent types allows us to run
repr inside the type
BinOp to compute the type of the function it must accept.
Now, we have a way to represent expressions that are guaranteed to be type safe.
With this, we can make our
typeheck function convert an
Expr to a
Wait a minute, though! We can’t just return
SafeExpr: it’s a type constructor!
We need to somehow return
SafeExpr a, where
a is a value of type
it doesn’t make sense for the return type to have a new type variable that didn’t
occur in the rest of the type signature. It would be ideal if we could return both
the type of the expression, and a
SafeExpr of that type.
In fact, we can!
Idris has something called dependent pairs, which are like normal pairs, but in which the type of the second element depends on the value of the first element. The canonical example of this is a pair of (list length, list of that many elements). For instance, in Idris, we can write:
listPair : (n : Nat ** Vec n Int)
In the above snippet, we declare the type for a pair of a natural number (
n : Nat) and a list of
Vect n Int), where the number of elements in the list is equal to the natural number. Let’s
try applying this to our problem. We want to return an
ExprType, and a
depends on that
ExprType. How about this:
Given an expression, we return either an error (
String) or a dependent pair, which
n and a
SafeExpr that evaluates to a value of type
We can even start implementing this function, starting with literals:
Note the use of
IntLiteral always produces
SafeExpr IntType, we allow
Idris to infer that
IntType must be the first element of the tuple. This is easy
enough, because a boolean, integer, or string literal can never be type-incorrect.
The interesting case is that of a binary operation. Is
"hello" * 3 invalid? It might
be, but some languages evaluate the multiplication of a string by a number as repeating
the string that many times:
"hellohellohello". It is up to us, the language designers,
to specify the set of valid operations. Furthermore, observe that
takes a function as its first argument, not an
Op. To guarantee that we can,
in fact, evaluate a
BinOperation to the promised type, we require that the means
of performing the evaluation is included in the expression. Thus, when we convert
SafeExpr, we need to convert an
Op to a corresponding function. As we can see
Op can correspond to a different function
depending on the types of its inputs. To deal with this, we define a new function
which takes an
Op and two expression types, and returns either an error (if the
be applied to those types) or a dependent pair containing the output type of the operation
and a function that performs the required computation. That’s a mouthful; let’s look at the code:
(+) is applied to two integers, this is not an error, and the result is also
an integer. To perform addition, we use Idris’ built-in function
same is true for all other arithmetic operations in this example. In all other
cases, we simply return an error. We can now use
typecheckOp in our
Here, we use do-notation to first type check first the left, then the right subexpression.
Since the result of type checking the subexpressions gives us their output types,
we can feed these types, together with
typecheckOp to determine the output
type and the applicable evaluation function. Finally, we assemble the new
from the function and the two translated subexpressions.
Alright, we’ve done all this work. Is it worth it? Let’s try implementing
That’s it! No
Maybe, no error cases.
eval is completely total, but doesn’t require
error handling because it knows that the expression it is evaluating is type-correct!
Let’s run all of this. We’ll need some code to print the result of evaluating an expression. Here’s all that:
And the output is:
>>> idris TypesafeIntr.idr -o typesafe
That’s right! What about a type-incorrect example?
BinOp Add (IntLit 6) (BinOp Multiply (IntLit 160) (StringLit "hi"))
The program reports:
Type error: Invalid binary operator application
In this post, we learned that type checking can be used to translate an expression into a more strongly-typed data type, which can be (more) safe to evaluate. To help strengthen the types of [note: You may be thinking, "but where did the if-expressions go?". It turns out that making sure that the branches of an if-expression are of the same type is actually a fairly difficult task; the best way I found was enumerating all the possible "valid" combinations of types in a case-expression. Since this is obviously not the right solution, I decided to publish what I have, and look for an alternative. If I find a better solution, I will write a follow-up post. ] we used the Idris language and its support for dependent types and Generalized Algebraic Data Types (GADTs). I hope this was interesting!
As usual, you can find the code for this post in this website’s Git repository. The source file we went through today is found here.