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 [note: 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 reasoning about type coercions, which make expressions like "Hello"+3 valid. ] such as:

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.

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, and, and 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 data types, BoolExpr, ArithExpr, and finally, a more general Expr' that can be constructed from the first two. Operations such as and and or 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 Expr into a more safe Expr':

typecheck :: Expr -> Either TypeError (Expr', ExprType)


However, we typically also want the branches of an if-expression to both have the same type - if x then 3 else False may work sometimes, but not always, depending of the value of x. How do we encode this? Do we have two constructors, IfElseBool and 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 BoolExpr and ArithExpr, we would be able to have Expr BoolType and Expr IntType, which would share the IfElse constructor…

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:

 1 2 3 4  data ExprType = IntType | BoolType | StringType

Then, we can define a SafeExpr type family, which is indexed by ExprType. Here’s the [note: I should probably note that the definition of SafeExpr is that of a Generalized Algebraic Data Type, or GADT for short. This is what allows each of our constructors to produce values of a different type: IntLiteral builds SafeExpr IntType, while BoolLiteral builds SafeExpr BoolType. ] which we will discuss below:

 23 24 25 26 27  data SafeExpr : ExprType -> Type where IntLiteral : Int -> SafeExpr IntType BoolLiteral : Bool -> SafeExpr BoolType StringLiteral : String -> SafeExpr StringType BinOperation : (repr a -> repr b -> repr c) -> SafeExpr a -> SafeExpr b -> SafeExpr c

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 for SafeExpr. One such constructor is IntLiteral, which takes a value of type 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 BoolLiteral and 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 SafeExprs producing a and b, and combines the values of those expressions using the function to generate a value of type c. Since the whole expression returns c, BinOperation builds a value of type SafeExpr c.

That’s almost it. Except, what’s up with repr? We need it because SafeExpr is parameterized by a value of type ExprType. Thus, a, b, and c are all values in the definition of BinOperation. However, in a function input->output, both input and output have to be types, not values. Thus, we define a function repr which converts values such as IntType into the actual type that eval would yield when running our expression:

 6 7 8 9  repr : ExprType -> Type repr IntType = Int repr BoolType = Bool repr StringType = String

The power of dependent types allows us to run repr inside the type of 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 SafeExpr. 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 ExprType. But 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 integers (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 SafeExpr which depends on that ExprType. How about this:

 36  typecheck : Expr -> Either String (n : ExprType ** SafeExpr n)

Given an expression, we return either an error (String) or a dependent pair, which contains some ExprType n and a SafeExpr that evaluates to a value of type n. We can even start implementing this function, starting with literals:

 37 38 39  typecheck (IntLit i) = Right (_ ** IntLiteral i) typecheck (BoolLit b) = Right (_ ** BoolLiteral b) typecheck (StringLit s) = Right (_ ** StringLiteral s)

Note the use of _. Since 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 BinOperation 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 Expr to SafeExpr, we need to convert an Op to a corresponding function. As we can see with "hello"*3 and 163*2, an Op can correspond to a different function depending on the types of its inputs. To deal with this, we define a new function typecheckOp, which takes an Op and two expression types, and returns either an error (if the Op can’t 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:

 29 30 31 32 33 34  typecheckOp : Op -> (a : ExprType) -> (b : ExprType) -> Either String (c : ExprType ** repr a -> repr b -> repr c) typecheckOp Add IntType IntType = Right (IntType ** (+)) typecheckOp Subtract IntType IntType = Right (IntType ** (-)) typecheckOp Multiply IntType IntType = Right (IntType ** (*)) typecheckOp Divide IntType IntType = Right (IntType ** div) typecheckOp _ _ _ = Left "Invalid binary operator application"

When (+) 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 (+). The 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 typecheck function:

 40 41 42 43 44  typecheck (BinOp o l r) = do (lt ** le) typecheck l (rt ** re) typecheck r (ot ** f) typecheckOp o lt rt pure (_ ** BinOperation f le re)

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 o, to typecheckOp to determine the output type and the applicable evaluation function. Finally, we assemble the new SafeExpr from the function and the two translated subexpressions.

Alright, we’ve done all this work. Is it worth it? Let’s try implementing eval:

 46 47 48 49 50  eval : SafeExpr t -> repr t eval (IntLiteral i) = i eval (BoolLiteral b) = b eval (StringLiteral s) = s eval (BinOperation f l r) = f (eval l) (eval r)

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:

 52 53 54 55 56 57 58 59 60 61 62 63 64  resultStr : {t : ExprType} -> repr t -> String resultStr {t=IntType} i = show i resultStr {t=BoolType} b = show b resultStr {t=StringType} s = show s tryEval : Expr -> String tryEval ex = case typecheck ex of Left err => "Type error: " ++ err Right (t ** e) => resultStr $eval {t} e main : IO () main = putStrLn$ tryEval \$ BinOp Add (IntLit 6) (BinOp Multiply (IntLit 160) (IntLit 2))

And the output is:

>>> idris TypesafeIntr.idr -o typesafe
>>> ./typesafe
326


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


Awesome!

### Wrapping Up

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.