ML Modules in C# - Sorely Missing Polymorphic Type Constructors

As Chung-chieh Shan pointed out, my encoding of modules in C# is somewhat limited. In particular, I cannot abstract over type constructors, which is to say, C# is missing generics over generics. Consider the Orc.NET interpreter:

class Orc {
  class Exp<T> { ... }

  public Exp<U> Seq<T,U>(Exp<T> e1, <U>) { ... }
  public Exp<T> Par<T>(Exp<T> e1, Exp<T>) { ... }
  public Exp<T> Where<T>(Exp<T> e1, Exp<Promise<T>>) { ... }
}

This is the result of my translation, which was necessitated by the "Where" method. Where introduces a dependency which currently cannot be expressed with ordinary C# constraints, so the module encoding is necessary.

The above interface is a direct, faithful implementation of the Orc semantics. The implementation I currently have is an interpreter for those semantics. What if I want to provide a compiler instead? The interface should remain the same, but the implementation should differ. This is a classic abstraction problem.

It's clear that the implementations can be changed at compile-time, but providing a different Orc implementation at runtime, something very natural for OO languages, does not seem possible. The reason is that C# lacks the requisite polymorphism.

A generic type is a type constructor, which means that from an abstract type definition, Exp<T>, you can construct a concrete type by providing a concrete type T, such as Exp<int>. But, if Exp<T> is not a concrete type until you provide it a concrete type T, what is the type of the abstract definition Exp<T>?

Reasoning about type constructors requires the introduction of something called "kinds". As types classify values, so kinds classify types. The set of values, or concrete types, is of kind *. The set of type constructors is of kind *⇒*, which is to say they are "compile-time functions" accepting a concrete type and producing a concrete type.

Now consider that we have multiple Exp<T> implementations, say Interpreter.Orc.Exp and Compiler.Orc.Exp, all with different innards, and all define the same operations and thus are theoretically interchangeable. We would somehow like to abstract over the different implementations of Exp<T> so that we can use whichever one is most appropriate. In other words, we want to make our code polymorphic over a set of generic types Exp<T>.

This necessitates type constructor constructors, the kind *⇒*⇒*, which accepts a type such as Orc.Compiler.Exp, and produces a type constructor Exp<T>.

At the moment, I can't think of a way to encode such polymorphism in C#. Functional languages provide this level of polymorphism, and some even provide higher-order kinds, meaning type constructors can be arguments to other type constructors, thus achieving entirely new levels of expressiveness.