The DOT calculus

A calculus is a kind of mini-language that is small enough to be studied formally. Translated to Scala notation, the language covered by DOT is described by the following abstract grammar:

Value       v  =  (x: T) => t            Function
                  new { x: T => ds }     Object

Definition  d  =  def a = t              Method definition
                  type A = T             Type

Term        t  =  v                      Value
                  x                      Variable
                  t1(t2)                 Application
                  t.a                    Selection
                  { val x = t1; t2 }     Local definition

Type        T  =  Any                    Top type
                  Nothing                Bottom type
                  x.A                    Selection
                  (x: T1) => T2          Function
                  { def a: T }           Method declaration
                  { type T >: T1 <: T2 } Type declaration
                  T1 & T2                Intersection
                  { x => T }             Recursion

The grammar uses several kinds of names:

x      for (immutable) variables
a      for (parameterless) methods
A      for types

The full calculus adds to this syntax formal typing rules that assign types T to terms t and formal evaluation rules that describe how a program is evaluated. The following type soundness property was shown with a mechanized, (i.e. machine-checked) proof:

If a term t has type T, and the evaluation of t terminates, then the result of the evaluation will be a value v of type T.

Difficulties

Formulating the precise soundness theorem and proving it was unexpectedly hard, because it uncovered some technical challenges that had not been studied in depth before. In DOT - as well as in many programming languages - you can have conflicting definitions. For instance, you might have an abstract type declaration in a base class with two conflicting aliases in subclasses:

trait Base { type A }
trait Sub1 extends Base { type A = String }
trait Sub2 extends Base { type A = Int }
trait Bad extends Sub1 with Sub2

Now, if you combine Sub1 and Sub2 in trait Bad you get a conflict, since the type A is supposed to be equal to both String and Int. If you do not detect the conflict and assume the equalities at face value you get String = A = Int, hence by transitivity String = Int! Once you are that far, you can of course engineer all sorts of situations where a program will typecheck but cause a wrong execution at runtime. In other words, type soundness is violated.

Now, the problem is that one cannot always detect these inconsistencies, at least not by a local analysis that does not need to look at the whole program. What's worse, once you have an inconsistent set of definitions you can use these definitions to "prove" their own consistency - much like a mathematical theory that assumes true = false can "prove" every proposition including its own correctness.

The crucial reason why type soundness still holds is this: If one compares T with an alias, one does so always relative to some path x that refers to the object containing T. So it's really x.T = Int. Now, we can show that during evaluation every such path refers to some object that was created with a new, and that, furthermore, every such object has consistent type definitions. The tricky bit is to carefully distinguish between the full typing rules, which allow inconsistencies, and the typing rules arising from runtime values, which do not.

Why is This Important?

There are at least four reasons why insights obtained in the DOT project are important.

1. They give us a well-founded explanation of nominal typing. Nominal typing means that a type is distinguished from others simply by having a different name. For instance, given two trait definitions

   trait A extends AnyRef { def f: Int }
   trait B extends AnyRef { def f: Int }
   

   we consider A and B to be different types, even though both traits have the same parents and both define the same members. The opposite of nominal typing is structural typing, which treats types that have the same structure as being the same. Most programming languages are at least in part nominal whereas most formal type systems, including DOT, are structural. But the abstract types in DOT provide a way to express nominal types such as classes and traits. The Wadlerfest paper contains examples that show how one can express classes for standard types such as Boolean and List in DOT.

2. They give us a stable basis on which we can study richer languages that resemble Scala more closely. For instance, we can encode type parameters as type members of objects in DOT. This encoding can give us a better understanding of the interactions of subtyping and generics. It can explain why variance rules are the way they are and what the precise typing rules for wildcard parameters [_ <: T], [_ >: T] should be.

3. DOT also provides a blueprint for Scala compilation. The new Scala compiler Dotty has internal data structures that closely resemble DOT. In particular, type parameters are immediately mapped to type members, in the way we propose to encode them also in the calculus.

4. Finally, the proof principles explored in the DOT work give us guidelines to assess and treat other possible soundness issues. We now know much better what conditions must be fulfilled to ensure type soundness. This lets us put other constructs of the Scala language to the test, either to increase our confidence that they are indeed sound, or to show that they are unsound. In my next blog I will present some of the issues we have discovered through that exercise.