Martin Odersky
What do you get if you boil Scala on a slow flame and wait until all incidental features evaporate and only the most concentrated essence remains? After doing this for 8 years we believe we have the answer: it’s DOT, the calculus of dependent object types, that underlies Scala.
A paper on DOT will be presented in April at Wadlerfest, an event celebrating Phil Wadler’s 60th birthday. There’s also a prior technical report (From F to DOT) by Tiark Rompf and Nada Amin describing a slightly different version of the calculus. Each paper describes a proof of type soundness that has been machinechecked for correctness.
The DOT calculus
A calculus is a kind of minilanguage 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. machinechecked) proof:
If a term
t
has typeT
, and the evaluation oft
terminates, then the result of the evaluation will be a valuev
of typeT
.
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.

They give us a wellfounded 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
andB
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 asBoolean
andList
in DOT. 
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. 
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.

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.