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+---
+layout: blog
+title: Scaling DOT to Scala - Soundness
+author: Martin Odersky
+authorImg: /images/martin.jpg
+---
+
+In my [last
+blog post](http://www.scala-lang.org/blog/2016/02/03/essence-of-scala.html)
+I introduced DOT, a minimal calculus that underlies much of Scala.
+DOT is much more than an academic exercise, because it gives us
+guidelines on how to design a sound type system for full Scala.
+
+## Recap: The Problem of Bad Bounds
+
+As was argued in the previous blog post, the danger a path-dependent type
+system like Scala's faces is inconsistent bounds or aliases. For
+instance, you might have a type alias
+
+      type T = String
+
+in scope in some part of the program, but in another part the same
+type member `T` is known as
+
+      type T = Int
+
+If you connect the two parts, you end up allowing assigning a `String`
+to an `Int` and vice versa, which is unsound - it will crash at
+runtime with a `ClassCastException`. The problem is that there
+is no obvious, practical, compile time analysis for DOT or
+Scala that ensures that all types have good bounds. Types can contain
+abstract type members with bounds that can be refined elsewhere and
+several independent refinements might lead together to a bad bound
+problem.  Barring a whole program analysis there is no specific
+point in the program where we can figure this out straightforwardly.
+
+In DOT, the problem is resolved by insisting that every path prefix `p`
+of a type `p.T` is at runtime a concrete value. That way, we only have
+to check for good bounds when objects are _created_ with `new`, and
+that check is easy: When objects are created, we know their class and
+we can insist that all nested types in that class are aliases or
+have consistent bounds. So far so good.
+
+## Loopholes Caused by Scaling Up
+
+But if we want to scale up the DOT result for full Scala, several
+loopholes open up. These come all down to the fact that the prefix of
+a type selection might _not_ be a value that's constructed with a
+`new` at run time.  The loopholes can be classified into three
+categories:
+
+ 1. The prefix value might be lazy, and never instantiated to anything, as in:
+
+        lazy val p: S = p
+        ... p.T ...
+
+    Note that trying to access the lazy value `p` would result in an infinite loop. But using `p` in a type does not force its evaluation, so we might never evaluate `p`. Since `p` is not initialized with a `new`, bad bounds for `T` would go undetected.
+
+ 2. The prefix value might be initialized to `null`, as in
+
+        val p: S = null
+        ... p.T ...
+
+    The problem here is similar to the first one. `p` is not initialized
+    with a `new` so we know nothing about the bounds of `T`.
+
+ 3. The prefix might be a type `T` in a type projection `T # A`, where `T`
+    is not associated with a runtime value.
+
+We can in fact construct soundness issues in all of these cases. Look
+at the discussion for issues [#50](https://github.com/lampepfl/dotty/issues/50)
+and [#1050](https://github.com/lampepfl/dotty/issues/1050) in the
+[dotty](https://github.com/lampepfl/dotty/issues/1050) repository
+on GitHub. All issues work fundamentally in the same way: Construct a type `S`
+which has a type member `T` with bad bounds, say
+
+    Any <: T <: Nothing
+
+Then, use the left subtyping to turn an expression of type `Any` into
+an expression of type `T` and use the right subtyping to turn that
+expression into an expression of type `Nothing`:
+
+    def f(x: Any): p.T = x
+    def g(x: p.T): Nothing = x
+
+Taken together, `g(f(x))` will convert every expression into an
+expression of type `Nothing`. Since `Nothing` is a subtype of every
+other type, this means you can convert an arbitrary expression to have
+any type you choose. Such a feat is an impossible promise, of
+course. The promise is usually broken at run-time by failing with a
+`ClassCastException`.
+
+## Plugging the Loopholes
+
+To get back to soundness we need to plug the loopholes. Some of the
+necessary measures are taken in pull request [#1051](https://github.com/lampepfl/dotty/issues/1051).
+That pull request
+
+ - tightens the rules for overrides of lazy values: lazy values
+   cannot override or implement non-lazy values,
+ - tightens the rules which lazy values can appear in paths: they
+   must be final and must have concrete types with known consistent bounds,
+ - allows type projections `T # A` only if `T` is a concrete type
+   with known consistent bounds.
+
+It looks like this is sufficient to plug soundness problems (1) and
+(3). To plug (2), we need to make the type system track nullability in
+more detail than we do it now. Nullability tracking is a nice feature
+in its own right, but now we have an added incentive for implementing
+it: it would help to ensure type soundness.
+
+There's one sub-case of nullability checking which is much harder to do
+than the others. An object reference `x.f` might be `null` at run time
+because the field `f` is not yet initialized. This can lead to a
+soundness problem, but in a more roundabout way than the other issues
+we have identified. In fact, Scala guarantees that in a program that
+runs to completion without aborting, every field will eventually be
+initialized, so every non-null field will have good bounds. Therefore,
+the only way an initialized field `f` could cause a soundness problem
+is if the program in question would never get to initialize `f`,
+either because it goes into an infinite loop or because it aborts with
+an exception or `System.exit` call before reaching the initialization
+point of `f`. It's a valid question whether type soundness guarantees
+should extend to this class of "strange" programs. We might want to
+draw the line here and resort to runtime checks or exclude "strange"
+programs from any soundness guarantees we can give. The research community
+has coined the term [soundiness](http://soundiness.org/) for
+this kind of approach and has [advocated](http://cacm.acm.org/magazines/2015/2/182650-in-defense-of-soundiness/fulltext) for it.
+
+The necessary restrictions on type projection `T # A` are problematic
+because they invalidate some idioms in type-level programming. For
+instance, the cute trick of making Scala's type system Turing complete
+by having it [simulate SK
+combinators](https://michid.wordpress.com/2010/01/29/scala-type-level-encoding-of-the-ski-calculus/)
+would no longer work since that one relies on unrestricted type
+projections. The same holds for some of the encodings of type-level
+arithmetic.
+
+To ease the transition, we will continue for a while to allow unrestricted type
+projections under a flag, even though they are potentially
+unsound. In the current dotty compiler, that flag is a language import
+`-language:Scala2`, but it could be something different for other
+compilers, e.g. `-unsafe`.  Maybe we can find rules that are less
+restrictive than the ones we have now, and are still sound.  But one
+aspect should be non-negotiable: Any fundamental deviations from the
+principles laid down by DOT needs to be proven mechanically correct
+just like DOT was. We have achieved a lot with the DOT proofs, so we
+should make sure not to back-slide. And if the experience of the past
+10 years has taught us one thing, it is that the meta theory of type
+systems has many more surprises in store than one might think. That's
+why mechanical proofs are essential.
+
+
+
+
+
+
+