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When can Liskov be lifted?

Scalaz avoids variance in the sense of the Scala type parameter annotation, with its associated higher-kind implications, except where it has historically featured variance; even here, variance is vanishing as unsoundness in its released implementations is discovered.

There is a deeply related concept in Scalaz’s typeclasses, though: covariant and contravariant functors. Functor is traditional shorthand for covariant functor, whereas Contravariant represents contravariant functors.

These concepts are related, but neither subsumes the other. A Functor instance does not require its parameter to be Scala-covariant. A type can be Scala-covariant over a parameter without having a legal Functor instance.

Liskov

Liskov, also known as <~< and very close to Scala’s own <:<, represents a subtyping relationship, and is defined by the ability to lift it into Scala-covariant and Scala-contravariant parameter positions, like so:

def liftCo[F[+_], A, B](a: A <~< B): F[A] <~< F[B]
def liftCt[F[-_], A, B](a: A <~< B): F[B] <~< F[A]

As Liskov is, soundly, Scala-variant, this can be implemented without a cast. However, it can only be called with Scala-covariant F.

By definition, applying an A <~< B to a value of type A should yield a value of type B, but must also do nothing but return the value; in other words, it is an operational identity. Despite the limitation of liftCo, for functorial values that are parametrically sound, even for Scala-invariant F, it is operationally sound to lift Liskov, though impossible to implement without exploiting Scala soundness holes:

def liftCvf[F[_]: Functor](a: A <~< B): F[A] <~< F[B]

For example, this is sound for scalaz.IList.

But IList[Int] isn’t a subtype of IList[Any]!

Sure, as far as Scala is concerned. But Liskov is all about making claims that can’t directly be proven due to the language’s limitations. Haskell allows you to constrain functions with type equalities, which is very important when working with type families; Scala doesn’t, so we get Leibniz instead.

A type is a set of values. Where Y is a supertype of X, every value in X is in Y. Since IList[String]("hi", "there") has the same representation as IList[Any]("hi", "there"), they are the same value. This is true for all IList[String]s, but the opposite is not true; therefore, IList[Any] is an IList[String] supertype, regardless of what Scala knows.

So doing a casting Liskov lift, like that into IList, is essentially “admitted” in a proof system sense. You are saying, “I can’t prove that this subtype relationship holds, but it does, so assume it.”

To decide whether an admitted A <~< B is sound: suppose that the compiler admits that subtyping relationship. Can it then draw incorrect conclusions, about the sets of values, derived from that assumption? This is the cardinal rule.

By extension, to decide whether an F permits Liskov lifting: does the above rule pass given F[A] <~< F[B] for all A, B where B is a supertype of A?

Parametrically sound covariance

Because a Liskov must be an operational identity, it is essential that, given any value of F[A], for all supertypes B of A, the representation of F[B] must be identical. You can determine this by analyzing the subclasses of F as an algebraic data type, where the key test is to ensure that A never appears in the primitive contravariant position: as the parameter to a function. This test is not quite enough to prove that Liskov lifting is sound, but it gets us most of the way.

For example, an IList of "hi" and "there" has exactly the same representation whether you instantiated the IList with String or with Any. So that is a good first test. If a class changes its construction behavior based on manifest type information, or its basic data construction functions violate the rules of parametricity, that is a good sign that the data type cannot follow these rules.

This data type analysis is recursive: a data type being variant in a parametrically sound way over a parameter requires that all appearances of that parameter in elements of your data type are also parametrically sound in that way. For example, if your F[A] contains an IList[A] in its representation, you may rely on IList’s parametrically sound covariance when considering F’s.

Any var, or var-like thing such as an Array, places its parameter in an invariant position, because it features a getter (return type) and setter (parameter type). So its presence in the data model invalidates Liskov lifting if the type parameter appears within it.

Obviously, runtime evidence of a type parameter’s value eliminates the possibility of lifting Liskov over that parameter.

You cannot perform this representation analysis without considering all subclasses of a class under consideration. For example, considering only HashSet, collection.immutable.Set appears to allow Liskov lifting. However, TreeSet, a subclass of Set, contains a function (A, A) => Ordering. If any representation contains a contradiction like this, Liskov lifting is unsafe. You cannot constrain Liskov application by a runtime test.

If you permit open subclassing, you must either declare the requirement to preserve parametric covariance, or accept that it will be violated, and so forbid Liskov lifting.

Data that doesn’t use a type parameter doesn’t affect its parametric soundness. For example, here A is invariant, but B is covariant:

final case class VA[A, B](xs: Set[A], ys: IList[B])

GADTs

Some features of Scala resist simple ADT analysis, so must be considered separately from the above. Despite their sound covariance considering only the representational rules in the previous section, they still break the cardinal rule by allowing the compiler to make invalid assumptions about the sets of values. A “recoverable phantom” implies a type relationship that forbids Liskov-lifting, for example:

sealed trait Gimme[A]
case object GimmeI extends Gimme[Int]

In pattern matching, given a Gimme[A] over unknown A, matching GimmeI successfully recovers the type equality A ~ Int; therefore, Liskov-lifting is unsound for Gimme. For example, lifting Int <~< Any, applying to GimmeI, and matching, gives us Any ~ Int, which is nonsense.

We can reason about this type equality as a value member of GimmeI of type Leibniz[⊥, ⊤, A, Int], which places A in a representationally invariant position.

Some other GADTs invalidate covariance. For example:

sealed trait P[A]
case class PP[A, B](a: P[A], b: P[B]) extends P[(A, B)]

The pattern match of a P[A] to PP[_,_] can theoretically determine A ~ (x, y) forSome {type x; type y}, so Liskov cannot be lifted into P.

However, not all GADTs invalidate Liskov-lifting:

sealed trait AM[A]
case class FAM[A, B](fa: AM[A], f: A => B) extends AM[B]

Matching AM[A] to FAM[_,_] reveals nothing about A; its use of GADTs only introduces a new existential unrelated to A. Considering only B, as the A parameter is called in FAM, its covariance is sound in FAM, so Liskovs can be lifted into AM.

Contravariance

Liskovs can also be lifted into parametrically sound contravariant positions. This looks a bit like:

def liftCtf[F[_]: Contravariant, A, B](a: A <~< B): F[B] <~< F[A]

Analysis of parametrically sound contravariance is essentially the same as that for covariance. The only difference is that, for F[A], A can only appear in the primitive contravariant position: the function parameter type.

With regard to recursion, the “flipping” behavior of Scala-contravariance applies. For example, this data type is soundly contravariant over A:

final case class IOf[A](f: IList[A] => Unit)

IList is soundly covariant over A, and IList[A] appears in soundly contravariant position, making A contravariant. Meanwhile, A is soundly covariant in this data type built upon IOf:

final case class IOf2[A](f: IOf[IOf[A]])

Some surprises

Despite the unsoundness of Liskov-lifting into Gimme earlier, it may seem surprising that Scala allows:

sealed trait GimmeC[+A]
case object GimmeCI extends Gimme[Int]

Moreover, this isn’t a bug; it’s perfectly sound. That is because, while matching GimmeI causes Scala to infer A ~ Int, it won’t do that for GimmeCI! Scala can soundly determine that A ⊇ Int when it matches GimmeCI, but I do not think it even goes so far as to do that as of this writing. We can’t blame Scala for this difference; Scala has declared up front that its type system encodes what it believes, and is our responsibility to follow the cardinal rule of not violating its assumptions if we lift Liskov into Gimme.

As stated earlier, Liskov cannot be lifted into collection.immutable.Set; TreeSet exists to trivially demonstrate the problem, but even if TreeSet was not there, we would not be able to honestly do it because c.i.Set is open to new subclasses that could perform similar violations. However, despite lacking a Functor, scalaz.ISet does allow Liskov-lifting. Do the ADT analysis yourself, if you like. Well, so, once you convert your ISet[Int] to ISet[Any], you can’t do many operations on it, but that’s neither here nor there.

Should this function exist?

The Scalaz community has settled on a definition of covariant-functoriality that conforms with the principle of parametric soundness. The rejection of Functor instances for the scala.collection.* classes, which have subclasses with mutable values over their parameters, and for collection.immutable.Set, which has the TreeSet case stated above and violates parametricity in the construction of HashSets, speak to this. As far as I know, Scalaz contains no Functors that are both Scala-invariant and violate the rules delineated above.

So how do you feel about the provision of a combinator of the type of liftCvf for Scalaz’s Functor?

Licensing

Unless otherwise noted, all content is licensed under a Creative Commons Attribution 3.0 Unported License.

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