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When are two methods alike?

This is the second of a series of articles on “Type Parameters and Type Members”. If you haven’t yet, you should start at the beginning, which introduces code we refer to throughout this article without further ado.

In the last part, we just saw two method types that, though different, are effectively the same: those of plengthT and plengthE. We have rules for deciding when an existential parameter can be lifted into a method type parameter—or a method type parameter lowered to an existential—but there are other pairs of method types I want to explore that are the same, or very close. So let’s talk about how we determine this equivalence.

A method R is more general than or as general as Q if Q may be implemented by only making a call to R, passing along the arguments. By more general, we mean R can be invoked in all the situations that Q can be invoked in, and more besides. Let us call the result of this test $R <:_m Q$ (where $<:_m$ is pronounced “party duck”); if the test of Q making a call to R fails, then $\neg(R <:_m Q)$.

If $Q <:_m R$ and $R <:_m Q$, then the two method types are equivalent; that is, neither has more expressive power than the other, since each can be implemented merely by invoking the other and doing nothing else. We write this as $Q \equiv_m R$. Likewise, if $R <:_m Q$ and $\neg(Q <:_m R)$, that is, Q can be written by calling R, but not vice versa, then R is strictly more general than Q, or $R <_m Q$.

What the concrete method—the one actually doing stuff, not invoking the other one—does is irrelevant, for the purposes of this test, because this is about types. That matters because sometimes, in Scala, as in Java, the body will compile in one of the methods, but not the other. Let’s see an example that doesn’t compile.

import scala.collection.mutable.ArrayBuffer

def copyToZero(xs: ArrayBuffer[_]): Unit =
  xs += xs(0)

TmTp2.scala:9: type mismatch;
 found   : (some other)_$1(in value xs)
 required: _$1(in value xs)
    xs += xs(0)
            ^

Likewise, the Java version has a similar problem, though the error message doesn’t give as good a hint as to what’s going on.

import java.util.List;

void copyToZero(final List<?> xs) {
    xs.add(xs.get(0));
}

TmTp2.java:11:  error: no suitable method found for add(CAP#1)
        xs.add(xs.get(0));
          ^

Luckily, in both Java and Scala, we have an equivalent method type, from lifting the existential (misleadingly called wildcard in Java terminology) to a method type parameter.

We can apply this transformation to put the method implementation somewhere it will compile.

def copyToZeroE(xs: ArrayBuffer[_]): Unit =
  copyToZeroP(xs)

private def copyToZeroP[T](xs: ArrayBuffer[T]): Unit =
  xs += xs(0)

Similarly, in Java,

void copyToZeroE(final List<?> xs) {
    copyToZeroP(xs);
}

<T> void copyToZeroP(final List<T> xs) {
    final T zv = xs.get(0);
    xs.add(zv);
}

The last gives a hint as to what’s going on, both here and in the compiler errors above: in copyToZeroP’s body, the list element type has a name, T; we can use the name to create variables, and the compiler can rely on the name as well. The compiler, ideally, shouldn’t care about whether the name can be written, but that one of the above compiles and the other doesn’t is telling.

If you were to define a variable to hold the result of getting the first element in the list in either version of copyToZeroE, how would you do that? In Java, the reason this doesn’t work is straightforward: you would have to declare the variable to be of type Object, but that type isn’t specific enough to allow the variable to be used as an argument to xs.add.

Scala’s type-inferred variables don’t help here; Scala considers the existential type to be scoped to xs, and makes the definition of zv independent of xs by breaking the type relationship, and crushing the inferred type of zv to Any.

def copyToZeroE(xs: ArrayBuffer[_]): Unit = {
  val zv = xs(0)
  xs += zv
}

TmTp2.scala:19: type mismatch;
 found   : zv.type (with underlying type Any)
 required: _$1
    xs += zv
          ^

When we call the type-parameterized variant to implement the existential variant, with the real implementation residing in the former, we are just helping the compiler along by using the equivalent method type; in the simpler case of the former, both scalac and javac manage to infer that the type T should be the (otherwise unspeakable) existential. Method equivalence and generality make it possible to write methods, safely, that could not be written directly.

Why are existentials harder to think about?

I think we, as humans, may have even more difficulty with the lack of names for existentials than the compilers do. The name “unspeakable”, which I have borrowed from Jon Skeet’s C# in Depth, is telling: even in our heads, our thought processes are shaped by language. We tame the mathematics of programming with symbols, with names. Existentials and their “unspeakable” names rob us of the tools to talk about them, to think about them.

Java has done its practitioners two great disservices here. One: by calling its existentials “wildcards”. They are not “wildcards”, in any commonly or uncommonly understood sense. If you suppose your preexisting notions of “wildcards” to apply to these much more exotic creatures, you will confidently stroll into the darkness until you trip and fall off a cliff. They are only superficially “wildcards”. The effect of this sorry attempt at avoiding new terminology is chiefly to cheat Java programmers out of learning what’s really going on. (We will explore some of this more exotic behavior in a later post.)

Two: by encouraging use of existential signatures like mdropFirstE over parameterized versions like mdropFirstT that do not require the same kind of mental gymnastics.

For lifting these type parameters is how we can reclaim the power we lost in the debacle of the unspeakable names. We name them, and in so doing can once more talk and think about them without exhausting ourselves by gesticulating wildly, comforting ourselves with fairytales of “wildcards”. Because in parameter lifting, we have found a true analogy.

When are two methods less alike?

Now, let’s examine another pair of methods, and apply our test to them.

Let’s say we want to write the equivalent of this method for MList.

def pdropFirst[T](xs: PList[T]): PList[T] =
  xs match {
    case PNil() => PNil()
    case PCons(_, t) => t
  }

According to the PList ⇔ MList conversion rules given in the previous article, section “Why all the {type T = ...}?”, the equivalent for MList should be

def mdropFirstT[T0](xs: MList {type T = T0})
  : MList {type T = T0} =
  xs.uncons match {
    case None => MNil()
    case Some(c) => c.tail
  }

Let us try to drop the refinements. That seems to compile:

def mdropFirstE(xs: MList): MList =
  xs.uncons match {
    case None => MNil()
    case Some(c) => c.tail
  }

It certainly looks nicer. However, while mdropFirstE can be implemented by calling mdropFirstT, passing the type parameter xs.T, the opposite is not true; mdropFirstT $<_m$ mdropFirstE, or, mdropFirstT is strictly more general.

In this case, the reason is that mdropFirstE fails to relate the argument’s T to the result’s T; you could implement mdropFirstE as follows:

def mdropFirstE[T0](xs: MList): MList =
  MCons[Int](42, MNil())

The stronger type of mdropFirstT forbids such shenanigans. However, I can just tell you that largely because I’m already comfortable with existentials; how could you figure that out if you’re just starting out with these tools? You don’t have to; the beauty of the equivalence test is that you can apply it mechanically. Knowing nothing about the mechanics of the parameterization and existentialism of the types involved, you can work out with the equivalence test that mdropFirstT $<_m$ mdropFirstE, and therefore, that you can’t get away with simply dropping the refinements.

Method likeness and subtyping, all alike

If you know what the symbol <: means in Scala, or perhaps you’ve read SLS §3.5 “Relations between types”, you might think, “gosh, method equivalence and generality look awfully familiar.”

Indeed, the thing we’re talking about is very much like subtyping and type equality! In fact, every type-equal pair of methods M₁ and M₂ also pass our method equivalence test, and every pair of methods M₃ and M₄ where $M_3 <: M_4$ passes our M₄-calls-M₃ test. So $M_1 \equiv M_2$ implies $M_1 \equiv_m M_2$, and $M_3 <: M_4$ implies $M_3 <:_m M_4$.

We even follow many of the same rules as the type relations. We have transitivity: if M₁ can call M₂ to implement itself, and M₂ can call M₃ to implement itself, obviously we can snap the pointer and have M₁ call M₃ directly. Likewise, every method type is equivalent to itself: reflexivity. Likewise, if a method M₁ is strictly more general than M₂, obviously M₂ cannot be strictly more general than M₁: antisymmetricity. And we even copy the relationship between ≡ and <: themselves: just as $T_1 \equiv T_2$ implies $T_1 <: T_2$, so $R \equiv_m Q$ implies $R <:_m Q$.

Scala doesn’t understand the notion of method equivalence we’ve defined above, though. So you can’t, say, implement an abstract method in a subclass using an equivalent or more general form, at least directly; you have to override the Scala way, and call the alternative form yourself, if that’s what you want.

I do confess to one oddity in my terminology: the method that has more specific type is the more general method. I hope the example of mdropFirstT $<:_m$ mdropFirstE justifies my choice. mdropFirstT has more specific type, and rejects more implementations, such as the one that returns a list with 42 in it above. Thus, it has fewer implementations, in the same way that more specific types have fewer values inhabiting them. But it can be used in more circumstances, so it is “more general”. The generality in terms of when a method can be used is directly proportional to the specificity of its type.

Java’s edge of insanity

Now we have enough power to demonstrate that Scala’s integration with Java generics is faulty. Or, more fairly, that Java’s generics are faulty.

Consider this method type, in Scala:

def goshWhatIsThis[T](t: T): T

This is a pretty specific method type; there are not too many implementations. Of course you can always perform a side effect; we don’t track that in Scala’s type system. But what can it return? Just t.

Specifically, you can’t return null:

TmTp2.scala:36: type mismatch;
 found   : Null(null)
 required: T
  def goshWhatIsThis[T](t: T): T = null
                                   ^

Well now, let’s convert this type to Java:

public static <T> T holdOnNow(T t) {
    return null;
}

We got away with that! And, indeed, we can call holdOnNow to implement goshWhatIsThis, and vice versa; they’re equivalent. But the type says we can’t return null!

The problem is that Java adds an implicit upper bound, because it assumes generic type parameters can only have class types chosen for them; in Scala terms, [T <: AnyRef]. If we encode this constraint in Scala, Scala gives us the correct error.

def holdOnNow[T <: AnyRef](t: T): T = TmTp2.holdOnNow(t)

def goshWhatIsThis[T](t: T): T = holdOnNow(t)

TmTp2.scala:38: inferred type arguments [T] do not conform
 to method holdOnNow's type parameter bounds [T <: AnyRef]
  def goshWhatIsThis[T](t: T): T = holdOnNow(t)
                                   ^

This is forgivable on Scala’s part, because it’d be annoying to add <: AnyRef to your generic methods just because you called some Java code and it’s probably going to work out fine. I blame null, and while I’m at it, I blame Object having any methods at all, too. We’d be better off without these bad features.

In the next part, “What happens when I forget a refinement?”, we’ll talk about what happens when you forget refinements for things like MList, and how you can avoid that while simplifying your type-member-binding code.

This article was tested with Scala 2.11.7 and Java 1.8.0_45.

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

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