joanne3634 / class12

Notes and Code for Class 12

Class 12

Parametric Polymorphism and Generic Programming

We earlier discussed parametric polymorphism as a way to make a language more expressive, while still maintaining static type safety. A function or a class can be written such that it can handle values identically without depending on their type.

Generic programming is a style of programming in which algorithms are written in terms of types to-be-specified-later. These type dependencies are expressed using type parameters that are then instantiated when needed for specific types. In other words, generics allow you to abstract over types.

Why do we want such a language feature?

• We have a compiler for a reason. It finds bugs for us at compile time before we even run the program. Run-time bugs are generally much more difficult to find and resolve.

• Generic programming gives us another way of expressing some constraints to the compiler and therefore to prevent certain types of bugs.

• Generic programming also allows us to write less code, which is always a good thing.

Benefits:

• Stronger type checks at compile time: the compiler applies strong type checking to generic code and issues errors if the code violates type safety.

• Elimination of dynamic casts: values can be inserted and extracted from generic data structures without dynamic type checks.

• Enabling programmers to implement generic algorithms: programmers can implement generic algorithms that work on collections of different types, can be customized, and are type safe.

Scala Generics

Scala Generics provide an example of a generic programming language construct. Generics enable types to be parameters when defining classes and methods. Like formal parameters used in method declarations, type parameters provide a way for you to re-use the same Scala code with different types. Common examples are found in the Scala API.

For example, similar to OCaml's type 'a list that represents lists of elements of some type parameter 'a, Scala provides a type List[A] for representing lists of elements of some type parameter A. The following declares a list of Int values l:

  val l: List[Int] = List(1, 2, 3, 4)

A Generic Functional Queue Implementation

We study Scala generics by implementing a generic functional queue data structure similar to the one we implemented in OCaml. The following discussion builds on Ch. 19 of OSV, so I encourage you to read that chapter for additional details. Though, we will cover some orthogonal aspects of generics that are not covered in the book.

Our queue data structure supports three operations:

• enqueue to insert a new element at the back of the queue

• dequeue to remove the first element from the front of the queue

• and isEmpty to check whether the queue is empty.

Remember that a functional queue data structure differs from a queue that is implemented in an imperative style in that the enqueue and dequeue operations do not mutate the state of the queue object in-place, but instead return new queue objects that capture the updated state of the queue after the operation has been performed. The state of the original queue object remains unchanged. Such data structures are therefore also called persistent. This design choice suggests the following signature for our Queue class:

abstract class AbstractQueue[A] {
  def enqueue(x: A): AbstractQueue[A]

  def dequeue: (A, AbstractQueue[A])

  def isEmpty: Boolean
}

This class is generic in the type parameter A. This type parameter abstracts from the type of the elements stored in the queue. Note that dequeue returns a pair consisting of the dequeued element and a queue object that represents the new state of the queue holding the remaining elements of the queue.

The keyword abstract indicates that the class Queue cannot be instantiated directly (because we did not actually specify how the three methods are implemented).

A straightforward implementation of this data structure is to simply use a List[A] to represent the elements of the queue in dequeue order. Here is an implementation of this idea:

class Queue[A] private (
    private val queue: List[A]
  ) {
  
  def enqueue(x: A): Queue[A] = new Queue[A](queue :+ x)

  def dequeue: (A, Queue[A]) = {
    require(!isEmpty, "Queue.dequeue on empty queue")
    val x :: queue1 = queue
    (x, new Queue(queue1))
  }

  def isEmpty: Boolean = queue.isEmpty

  override def toString: String = {
    s"Queue${queue.toString.drop(4)}"
  }
}

The field queue that we use to hold the values stored in the queue is declared private so that the internal representation of a Queue instance is hidden from its clients. For the same reason, the primary constructor of class Queue is declared private. This is achieved by adding the keyword private in front of the parameter list of the class. As a consequence, only instances of class Queue and its companion object can construct Queue instances directly.

In the implementation of enqueue, the expression queue :+ x creates a new list by appending x at the end of the list queue, leaving queue itself unchanged. The implementation of dequeue first ensures that the queue is non-empty and then decomposes queue into its head x and tail queue1 using pattern matching and then returns x and the new Queue instance holding queue1. The implementation of isEmpty simply delegates the emptiness check to the list queue.

The companion object then provides two factory methods for constructing queues to hide the queue's internal representation:

object Queue {
  def empty[A]: Queue[A] = new Queue(Nil)

  def apply[A](xs: A*): Queue[A] = new Queue(xs.toList)
}

The type A* of the parameter xs in method apply indicates that apply can be called with a variable length list of parameters of type A. Here is how a client can use this data structure:

scala> val q = Queue(1, 2, 3, 4)
q: Queue[Int] = Queue(1,2,3,4)

scala> val (d, q1) = q.dequeue
d: Int = 1, q1: Queue[Int] = Queue(2,3,4)

scala> q.enqueue(5)
res1: Queue[Int] = Queue(1,2,3,4,5)

Observe that the data structure is indeed persistent: the call q.dequeue does not modify the state of q so that the subsequent call q.enqueue(5) adds the new element at the end of the original queue.

For a generic class such as Queue[A], Queue itself is not a type. Only the instances Queue[Int], Queue[String], ... of Queue[A] obtained by instantiating the type parameter A with another type are themselves types. Instead, Queue is called a type constructor. This is different to generics in Java where a generic class C[A] gives rise to a type C that is referred to as a raw type.

Type Erasure and Specialization

How are generics implemented by the Scala compiler? The Scala compiler uses a technique called type erasure. That is, the type parameter annotations in generic classes and methods are only needed at compile time when the program is type checked. Once the compiler has determined that all generic classes are implemented and used correctly, these annotations are erased from the program. Each value of some generic type A is simply treated as a value of type Any, i.e., as belonging to some arbitrary object type. Essentially, the type parameters go away and the compiler generates exactly the same byte code that we would get if we wrote the generic class using type Any instead of using a generic type parameter.

The advantage of this approach is that the compiler can generate a single byte code version of a generic class that can be shared by all its instantiations. In particular, if we call the enqueue method on a Queue[String] and a Queue[Int] in our program, the exact same byte code is executed by these two method calls at run time.

One disadvantage of type erasure is that generic classes cannot create instances of their generic type parameters. E.g., in our queue implementation we would not be allowed to write something like new A(). For this to work, the compiler would need to dynamically dispatch the constructor call to create the A instance. However, constructors are not dynamically dispatched.

Another disadvantage of type erasure is that the generated byte code uses the type Any to represent the values of the instantiated types uniformly for all instantiations. This is OK for actual object types since they are uniformly represented as pointers to heap allocated objects. However, for types that represent primitive values such as Int and Double which are allocated on the stack, the compiler needs to perform an implicit type conversion whenever a value of the type parameter is passed to a method of the generic class or retrieved from it. For instance, when we call q.enqueue(x) on a queue q of type Queue[Int] and an argument x of type Int, then the generic byte code generated for enqueue expects a 64-bit pointer to a heap allocated object as argument. However, x stores an Int value, which only takes 32 bits.

To perform the type conversion, the compiler uses a technique called auto boxing. Before passing x to enqueue, the compiler generates auxiliary glue code that creates a new heap allocated object and stores the value of x in a field of that object. Instead of passing x directly to enqueue, it instead passes the pointer to the wrapper object. Similarly, when we call q.dequeue, there will be auxiliary glue code that retrieves the wrapper object from the queue and unboxes it by extracting the contained Int value. While these conversions are opaque to the programmer, they incur a constant time and space overhead per conversion that you should be aware of.

If you write performance critical applications, you may want to avoid the additional cost of auto-boxing that is incurred by using type erasure on generic classes instantiated by value types. For instance, suppose we want to use our Queue data structure in performance critical code that instantiates Queue for the value types Int and Double. To avoid the performance overhead, we can tell the compiler to specialize the generated byte code for these two types by prefixing the type parameter in the generic class with the @specialized annotation as follows:

class Queue[@specialized(Int, Double) A] private (
    private val queue: List[A]
  ) {
  ...
}

Instead of generating a single byte code implementation of Queue that is shared by all types A, the compiler will now generate three versions: one for type Int, one for type Double, and one generic version for all remaining types. Since we have specialized byte code versions of Queue for types Int and Double, the values of these types can be passed to their Queue versions directly without auto-boxing, thus eliminating the performance overhead. All remaining value types (Boolean, Char, Byte, ...) will still fall back to the generic byte code version and require auto-boxing as before.

The disadvantage of specialization is that compilation time will increase. Moreover, if we are instantiating the generic class for different specialized types within the same program, then the JVM will have to load the byte code for each of those versions into memory at run-time. This incurs a constant space overhead. However, this overhead is usually negligible compared to the space overhead caused by auto-boxing. Therefore, most of the generic classes provided by the Scala API are specialized for all primitive value types.

Type erasure and specialization are common techniques used by compilers to implement parametric polymorphism in programming languages. For instance, both Java and C# compilers use type erasure to implement generics in the respective languages. On the other hand, functors in languages in the ML family and C++ templates, which is the programming feature that provides parametric polymorphism in C++, are implemented using specialization. That is, in OCaml and C++ each instantiation of a functor or template will yield a separate native code version that is specialized for the particular instantiation. The Scala compiler takes a middle way between the two approaches.

Variance

Next, we will study how generics relate to subtyping. For instance, suppose we have the types Duck, Sparrow, and Bird where Duck and Sparrow are subtypes of Bird. If we have a queue instance of type Queue[Duck], can we also use it in situations where we need an instance of type Queue[Bird]? That is, is Queue[Duck] a subtype of Queue[Bird]? If that's the case, then for example the following code would be OK:

val ducks: Queue[Duck] = Queue(new Duck)
val birds: Queue[Bird] = ducks.enqueue(new Sparrow)

More generally, suppose we have a generic class C[A] that parameterizes over a type parameter A. Suppose further that we have two concrete types S and T such that S is a subtype of T, S <: T. The question is: what does this tell us about the subtype relationship between C[S] and C[T]. We refer to this relationship as the variance of the type constructor C with respect to its type parameter A. We distinguish three cases:

• C[A] is covariant in A: if S <: T, then C[S] <: C[T]. That is, the subtype relationship between the argument types is preserved by the instantiation of C.

• C[A] is contravariant in A: if S <: T, then C[T] <: C[S]. That is, the subtype relationship between the argument types is inverted by the instantiation of C.

• C[A] is invariant in A: neither C[S] <: C[T] nor C[T] <: C[S] holds in general if S <: T. That is, there is no subtype relationship between the instantiations regardless of how S and T are related.

Whether C is covariant, contravariant, or invariant in its type parameter A depends on the implementation details of C. Covariant and contravariant generics provide additional flexibility to clients compared to generics that are invariant. For instance, if C[A] is covariant in A, clients can use a C[S] whenever a C[T] is expected. On the other hand, if C[A] is invariant in A, then this is not possible.

Since variance depends on implementation details, it is the programmer's responsibility to decide whether this property should be exposed to the clients of a generic class. The implementer of the generic class might decide against exposing the fact that the implementation is co- or contravariant to retain more flexibility for future changes to the implementation. For instance, a future optimization of the implementation might turn a covariant implementation into an invariant one, breaking client code that relied on the variance of the implementation. Functional implementations tend to be more resilient in terms of modifications that affect variance whereas implementations that expose state to the client are typically invariant.

By default the compiler treats all type parameters of a generic class as invariant. If the programmer decides that a particular type parameter should be treated covariantly, they can express this with a variance annotation by preceding the binding occurrence of the type parameter in the head of the class declaration with the symbol +. Likewise, if the type parameter is to be treated contravariantly, then it should be annotated with the symbol -. The compiler will statically check whether these variance annotations are indeed compatible with the implementation of the class.

Let us study these issues in more depth using our example of the functional queue Queue[A]. First, is the implementation of Queue[A] covariant, contravariant, or invariant in A? It is easy to see that it can't be contravariant. For otherwise, clients would be allowed to do the following:

val q: Queue[Any] = Queue[Any](1) // OK because Int <: Any
val qs: Queue[String] = q // OK because String <: Any and Queue is
                          // supposedly contravariant, i.e. Queue[Any] <: Queue[String]

val (s: String, _) = qs.dequeue // OK because qs is of type Queue[String]
s.charAt(0) // bzzzz! would try to call method charAt on an Int

On the other hand, we can argue formally that Queue is covariant in A. Suppose we have S <: T and a Queue[S] instance q. To see that q can be used in a context that views q as a Queue[T], we have to analyze the ways in which that context can interact with q. There are only two possible interactions that we need to consider:

1. the context calls q.dequeue: this call will return a pair (x, q1) where x is of type S and q1 is the remainder of the queue, which is again of type Queue[S]. Since the caller views q as a Queue[T] it expects a pair of type (T, Queue[T]). Since S <: T we know that x is also a T instance. Moreover, by our assumption that Queue is covariant, we can again conclude that q1 is also of type Queue[T] (formally, this argument uses a proof technique called coinduction). So the return value of the call is indeed of type (T, Queue[T]) as expected by the context.

2. the context calls q.enqueue(y) for some y of type T: first note that our implementation of enqueue does not modify the state of q, so q itself is unaffected by the fact that we pass a T instead of an S. Second, enqueue returns a new queue that stores y together with the elements stored in q. Since q is of type Queue[S], all elements stored in q.queue are of type S. Further, since S <: T, all those elements are also T instances. Thus, all elements in queue of the new queue are T instances and hence the new queue is of type Queue[T]. Therefore, this case is also OK.

As we can see, it is a non-trivial task to formally prove the correctness of the variance annotations in the interface of a generic class to ensure that the specified interface between the class' client and implementation code is consistent. Fortunately, the type checker in the Scala compiler automates this task for us.

Thus, let us see what the type checker thinks about our claim that Queue[A] is covariant in A. We can annotate the binding occurrence of A in Queue with a covariance annotation as follows:

class Queue[+A] private (
    private val queue: List[A]
  ) {
  ...
}

Running this through the type checker yields the following type error:

Error:(8, 15) covariant type A occurs in contravariant position in type A of value x
def enqueue(x: A) = new Queue[A](queue :+ x)
            ^

It seems that we missed something in our proof (attempt)! Before we analyze this problem further, let us first look at a simpler example to better understand the reasoning that is done by the type checker to prove the correctness of variance annotations.

Covariant and Contravariant Occurrences of Type Parameters

Consider the following simple generic class Cell[A] that describes objects providing read and write access to a mutable field content of type A:

class Cell[A](private[this] var content: A) {
  def read: A = content
  
  def write(x: A): Unit = {
    content = x
 }
}

Note that the access modifier private[this] specifies that Cell instances only have access to their own content field. That is neither the companion object of Cell nor other cell instances can access the content field of this.

We distinguish between covariant and contravariant type annotations within the implementation of a generic class (relative to the client interface). A type annotation is covariant if it describes values that may flow from an instance of the class to its client (e.g. the return type of a public method). An example of a covariant occurrence of A in Cell is the use of A in the return type of the method Cell.read. If on the other hand, the type annotation describes values that flow from the client to an instance of the class (e.g., a parameter type of a public method), then this annotation is contravariant. An example of a contravariant occurrence of A in Cell is the use of A in the type of the parameter x of method Cell.write.

You may also think of covariant occurrences as describing values that are read by the client from the instance and contravariant occurrences as describing values that are written by the client into the instance.

Intuitively, covariant type annotations express guarantees that the instances of the generic class promise their clients. From the clients' perspective, a covariant type annotation A can therefore be viewed as expressing lower bounds on types with respect to subtyping: the clients may safely assume that the values returned by method calls are an instance of any supertype of A. From the perspective of the implementation of the class, they express upper bounds with respect to subtyping: the method body must return an instance of some subtype of A. E.g. consider the types Mallard <: Duck <: Bird. If the return type of a method is Duck, the implementation may return a Mallard as a special kind of Duck, but it can't return an arbitrary Bird. Likewise, the client may treat the return value as an arbitrary Bird, but it can't assume that it is a Mallard if the method only promises to return a Duck.

On the other hand, contravariant type annotations express assumptions that the instances of the generic class make about values provided by the clients. Hence, they express lower bounds with respect to subtyping from the perspective of the implementation of the class. From the clients' perspective, they can be viewed as expressing upper bounds: the clients are required to provide values of the specified parameter types or any of their subtypes in method calls. E.g. if a method expects a parameter of type Duck, the client can provide a Mallard but not an arbitrary Bird. On the other hand, the method body may treat the parameter as an arbitrary Bird if the type of the parameter is Duck but it can't assume that the argument is a Mallard.

A covariant type parameter of a generic class is only allowed to occur covariantly in type annotations within the class and dually for contravariant type parameters. For instance, if we tried to make the type parameter A of Cell covariant, the compiler would produce the following type error:

error: (4, 12) covariant type A occurs in contravariant position in type A of value x
def write(x: A): Unit = {
          ^

To see why using covariant type parameters in contravariant positions is problematic, consider the following client code. This code would be well typed if we allowed Cell to be covariant in A:

val c: Cell[String] = new Cell("Hello")
val c1: Cell[Any] = c // OK because String <: Any and Cell is covariant
c1.write(1) // OK because c1 is of type Cell[Any] and 1: Int <: Any
c.read.charAt(0) // OK because c: Cell[String] and hence c.read: String

However, if we executed this code, then the third line would modify the contents of cell c since c and c1 alias the same instance, writing a boxed Int value 1 into the contents field of that Cell instance. Thus, on the fourth line we would be trying to call charAt on that boxed Int instance in the last line. However, Int has no such method. So when we would execute the usual instructions for a vtable look-up, in the best case, we would jump to a completely different method that happens to have its pointer stored in the same vtable slot as charAt in String's vtable and then start executing that method. In the worst case, we would do an out of bounds read on the vtable and jump to some random address in memory, starting to interpret the data stored at that point as if it were the instructions of a method.

Note that this situation is purely hypothetical. The compiler won't actually allow us to execute this code. It will refuse to compile the class Cell with a covariant type parameter A. However, if we remove the method write from Cell so that values of type A can only be read but not written, then Cell can be made covariant in A. Thus, the following code compiles:

class Cell[+A](private[this] var content: A) {
  def read: A = content
}

Likewise, if we keep write but remove read from Cell so that values of type A are only written but not read, then Cell becomes contravariant in A:

class Cell[-A](private[this] var content: A) {
  def write(x: A): Unit = {
    content = x
  }
}

If we keep both read and write, then Cell is invariant in A.

Note that covariance and contravariance annotations can also be used in combination within the same generic class (i.e. if a class has multiple type parameters, some of them may be covariant and others contravariant).

In general, in order to determine whether a generic class C[A] is covariant, contravariant, or invariant in its type parameter, we have to consider how A is being used within C because this determines how other objects can interact with the instances of C.

Each type expression T that occurs in a class C has an associated variance:

• If T is a parameter type of a method, then this occurrence of T is in contravariant position.

• If T is the return type of a method or the type of a val field, then this occurrence of T is in covariant position.

• If T is the type of a var field, then this occurrence of T is in invariant position.

• If T occurs in a larger type expression in the ith position below a type constructor C2[T1, ..., T, ..., TN] for some i between 1 and N, then

  • If C2 is covariant in its ith type parameter, then T occurs in a position of same variance as C2[T1, ..., T, ..., TN].

  • If C2 is contravariant in its ith type parameter, then T occurs in a position of opposite variance as C2[T1, ..., T, ..., TN] (e.g. if C2[T1, ..., T, ..., TN] occurs in contravariant position, then T occurs in covariant position).

  • If C2 is invariant in its ith type parameter, then T occurs in invariant position, irregardless of the variance of the position of C2[T1, ..., T, ..., TN].

For instance, consider again our implementation of Queue:

class Queue[A] private (
    private val queue: List[A]
  ) {
  ...
}

Here, C[A] is Queue[A] and consider the type expression List[A] in Queue. This type expression occurs as the type of a private val field. This occurrence of List[A] is in a covariant position. The type constructor List is covariant in its type parameter (see here). Hence, the occurrence of A below List is also in covariant position.

A type parameter A of a generic class C can be made covariant if it only appears in covariant positions within C and similarly, it can be made contravariant if it only appears in contravariant positions. In all other cases, A must be invariant.

Note that the above rules for determining variance of type expressions do not apply to instance private members. That is, if a member is instance private, then all the types in its declaration can be ignored when determining the variance of a type parameter of the surrounding class. This is because only the current instance itself can interact with these members. However, when we determine variance of a type parameter, we only need to concern ourselves with interactions that involve at least two distinct objects (i.e. one object accessing or calling a member of another object).

Lower Bounds on Type Parameters

Coming back to our implementation of Queue[A], we now understand why the compiler does not allow us to make the type parameter A covariant: the enqueue method takes an argument of type A and hence A occurs in a contravariant position.

However, we may wonder why the variance mismatch is problematic in this case. After all, unlike the case of Cell[A].write where the class relies on the fact that the client passes a A object to write, when a client calls Queue[A].enqueue, the queue instance does not really care whether the object being passed to enqueue belongs to a supertype of A instead of a subtype of A, as we argued in our earlier proof attempt.

The flaw in our reasoning is that we have ignored the fact that classes are open to extension by inheritance. That is, even though Queue[A].enqueue can be called safely with arguments that belong to a supertype of A, this may not be true for subclasses of Queue that override enqueue. As an example, suppose we added a method to the companion object of Queue that created a subclass of Queue[String] overriding enqueue with a new implementation that depends on enqueue being provided with a String value:

object Queue {
  ...
  
  class SillyQueue extends Queue[String](Nil) {
    override def enqueue(x: String): Queue[String] = {
      println(x.length) // relies on x being of type String
      super.enqueue(x) // delegate actual work to superclass
    } 
  }
  
  def silly: Queue[String] = new SillyQueue
}

Now, if Queue was allowed to be covariant in A, the following client code would compile:

val q: Queue[Any] = Queue.silly // OK because Queue.silly returns a
                                   // Queue[String] and Queue is covariant
q.enqueue(1) // OK because q: Queue[Any], 1: Int, and Int <: Any

At run-time, the call q.enqueue(1) would go to the overridden version of enqueue in the class SillyQueue. This version of enqueue would then try to access field length of an Int value. However, Int values do not have such a field.

In order to make Queue covariant in A, we need to modify the type signature of Queue[A].enqueue. As we observed, the implementation of enqueue does not actually rely on the fact that the provided argument is of type A. So we simply need to make this explicit in the type signature of enqueue. The overriding implementations of enqueue in the subclasses of Queue[A] then can no longer rely on their parameter being of type A either. We achieve this by making enqueue itself generic in its type parameter:

class Queue[+A] ( ... ) {
 ...
 
 def enqueue[B](x: B): Queue[?] = new Queue[?](queue :+ x)
}

We have replaced the type A of the parameter x in enqueue with a new type parameter B that is instantiated for each call to enqueue. Hence, B is a type parameter of the method enqueue.

Now A no longer occurs contravariantly. However, one question remains: what should be the element type ? of the queue returned by enqueue? It can't be A because x is of type B, which may not be a subtype of A. The only other sensible option is B. However, for this to work, the client needs to guarantee that B is a supertype of A for otherwise the list queue :+ x will not be of type List[B]. We can express this constraint by adding a lower bound on the type parameter B of enqueue:

class Queue[+A] ( ... ) {
 ...
 
 def enqueue[B >: A](x: B): Queue[B] = new Queue[B](queue :+ x)
}

The notation B >: A in the declaration of the type parameter B expresses that enqueue can only be called with values of types B that are supertypes of the element type A of the queue instance. Note that the types occurring in lower bound constraints are treated covariantly (why?).

With these modifications, we obtain a covariant implementation of Queue. In particular, the following client code will now compile and work as expected:

class A {
  override def toString: String = "A"
}

val q = new Queue(new A) // creates Queue[A]
val q1 = q.enqueue("Hello") // OK because there exists a common supertype of A and String: AnyRef
                            // That is, the compiler will infer type Queue[AnyRef] for q1
val (o, _) = q1.dequeue // o will now point to the A instance created earlier
println(o.toString) // OK because o: AnyRef and AnyRef has a toString method

Upper Bounds on Type Parameters

Similar to lower bounds, we can also add constraints to type parameters that express upper bounds with respect to subtyping. This is useful in cases where a generic class has certain minimal requirements on the capabilities of the types used for instantiating its type parameters. As an example consider a simple class hierarchy consisting of an abstract class Animal and concrete subclasses Cat, Dog, and Lion:

abstract class Animal {
 def makeNoise: Unit
}

class Cat extends Animal {
  override def makeNoise = println("Meow!")
}

class Dog extends Animal {
  override def makeNoise = println("Woof!")
}

class Lion extends Animal {
  override def makeNoise = println("Roar!")
}

Each subclass overrides the abstract method makeNoise of Animal with an implementation specific to that animal.

Suppose now that we want to implement a container for animals such that the container itself provides a method makeNoise, which simply calls makeNoise on all the contained animals. The following code realizes such an implementation:

class Animals[A <: Animal](private val animals: List[A]) {
  def makeNoise: Unit = for (a <- animals) a.makeNoise
}

The notation A <: Animal expresses an upper bound on the type parameter A of the generic container class Animals. This upper bounds guarantees that the call a.makeNoise in the implementation of Animals[A].makeNoise is always safe because we know that a must be an instance of some class that is a subtype of Animal.

The following client code then works as expected (the type annotations are only added for documentation - they would be automatically inferred by the compiler):

val dogs: Animals[Dog] = new Animals(List(new Dog))
  
dogs.makeNoise // prints Woof!
  
val cats: Animals[Cat] = new Animals(List(new Cat))
  
cats.makeNoise // prints Meow!
  
val zoo: Animals[Animal] = new Animals(List(new Dog, new Cat, new Lion))
  
zoo.makeNoise // prints Woof! Meow! Roar!