The Expression Problem in Java (Litterature Review)

Previously: The Visitor Pattern in Java 8

Last time I presented a way to implement the visitor pattern, by taking advantage of Java 8's default interface methods.

In the process I said this was a partial solution to the expression problem, which was defined as:

The goal is to define a datatype by cases, where one can add new cases to the datatype and new functions over the datatype, without recompiling existing code, and while retaining static type safety (e.g., no casts).

Recall that in the context of Java, we can think of a datatype as an interface or parent class, and of a case as a class implementing/extending the interface or parent. When using this interpretation we will call the cases "data classes", which is a bit less awkward.

On the other hand, some of the papers we will review will take another interpretation in order to produce an interesting solution.

The solution I presented last time is partial, because it is not strictly type-safe: it uses a cast.

Today, I want to look at the solutions that have been proposed in the litterature, and try to extract their guiding insights, and show their respective strengths and shortcomings.

The Contenders

While the litterature on the expression problem in Java-like object-oriented languages is surprisingly rich, I want to focus specifically on three papers which I think covers the space of interesting solutions:

Since the first paper actually presents four solutions, that gives us 6 solutions to review. I'll also throw my partial solution into the mix for comparison's sake.

The Problem's Raison d'Être: Ambiguous Call Sites

For the expression problem to be interesting at all, it has to involve ambiguous call sites: the same piece of code has to perform a method call which could be dispatched to a specialized method for any of datatype cases.

Said otherwise, if every piece of code is statically typed and doesn't involve any kind of polymorphism (e.g. inheritance or generics), then plain static overloading is enough, and you don't have a problem in the first place.

Therefore, to build a type-safe solution to the expression problem, two big avenues are open.

The first one has to be built into the compiler: the compiler will check that implementations of operations (which can be added by anyone, not just the original author of the datatype) exist for every data class. But this doesn't seem to exist. I said as much in the previous post:

In theory, there is nothing that prevents solving the expression problem at the language level. In an ideal world, we'd just be able to add abstract extension methods that have to be implemented for all classes implementing the interface. The linker would then verify that these methods were implemented for all such classes, and generate the proper virtual method tables. But no such object-oriented language exists.

The second avenue is to somehow carry the specialized implementation to the call sites. This is what every solution in the litterature does, each in its own way.

That is also how typeclasses work in Haskell. In this case, it's the typeclass instances that carry the operation's implementation to the call site.

In light of this, the statement of the expression problem is somewhat problematic, because it doesn't specify which shape the ambiguous call sites can take.

But I can think of two very interesting examples.

The first one is to apply one of the specialized methods on a list of data class instances, whose exact type is not known (said otherwise, which just know they are instances of the datatype).

Sadly, no type-safe solution in the litterature can do that. Our solution can, but again, it isn't type safe.

Interestingly, Haskell can do this only if using a compiler extension introducing existential types. An existential type is basically just a pair of a type and its associated typeclass instance for a given typeclass. The existential type just says "here is an instance of something that has an instance for the given typeclass". Then you have to use a list of existentially-typed values — which crucially you mean you can't reuse a pre-existing list that isn't existentially-typed. There has to be a way to (statically) retrieve the correct typeclass instance when constructing the list.

The second example is, fortunately, the one that is always used as a benchmark in the litterature: a tree structure where each node is a data class instance.

This example is easier because it is possible to inject type information while building the tree — something that is not possible with plain lists, but is exactly what we're doing when we're building an existentially-typed list.

The Benchmark Problem

In particular, the typical example uses trees that represent arithmetic expressions.

This benchmark example was there since the beginning, and is certainly partially responsible for the name of the expression problem.

Our datatype is an Exp type.

The cases for the datatype are:

Initially, we'll have a single operation: Print which prints a string representation of the expression to standard output. Later we'll add Eval, which evaluates the expression.

Norswap's Solution

To ease us into the problem, let's see a type-unsafe solution to the problem using my formulation of the visitor pattern.

Compared to the previous post, the solution has been simplified/crippled a little bit for the sake of brevity and better comparison. We no longer use implementation interfaces, which allow the composition of independently developped extensions (i.e. new data classes or operations).

None of Torgersen's solutions can handle composition. This is excusable, as our trick (using default methods in interfaces) wasn't available at the time the paper was written.

The Choice for Data-Structure Solutions

The nature of the expression problem is that each time we add a new data class, we need to add corresponding implementations for the existing operations. Conversely, each time we add a new operation, we need to implement it for all existing data classes.

Unfortunately, it's not as simple as just writing them. The "compiler" solution that neatly composes everything for us isn't available. Therefore, we will have to take care of the plumbing ourselves.

As long as we keep one dimension fixed, everything is easy. If we have a fixed set of operations, they can be encoded as an interface which we can just implement. If we have a fixed set of data classes, the simple visitor pattern suffices, and we can just implement the visitor interface to add new operations.

Things become tricky when we need to add both new operations and new data classes.

There are fundamentally two ways to do this.

Option 1: replacing the data classes. Each time we add a new operation, we need to extend all data classes so that they may handle the new operation. Operation's implementation will live inside the data classes.

This option means we need to control/parameterize the creation of our data structure. Whenever we add a new operation, we need to swap the classes that are being instantiated!

Option 2: replacing the operations. Each time we add a new data class, we need to extend all existing operations so that they may handle the new data class. Operation's implementation will typically live in some kind of visitor implementation.

This options means we need to control/parameterized the operation's call sites. Whenever we add a new data class, we need to swap the object that holds the operation's implementations, lest it doesn't work for the new data class.

My solution uses option 2.

Torgersen's 1st Solution: Data-Centered

This is the first solution in the "The Expression Problem Revisited: Four new solutions using generics" (link) paper by Mads Torgersen. Discussion of the other three solutions will follow.

This is a solution that takes option 1 from the last section: replacing the data classes. When adding a new operation, we subclass all existing data classes. The code that create data strutures needs to be parameterized.

There are two difficulties in this solution that not readily apparent when option 1 is stated briefly.

First, in order to make the solution type-safe, it is necessary to know which operations the nodes in the expression tree implement. This means there needs to be someway to "carry the type" to the nodes that are down in the tree.

In this solution, this is done via generics, and in particular the use of a F-bound: C extends Exp<C>. F-bounds are a crude way to encode "self-types" in Java. Basically it lets us use C as though it meant "the type of this class" (or, like here, the type of one of its superclasses or superinterfaces). However, to use an F-bound, you need to "fix" C. This is the role of all the classes whose name end with F, such as class LitF extends Lit<ExpF> implements ExpF. Unfortunately, that makes the solution more verbose as we need to actually add in all of these F classes.

The second difficulty — which is only hinted at in the paper — is the need to carry the node creation logic to the places where you would normally call a data class constructor. Since there may be a lot of different types of nodes, it makes sense to collect the creators in a factory.

The issue with that is that each time you add a new data case you need to extend the existing factory. Each time you add a new operation, you not only need to extend every data class, but also to create a whole new factory that return instances of these new classes.

So this works, but it's quite verbose and it's relatively annoying that we actually need to change the data classes being used when we add new operations.

Torgersen's 2nd Solution: Operation-Centered

This solution takes option 2 outlined above. Each time we add a new data class, we need to extend all existing operations so that they may handle the new data class. Operation's implementation live in a visitor implementation.

This is also what my own solution does, I will explain the difference below.

However, there is a big pitfall that comes from the need to be type-safe. For the initial data classes, there are no issues. But when a new data class is added, it is necessary to add a new visitor interface. This is as expected.

However, each data class instance has to know the requirements on the visitors it can handle. To use our previous examples, a Neg node can only handle visitors that implement NegVisitor.

But it doesn't stop there. If an Add node has a Neg child, it too should only accept visitors that implement NegVisitor — since they can invoke the visitor on their children.

Again, generics come to the rescue: we parameterize all data classes with <V extends Visitor> (for the initial data classes) or <V extends NegVisitor> (for Neg — same principle would apply if we added new data classes).

This doesn't entirely fix the problem. In the visit methods, it wouldnt work to call, for instance, add.left.accept(this). Why? Because there is no guarantee that this has type V.

Torgersen comes up with a really neat trick to solve this issue, which is to let the visitor accept itself as an extra parameter of type V. This parameter will be supplied by the accept methods: visitor.visit(visitor, this) (where this is a an instance of a data class such as Add<V>). Since visitor has type V there, this type-checks ok.

The cost? Once again, we can't reuse our expression trees. They now have to be parameterized differently depending on added data classes. So creation logic has to be parameterized by the proper visitor interface (note you can't really see this is our simplistic demo code). At least, we don't need verbose factories this time.

Torgersen's 3rd Solution: Operation-Centered with Object-Level Extensibility

This solution is an extension of the second solution. The goal is to relax the requirement on the control of all instantiation sites.

From the perspective of the previous solution, the goal here is to make it possible to reuse old Add<Visitor> and Lit<Visitor> trees that were instantiated without knowing that Neg existed. These node will still work if passed a NegVisitor (which extends Visitor)!

The author calls this ability to reuse old trees object-level extensibility.

And since the old trees were created before we added Neg, they couldn't contain Neg nodes, and so using plain Visitor implementations is fine as well.

Achieving object-level extensibility is actually pretty easy. In the data classes, just change the node's children type to Exp<? super V>. Without entering into the details, this means that Add<NegVisitor> may have children that with type Exp<Visitor> or Exp<NegVisitor>. On the other hand Add<Visitor> may not have a child of type Exp<NegVisitor>

This effectively enables reusing old trees in newer trees.

There is only one catch: your ability to rewrite the trees becomes limited. Since Add<Visitor> may not have children of type Exp<NegVisitor> this may hamper your ability to write involved tree rewrite logic that would need to assign a newer tree as a child of an older tree.

However, as the author correctly notes, there are plenty of use-cases (maybe even most of them) that do not involve such kind of tree rewriting.

If type-safety is a must-have for you and you don't need tree rewrites, this is pretty good. You'll still pay a cost of sorts by having to carry these annoyings type parameters everywhere.

Torgersen's 4th Solution: Type-Unsafe Hybrid

This one is interesting too.

Torgersen starts from the an operation-centered visitor solution (much like his second solution) but pairs each operation (i.e. visitor implementation) with an interface that defines the signature of the operation. Data classes can choose to implement this interface. If they do so, the operation will notice (via an instanceof check) and call the implementation — otherwise it falls back on the visitor pattern.

The solution isn't type-safe because Torgersen opts not to force the children of each data class node to encode their visitor interface. So instead of being typed as Exp<V> or Exp<? extends V> as in solution 2 and 3 respectively, they are simply typed as Exp.

The payload of foregoing type-safety here is that control over the creation logic is no longer necessary. You can finally have data classes whose types and implementations don't change depending on subsequent extensions. In that, it is similar to my solution.

If you know in advance all operations you need to implement, you can also avoid extending existing operations when you introduce a new data class, by having the data class implement the operations' associated interfaces.

Because the lack of type safety, when an expression accepts a visitor, it must verify that this visitor actually can handle the expression's data class or fall back on some general behaviour (at worse, throw a runtime exception, which is what my solution does).

To make all of this work, the solution involves some helper super-classes, which can be slightly confusing.

I'd also argue that the most use of generics in this solution is woefully unecessary — it just saves on inlining two lines of logic into every visitor, which you sort of have to do anyway because Java doesn't support instanceof on generic type arguments (which are erased). Hence I made a simplified solution that eliminates non-essential generics use, and simplifies the scaffolding considerably, making it much easier to understand, in my humble opinion.

Compared to my solution, this one is more complicated, but has the important benefit that data classes can be added without extending all operations individually, greatly reducing verbosity in that scenario.

Object Algebras

We now discussion the solution from the "Extensibility for the Masses: Practical Extensibility with Object Algebras" paper (link). This one is quite different from those we discussed previously, and conforms to neither of our two options — because it doesn't encode data as a data structure at all!

Instead, data is encoded as a tree of method calls. Here is an example:

interface Algebra<E> {
    E lit (int value);
    E add (E left, E right);

public static <E> E expression (Algebra<E> a) {
    return a.add(a.lit(1), a.lit(2)); // 1 + 2

The expression method encodes the expression tree 1 + 2 made of an "add node" with two "literal node" child. Of course, there are no such nodes — it's just a method!

To do anything useful with expression, we need to supply an Algebra<E> where E is an (unconstrainted) type used to represent the result of an operation on one of our "nodes". So for instance, a.lit(1) will return a value of type E. The add method returns a value of type E, but also takes as parameter two values of type E, corresponding to the result of "evaluating" its two operands.

To define an operation, we need to implement Algebra<E>. Here is a full implementation that uses the same example as previously:

So there our operations are printing and evaluation. We actually use two different techniques for these two operations.

In the case of PrintAlgebra, we implement Algebra<Print> where Print is a functional interface we defined with a print() method. Therefore, calling expression(new PrintAlgebra()) will return an object that can be used to print the expression.

This is not the most direct avenue — we could have opted to implement Algebra<String> instead and have lit and add return their string representation directly. In fact, we take this approach with EvalAlgebra which implements Algebra<Integer> — there, lit and add directly return the integer they evaluate to.

Finally, a really neat trick not mentionned in the paper is that we can build an actual data structure from the functional encoding. For this, simply make an algebra that implements Algebra<Exp> or NegAlgebra<Exp> (depending on your needs, and assuming Exp is the parent class) and have each method return the node it corresponds to.

Turning these data structures back into an algebra encoding is unfortunately not possible. One could imagine that Exp has a E visit(Algebra<E>) method that is overriden in data classes to simply call the corresponding algebra method. The problem happens when you have introduced new data cases. If you added a neg method in NegAlgebra<E>, now you need the signature to be E visit(NegAlgebra<E>). This is almost feasible, supposing we could parameterize Exp as follow: Exp<A extends Algebra> and then define the method E visit (A<E>). Unfortunately that would make A a higher-order type (i.e. a type that takes a parameter, here E) and Java doesn't have those.

Of course, you could just use Algebra<E> as a bound and add a cast in there, type-unsafe but effective.

When dealing with object algebra, one may be concerned that it's no longer possible to "build a data structure dynamically", i.e. that all data must be statically defined like in our expression method.

That is fortunately not the case. Since the algebra encoding of an expression is just method calls, any execution flow that calls algebra methods can yield expressions. And execution can contain conditions, loops, etc. One potential pitfall is that the whole construction logic needs to be re-run each time we want to run an operation of our data. If the construction logic is expensive, this can be a problem. Fortunately there is a solution: simply return a function object that encodes the expression:

public E expr1 (NegAlgebra<E> a) {
    return expensive_predicate()
        ? a.add(a.lit(1), a.lit(2))
        : a.add(a.lit(1), a.neg(a.lit(2)));

// use: expr1(my_algebra);
// slow!

public E Function<NegAlgebra<E>, E> expr2() {
    return expensive_predicate()
        ? a -> a.add(a.lit(1), a.lit(2))
        : a -> a.add(a.lit(1), a.neg(a.lit(2)));

// use: expr2.apply(my_algebra);
// fast!

When we pass an algebra to expr2, expensive_predicate() is not run — it is only run once when the expr2 is created.

Finally, object algebra make "tree reuse" easy. You can compose an expression built with an Algebra<E> and one built with a NegAlgebra<E> pretty easily: the trick is that they only interface using E, so as long as E is the same, anything goes. Of course, this means you have to use compatible algebras. It could be argued that is not type-safe (or that it is another advantage): nothing prevents you from using two algebra with different semantics together, passing the result of one to the other... as long as E is the same.

The paper mentions other interesting possibilities: multi-parameter algebra (mimicking type families), combinators for automatic combination of multiple algebra, as well as allowing extension of the E parameter (e.g. Eval<E extends Number> implements Algebra<E>), ...

There is a lot to like about object algebra, it's a really elegant technique — in fact it's the shortest implementation, and one of the most readable. It has many advantages, from the possibility of building a real data structure to "tree reuse". Perhaps its main weakness is being a bit too alien to be integrated in many code bases, where one will need to have actual data structures. For instance, in my Autumn parsing library, the parser combinators can't be canonically represented as function calls — while it is possible to implement parsing that way (by using objects similar to Print in our example), it would be incredibly slow. However it is possible to use an object algebra to build the parser combinator graph, and to reuse the encoding for visiting it. However, this is only possible because this graph is immutable (and so will always stay in sync with its functional encoding).

It's definitely a technique to keep in the back of your mind.

Covariant Return Types

Finally, we look at our last solution, from the paper "The Expression Problem, Trivially!" (link).

This solution is very very close to Torgersen's first (data-driven) solution, but the essential differences between both is that this solution foregoes the use of generics in favor of covariant return types. What are covariant return types? In Java, you can override a method by a method with a different return type, but only if that type is a subtype of the original return type, for instance:

abstract class A {
    abstract Object foo();

abstract class B extends A {
    @Override String foo();

This works because String is a subclass of Object. And if you didn't know, yes you can override an abstract method without implementing it — that's an essential feature needed in the covariant solution.

Let's have a look at the solution:

Whereas Torgersen's solution encodes the expression type as an F-shape bounded generic type parameter (C extends Exp<C>), and subsequently types children using this type (e.g. C left, right; in the Add class; the covariant solution defines the children as abstract method whose return type is Exp.

Both solution need a "fix class": in Torgersen's solution, the class fixes C: AddF extends Add<ExpF> and later EvalAddF extends EvalAdd<EvalExpF>. In the covariant solution, the methods are overriden with the actual expression type: simply Exp in the base case, but EvalExp in the "eval" extension.

This is rather neat, and absent some issue I didn't think of, seems strictly superior Torgersen's solution. It does however come with that solution's other pitfalls, including the need to parameterize the construction logic. You'll note we didn't include factories in our code for this solution, but we did in Torgersen's solution. Don't let this fool you: they are equally needed (or can equally be dispensed with) in both cases.

Discussion & Recommendations

I came out of this article having learned a lot more than I expected going in. The impetus for this article was that I couldn't clearly articulate how the different solutions worked and how they related to each other.

I also wanted to make the point that they were needlessly complex and that my solution was better. Having done the research, I wouldn't say that this is necessarily true. Hence the little discussion now to be had about what should be used when.

First off, you should try to determine your requirements as precisely as possible. In particular:

With that in mind...

First off, if you care about independent extensibility, you have no choice but to use my solution. The others might be modified to accomodate it, by using Java 8's default interface methods — but you'll have to figure that yourself. Do keep in mind that this aspect of it wasn't shown in the example code but is explained in the previous post.

Beware that independent extensibility does add a lot of boilerplate, and likely will in other solutions too. And remember my solution isn't type-safe.

If you need perfect type safety (do you really?) and you're using immutable trees, I would go for Torgensen's 3rd solution (operation-centered with object-level extensibility).

In general, I would try to think hard about whether object algebras can be used in your use case. In a sense, they're the most elegant solution. One big caveat: I would think twice about using them to build a data structure — now you have two representations to keep in sync, and double duties.

In general, I feel like the sweet spot for them is either small localized things, or a central paradigm around which everything revolves. I'd be uneasy about making an object algebra one of many big moving parts in a program. My programmer's intuition say this way lay clunky mixed-metaphors.

I would avoid using the data-centered solutions (Torgersen's 1st and the covariant solution) unless object instantiation is very tightly controlled or centralized in your program. Playing with factories is not super fun.

A few more observations:

My solution and Torgersen's 4th (hybrid) solution are pretty much tied. Mine is guaranteed to work with independent extensibility (with the proper boilerplate), but Torgersen's will also work fairly often. But for instance, it won't work if two people introduce new data classes and implement an old visitor for these classes — there is no easy/safe way to "merge" the two implementations. However, it would more natural to implement the operations directly into the data class in this case! Torgersen's solution can also lead to less boilerplate in the case where you never have to deal with independent extensibility.

The covariant solution strictly dominates Torgersen's 1st (data-centered) solution.