Types and Tailcalls

Solving the Expression Problem in Haskell and Java

published on June 14, 2014

After my last post on the expression problem, I thought that I would explore ways to solve it in the next post and that I would write that post shortly after. I knew how the solution worked in Haskell and that solutions existed for OO languages, so that post should not have been terribly hard to write. Well, here we are five months later and I am finally getting around to writing the post ;).

Expression Problem Recap

The term Expression Problem was coined by Philip Waldler in a mail to the Java Generics mailing list. The goal is to be able to define datatypes by cases and functions over these datatypes in a way that is extensible: one should be able to add both new cases and new functions without touching or recompiling old code and while maintaining static type safety.

As an example I’ll reuse the simple expression language from the last post. To represent such an expression language we will have a number of variants to capture the different types of expressions, for example literal integers, addition, and multiplication. To work with this representation we will have different functions to transform such expressions, for example evaluating or pretty-printing them.

Once we have defined the cases and functions how difficult will it be to add new cases and new functions? Statically type-checked functional languages make it easy to add new functions (see last post) while the object oriented languages make it easy to add new cases. The default approach in both languages does not make it easy1 to either add new cases or new functions. That means that the default approach in both languages does not solve the Expression Problem. However, it turns out that solutions are possible in both types of languages. This post will describe a possible solution in both Haskell and Java.

A Haskell Solution

The key to solving the Expression Problem in Haskell is to define typeclasses for the desired functions and make the datatypes instances of these typeclasses. We also define the different variants as their own datatypes, though this is not strictly necessary yet. For our expression language the setup looks as follows:

data Lit = Lit Int
data Add l r = Add l r

class Eval x where
  eval :: x -> Int

instance Eval Lit where
  eval (Lit x) = x

instance (Eval l, Eval r) => Eval (Add l r) where
  eval (Add l r) = eval l + eval r

The extension that is typically easy in functional languages is to add a new function over the datatype. With the setup as above, we now add a new typeclass which contains the function as a method and add instances for each of our datatypes. Compared to the standard approach in functional languages, this requires slightly more code, but is still fairly clear:

class PPrint x where
  pprint :: x -> String

instance PPrint Lit where
  pprint (Lit x) = show x

instance (PPrint l, PPrint r) => PPrint (Add l r) where
  pprint (Add l r) = "(" ++ pprint l ++ " + " ++ pprint r ++ ")"

OK, so adding new functions is still easy, how about adding new cases? Adding a new case is the interesting part, because this is the side of the Expression Problem which the standard approach in Haskell can’t handle. However, with the setup we have introduced above this becomes quite easy: we just add a new datatype and then add instances for each of our typeclasses:

data Mult l r = Mult l r

instance (Eval l, Eval r) => Eval (Mult l r) where
  eval (Mult l r) = eval l * eval r

instance (PPrint l, PPrint r) => PPrint (Mult l r) where
  pprint (Mult l r) = pprint l ++ " * " ++ pprint r

OK, so this approach lets us indeed add new cases and new functions without having to modify existing code. Note that we also have type safety: in the code below both eval and pprint can be called on both threePlus5 and threePlus5Times7 because these operations are defined on each of the datatypes. Had we forgotten to derive a typeclass instance for one of the cases Lit, Add or Mult the compiler would bark. The full code is available at this gist.

threePlus5 = Add (Lit 3) (Lit 5)
threePlus5Times7 = Mult threePlus5 (Lit 7)

main = do
  putStrLn $ pprint threePlus5 ++ " = " ++ show (eval threePlus5)
  putStrLn $ pprint threePlus5Times7 ++ " = " ++ show (eval threePlus5Times7)

A Java Solution

Solving the Expression Problem in classical (statically typed) OO languages is a bit more difficult. The solution I’ll present here is taken from the paper Extensibility for the masses (PDF) which has won the ECOOP 2012 best paper award. The idea is to use object algebras which implement so-called algebraic signatures. We will use the same example as above. The algebraic signature for the expression language looks as follows:2

signature E
    lit:  Int -> E
    add:  E x E -> E

The general idea is this: we will represent the above signature as an interface which is parameterized over E. To actually use objects created with this interface we’ll instantiate E to a concrete interface, for example to Eval and call the operations provided by this interface (eval()). However, code creating objects with the above interface does not need to know what E is and can thus be completely generic.

In case this is a bit confusing (it certainly was to me), let’s look at a piece of code which will hopefully make this idea somwhat clearer:

interface Alg1<E> {
    E lit(int x);
    E add(E l, E r);

class Impl1<E> {
    public static <E> E make3Plus5(Alg1<E> f) {
        return f.add(f.lit(3), f.lit(5));

interface Eval {
    int eval();

class ELit implements Eval {
    private int x;
    public ELit(int x) { this.x = x; }
    public int eval() { return x; }

class EAdd implements Eval {
    private Eval l, r;
    public EAdd(Eval l, Eval r) { this.l = l; this.r = r; }
    public int eval() { return l.eval() + r.eval(); }

class Alg1EvalFactory implements Alg1<Eval> {
    public Eval lit(int x) { return new ELit(x); }
    public Eval add(Eval l, Eval r) { return new EAdd(l, r); }

class Impl2 {
    static int eval3Plus5() {
        return Impl1.make3Plus5(new Alg1EvalFactory()).eval();

So we first define a generic interface called Alg1 which represents the algebraic signature above.3 Programs such as make3Plus5 can use this interface completely generically without needing to know what E acutally is.

Only when we acutally want to use the objects created from the Alg1 interface do we need to define a concrete interface such as Eval and classes that implement it. We also need a class that implements Alg1<E>, in the code above this is Alg1EvalFactory. An instance of this factory is passed to the generic program make3Plus5 which then produces an object which implements Eval so that we can call the eval() method on it.

Comparing this approach to the Haskell one there are some similarities: The interface Eval here plays the role of the typeclass Eval in the Haskell version and the classes ELit and EAdd correspond to the instance declarations. The piece that is missing from the Haskell version is the Alg1 interface and its implementation, but I think there are some similarities to what the Haskell compiler does behind the scenes.4

Now let’s check if we can extend this setup with both new functions and new variants. First, adding new functions is fairly easy: The interface Alg1 can stay unchanged, we merely need to create a new interface PPrint which will take the place of Eval and corresponding classes PLit and PAdd that implement this interface. To actually make use of Alg1 instantiated to this new interface we also need a new factory.

interface PPrint {
    public String pprint();

class PLit implements PPrint {
    private int x;
    public PLit(int x) { this.x = x; }
    public String pprint() { return Integer.valueOf(x).toString(); }

class PAdd implements PPrint {
    private PPrint l, r;
    public PAdd(PPrint l, PPrint r) { this.l = l; this.r = r; }
    public String pprint() { return "(" + l.pprint() + " + " + r.pprint() + ")"; }

class Alg1PPrintFactory implements Alg1<PPrint> {
    public PPrint lit(int x) { return new PLit(x); }
    public PPrint add(PPrint l, PPrint r) { return new PAdd(l, r); }

class Impl3 {
    static String pprint3Plus5() {
        return Impl1.make3Plus5(new Alg1PPrintFactory()).pprint();

This may look like a lot of code, but again, this roughly corresponds to the Haskell version. We did not need to duplicate any code (apart from the usual boilerplate that is required by Java). Also note that we were able to reuse make3Plus5 from above even though we’re now using a new operation on its result!

So we can add new functions over the datatype cases. To add new cases we need to extend the signature Alg1 to Alg2 to accomodate the new case. We then need to add classes that implement the concrete interfaces Eval and PPrint for this new cases. Furthermore, we also need new factories which implement the interface Alg2<Eval> and Alg2<PPrint>. Again, this is slightly more code than one would love to write, but it is completely extensible (note for example that we are reusing make3Plus5 unchanged with a factory that implements Alg2<E>):

interface Alg2<E> extends Alg1<E> {
    E mult(E l, E r);

class EMult implements Eval {
    private Eval l, r;
    public EMult(Eval l, Eval r) { this.l = l; this.r = r; }
    public int eval() { return l.eval() * r.eval(); }

class PMult implements PPrint {
    private PPrint l, r;
    public PMult(PPrint l, PPrint r) { this.l = l; this.r = r; }
    public String pprint() { return l.pprint() + " * " + r.pprint(); }

class Alg2EvalFactory extends Alg1EvalFactory implements Alg2<Eval> {
    public Eval mult(Eval l, Eval r) { return new EMult(l, r); }

class Alg2PPrintFactory extends Alg1PPrintFactory implements Alg2<PPrint> {
    public PPrint mult(PPrint l, PPrint r) { return new PMult(l, r); }

class Impl4<E> {
    // a client program using Alg2 (which uses a function using Alg1!)
    public static <E> E make3Plus5Times7(Alg2<E> f) {
        return f.mult(Impl1.make3Plus5(f), f.lit(7));

    public static int eval3Plus5Times7() {
        return make3Plus5Times7(new Alg2EvalFactory()).eval();

    public static String pprint3Plus5Times7() {
        return make3Plus5Times7(new Alg2PPrintFactory()).pprint();

For completeness, here is a main method which uses the above and gives the same output as the Haskell version. The full code can be found at this gist.

public class Main {
    public static void main(String[] args)
        System.out.println(Impl3.pprint3Plus5() + " = "
                           + Integer.valueOf(Impl2.eval3Plus5()).toString());
        System.out.println(Impl4.pprint3Plus5Times7() + " = "
                           + Integer.valueOf(Impl4.eval3Plus5Times7()).toString());


So this post gave a quick demonstration of how the Expression Problem can be solved both in Haskell and Java. I think it is pretty cool that the Expression Problem is actually solvable in a language like Java because I first thought that that wasn’t the case. On the one hand the Java version seems pretty heavyweight in terms of additional complexity. I therefore doubt that I would reach for this solution in practice unless I was certain in advance that solving the Expression Problem is important for a particular application and that it would justify the conceptual overhead. On the other hand this solution doesn’t feel conceptually much heavier than the visitor pattern and this solution solves both sides of the Expression Problem while the visitor pattern only solves one.

In the end I just wish I could use Haskell ;).

  1. Easy here means that no code needs to be changed / dublicated and type-safety is maintained.

  2. Note the similarity to type classes!

  3. The paper calls such interfaces object algebras and goes a bit into the category theoretical motivations for these terms which I’m ignoring here.

  4. Clearly the Haskell code is considerably easier to understand and - I would argue - also more elegant, but let’s not get into that.

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