# Sum Types, Visitors, and the Expression Problem

published on January 10th, 2014

I've heard that the visitor pattern is just a poor way of getting the benefit of sum types^[These are also known as disjoint union or variant types.] in functional programming circles several times. I must admit that I never had completely thought this through, but I was nevertheless a bit surprised when I saw that walking the AST in Rust was implemented by what looks like a use of the visitor pattern. Languages with sum types usually use pattern matching to achieve the same effect and I had always considered this a superior approach. In this blog post I try to understand the differences and similarities of the two approaches a little better.

To set the stage, both pattern matching and the visitor pattern solve one side of the expression problem, which is the problem of adding both variants of a data type and functions that act on those variants without changing or recompiling old code and without loosing type safety.

To make this a bit more concrete, consider a very simple expression languages consisting of numbers and addition as an example (no post on this topic can do without one!). We have two variants of expressions, (1) numbers and (2) addition. Let's assume that we want to compute the values represented by an expression as a first operation.

In Haskell a straightforward way of solving this problem is as follows

data Expr = Val Int | Add Expr Expr

eval :: Expr -> Int
eval (Val i)     = i
eval (Add e1 e2) = eval e1 + eval e2


If you're not familiar with Haskell, the first line defines a data type with two variants, it can either be a Val, which holds an Int, or it is an Add which holds two expressions. Val and Add are called constructors of Expr. The eval function pattern-matches and handles each case.

Now imagine that we do not only want to evaluate expressions but also pretty-print them. Adding operations is easy in Haskell, we just write a new function:

pprint :: Expr -> String
pprint (Val i)     = show i
pprint (Add e1 e2) = pprint e1 ++ " + " ++ pprint e2


In Java we might achieve something similar by introducing an Expr class:

interface Expr {
public int eval();
}

class Val implements Expr {
private final int v;
public Val(int v) { this.v = v; }
public int eval { return v; }
}

private final Expr l;
private final Expr r;
public Add(Expr l, Expr r) { this.l = l; this.r = r; }
public int eval { return l.eval() + r.eval(); }
}


But now, if we want to add the pprint operation, we have to touch every class. This is the side of the expression problem that functional languages tend to solve better than object oriented languages. However, the object oriented programming community has devised the visitor pattern as a way to solve this problem:

interface ExprVisitor {
public void visit(Val v);
}

interface Expr {
public void accept(ExprVisitor visitor);
}

class Val implements Expr {
private int v;
public Val(int v) { this.v = v; }
public int val()  { return v; }
public void accept(ExprVisitor visitor) { visitor.visit(this); }
}

class Add : public Expr {
private Expr l;
private Expr r;
public Add(Expr l, Expr r) { this.l = l; this.r = r; }
public Expr l() { return l; }
public Expr r() { return r; }
public void accept(ExprVisitor visitor) { visitor.visit(*this); }
}

class EvalVisitor implements ExprVisitor {
private int result = 0;
public int result() { return result; }
public void visit(Val val) { result = val.val(); }
int result_l = result;
result += result_l;
}
}


Ok, this is not exactly pretty, but let's not forget that this is the side of the problem where OO languages are not good at. At least we can pull something of. And now we are in a situation where we can add new operations pretty easily:

class PprintVisitor extends ExprVisitor {
private String result = "";
public String result() { return result; }
public void visit(Val val) override { result += val.val(); }
result += " + ";
}
}


This works, but the Haskell solution is clearly more elegant. Does the visitor pattern have any additional advantages? Well, neither approach solves the expression problem: if we want to add a new variant, say a Mult, then we have to change existing code in both cases.

I can't really think of an advantage for the visitor pattern. I've thought of two possibilities, default implementations and almost-but-not-quite-solving-the-expression-problem. But then I realized that the first problem is also similarly solvable in the pattern matching approach and that the second problem doesn't work without loosing type safety or duplicating code:^[Both maintaining type safety and not duplicating code are requirements in the expression problem.]

1. Default implementations are easy to implement with both approaches: in the visitor pattern defaults can be achieved by inherenting from a visitor with default implementations and overriding only certain methods. In the pattern-matching approach we would match all the constructors where we want to override the defaults and insert a wildcard match for the rest and call the default implementaiton on the bound variable.

2. Almost-but-not-quite-solving-the-expression-problem: I first thought that we could use some inheritance based trickery to solve the expression problem at least for new code. But none of these seems to work: If we add a new variant, say Mult, it can't derive from Expr because then it would have to implement Expr's accept method, which it can't sensibly do (because there is no right visit method in ExprVisitor).

Thus we must introduce a new interface Expr2. Expr2 cannot derive from Expr, lest we have the same problem as before. But the old variants don't derive from Expr2, so this is of limited use. Whichever way we twist or turn it, there is no easy way to solve the expression problem with this pattern.

So, as it stands, I can't really come up with an advantage for the visitor pattern over pattern matching. If you work in a language without sum types then it is certainly a great workaround, but in a language that does pattern matching seems much both more concise and more efficient.^[Due to the virtual method calls, which prevent inlining, I would expect the visitor pattern to be much slower than a direct function call.]