Discriminated unions exactly solve this problem (just "inside out" in the same way that Church encodings are algebraic datatypes "inside out"). It's only a matter of whether you have closed or open unions. That then just governs whether you have to nest types or not.

In fact, you can get even more code reuse than with your standard visitor pattern using openly recursive unions.

  // This is all pseudo code with open unions to reduce nesting.
  // Doable with closed unions, just more cluttered

  // Note these are all type aliases, since open unions imply structural types,
  // but nonetheless these are truly (open) discriminated unions

  // ------ BEGIN : In some external library
  type alias Evaluator = PrintEvaluator or ExecutionEvaluator

  // Open here is referring to open recursion rather than open unions
  // This gives us even more reuse than the standard visitor pattern
  type alias ExpressionOpen a = Literal Int or Addition a a

  stringEvaluator : ExpressionOpen String -> String
  stringEvaluator (Literal n) = intToString n
  stringEvaluator (Addition x y) = x ++ " + " ++ y

  executionEvaluator : ExpressionOpen Int -> Int
  executionEvaluator (Literal n) = n
  executionEvaluator (Addition x y) = x + y

  evaluate : Evaluator -> Fix ExpressionOpen -> String
  // This fold is a generalized version of the usual fold on list, meant to work on fixed point types
  evaluate PrintEvaluator expr = fold stringEvaluator expr
  evaluate ExecutionEvaluator expr = intToString (fold executionEvaluator expr)

  // ------ END : No longer in some external library

  // ------ BEGIN : My own code
  type alias ComplexEvaluator = HexadecimalEvaluator or Evaluator

  hexadecimalEvaluator : ExpressionOpen String -> String
  hexadecimalEvaluator (Literal n) = intToHexString n
  hexadecimalEvaluator (Addition x y) = x ++ " + " ++ y

  type alias ComplexExpressionOpen a = BasicExpressionOpen a or Multiplication a a

  extendedEvaluator : ComplexEvaluator -> Fix ExpressionOpen -> String
  // This may look like dynamic dispatch but it isn't
  // ": Evaluator" expands out to a case match on the union tags of Evaluator
  // In both branches of the match we then just call evaluate
  extendedEvaluator (evaluator : Evaluator) expr = evaluate evaluator expr
  extendedEvaluator HexadecimalEvaluator expr = fold hexadecimalEvaluator expr

  stringEvaluatorExtended : ComplexExpressionOpen String -> String
  stringEvaluatorExtended (expr : ExpressionOpen String) = stringEvaluator expr
  stringEvaluatorExtended (Multiplication x y) = x ++ " * " ++ y

  executionEvaluatorExtended : ComplexExpressionOpen Int -> Int
  executionEvaluatorExtended (expr : ExpressionOpen Int) = executionEvaluator expr
  executionEvaluatorExtended (Multiplication x y) = x * y

  hexadecimalEvaluatorExtended : ComplexExpressionOpen String -> String
  hexadecimalEvaluatorExtended (expr : ExpressionOpen String) = hexadecimalEvaluator expr
  hexadecimalEvaluatorExtended (Multiply x y) = x ++ " * " ++ y

  // If you want to extend both evaluator and expression at the same time you need
  // to repeat yourself a little
  // But you would need to repeat even more with the visitor pattern!
  // Namely none of your old evaluators work nor do any of your old expression trees!
  evaluateExtendedExpression : ComplexEvaluator -> Fix ComplexExpressionOpen -> String
  evaluateExtendedExpression PrintEvaluator expr = fold stringEvaluatorExtended expr
  evaluateExtendedExpression ExecutionEvaluator expr = fold executionEvaluatorExtended expr
  evaluateExtendedExpression HexadecimalEvaluator expr = fold hexadecimalEvaluatorExtended expr

  // ------ END : Finished with my own code

That doesn't look like an alternative to the visitor pattern, that basically IS the visitor pattern. If you replaced the magic of Fix T + fold with an equivalent to accept(), you would end up having written exactly the Visitor pattern just with pattern matching instead of virtual dispatch and sum types instead of subtypes.

Here is the exact same example in fully working C# (you can do the same in Java or C++, C# is just a little more terse; you could do it in Go as well, but it would be much uglier because you don't have generics).

  // ------ BEGIN : In some external library
  interface ExpressionOpen { T accept<T>(Evaluator<T> eval) => throw new NotImplementedException("Unsupported combination"); }
  record Literal(int n) : ExpressionOpen{
    public T accept<T>(Evaluator<T> eval) => eval.visitLiteral(this);
  }
  record Addition(ExpressionOpen x, ExpressionOpen y) : ExpressionOpen {
    public T accept<T>(Evaluator<T> eval) => eval.visitAddition(this);
  }

  interface Evaluator<T> {
    T visitLiteral(Literal l);
    T visitAddition(Addition a);
  }
  class StringEvaluator : Evaluator<string> {
    public virtual string visitLiteral(Literal l) => l.n.ToString();
    public string visitAddition(Addition a) => a.x.accept(this) + "+" + a.y.accept(this);
  }
  class ExecutionEvaluator : Evaluator<int> {
    public int visitLiteral(Literal l) => l.n;
    public int visitAddition(Addition a) => a.x.accept(this) + a.y.accept(this);
  }
  // ------ END : No longer in some external library

  // ------ BEGIN : My own code

  //note: no need for ComplexEvaluator here
  class HexadecimalEvaluator : StringEvaluator {
    public override string visitLiteral(Literal x) => x.n.ToString("x");

    // thanks to inheritance, the existing expressions can already accept this
    // so no need for an equivalent for extendedEvaluator;
    // we're also inheriting visit(Addition) but that is not fundamental
  }

  interface ComplexExpressionOpen : ExpressionOpen { T accept<T>(ComplexEvaluator<T> eval) => throw new NotImplementedException("Unsupported combination"); }
  record Multiplication(ExpressionOpen x, ExpressionOpen y) : ComplexExpressionOpen {
    public T accept<T> (ComplexEvaluator<T> eval) => eval.visitMultiplication(this);
  }

  //but we do need ComplexEvaluator to go with ComplexExpression
  interface ComplexEvaluator<T> : Evaluator<T> { T visitMultiplication(Multiplication x); }

  class StringEvaluatorExtended : StringEvaluator, ComplexEvaluator<string> {
    public string visitMultiplication(Multiplication m) => m.x.accept(this) + " * " + m.y.accept(this);
  }

  class ExecutionEvaluatorExtended : ExecutionEvaluator, ComplexEvaluator<int> {
    public int visitMultiplication(Multiplication m) => m.x.accept(this) * m.y.accept(this);
  }

  class HexadecimalEvaluatorExtended: HexadecimalEvaluator, ComplexEvaluator<string> {
    public string visitMultiplication(Multiplication m) => m.x.accept(this) + " * " + m.y.accept(this);
  }
  class HexadecimalEvaluatorExtended: HexadecimalEvaluator, ComplexEvaluator<string> {
    public string visit(Multiplication m) => m.x.accept(this) + " * " + m.y.accept(this);
    //if C# supported multiple inheritance, we wouldn't have needed to repeat ourselves here:
    //we could have just inherited both HexadecimalEvaluator and StringEvaluatorExtended
  }

  //we don't need any kind of boilerplate similar to evaluateExtendedExpression.

  // ------ END : Finished with my own code

You'll notice that (disregarding how verbose the language still is) we have actually written significantly less code - we haven't repeated a single line of code.

Also note that your example only works in languages which have discriminated unions, pattern matching, compiler macros, and support for recursive types. Without any of these, it becomes even more verbose and harder to extend than the actual visitor pattern.

  interface ExpressionOpen { T accept<T>(Evaluator<T> eval) => throw new NotImplementedException("Unsupported combination"); }

  interface ComplexExpressionOpen : ExpressionOpen { T accept<T>(ComplexEvaluator<T> eval) => throw new NotImplementedException("Unsupported combination"); }

Those don't really count though. Heck if you allow that I can write the code way shorter in true Haskell (not pseudo-Haskell) as well.

In particular

  //we don't need any kind of boilerplate similar to evaluateExtendedExpression.

That was purely for combinatorial safety. You could remove it altogether if you wanted.

The whole point of the visitor pattern is to prevent you from being able to put unsupported combinations together. If you get rid of that requirement you don't need types at all and there are far far easier ways of doing this than either with the visitor pattern or with discriminated unions (just do it all with nested dictionaries for a fraction of the code).

Throwing this exception is basically a cheat is it not?

> support for recursive types

I think every statically typed language I know has support for recursive types. Certainly C# does.

> pattern matching

Eh... I'm not really using pattern matching in any real sense here. They could be replaced by if statements. The only reason it's there is because there's total type erasure in Haskell so runtime dynamic dispatch isn't really a thing in the same way it is in C#.

> Those don't really count though. Heck if you allow that I can write the code way shorter in true Haskell (not pseudo-Haskell) as well.

We could get rid of those base cases; we would then have to make ComplexExpression not a subtype of ExpressionOpen, and that in turn would mean we can't extend any of the extended evaluators from the old evaluators. This would just add the same boilerplate you have (copying the StringEvaluator implementation in StringEvaluatorExtended etc), all to prevent multiply.accept(printEvaluator). I'm not sure this extra bit of type safety would be worth it, but it certainly depends on the specifics.

> The whole point of the visitor pattern is to prevent you from being able to put unsupported combinations together.

While that is an important part, this is not an all-or-nothing problem. Type safety is a spectrum, and we can choose where to give up some safety to gain something else.

> Type safety is a spectrum, and we can choose where to give up some safety to gain something else.

I agree, but I'm saying that if you're willing to give up the type safety of verified combinations, you can have a solution that is just as (or even more) extensible as the visitor pattern (or discriminated unions) that is fewer lines of code and might even perform faster. Namely you can just downcast in the evaluator to the cases you can handle and then just bail if the downcast fails. This is the exact same functionality, can be extended in the same way (and in fact in more ways), and is fewer lines of code.

As far as I can tell the only reason you'd want to use the visitor pattern (or discriminated unions) here is some notion of "purity," e.g. in the case of OO that you're violating some notion of the Liskov Substitution principle by explicitly downcasting and in the case of FP that you're violating parametric polymorphism. But neither notions of purity feel particularly convincing to me.

As an aside

> This would just add the same boilerplate you have

In fact I think in the C# version it would be even more code because you can't reuse the same match patterns; you'd have to repeat way more of each evaluator.

> As far as I can tell the only reason you'd want to use the visitor pattern (or discriminated unions) here is some notion of "purity," e.g. in the case of OO that you're violating some notion of the Liskov Substitution principle by explicitly downcasting and in the case of FP that you're violating parametric polymorphism. But neither notions of purity feel particularly convincing to me.

In an OO language, the reason you'd prefer the Visitor pattern to downcasting is that it allows extending behavior by simply declaring new subtypes, instead of having to edit the central evaluator. This is why I keep insisting that the Visitor pattern is about double virtual dispatch, not about discriminated unions.

> In fact I think in the C# version it would be even more code because you can't reuse the same match patterns; you'd have to repeat way more of each evaluator.

Here is the C# version without the default NotImplementedException escape hatch:

  // ------ BEGIN : In some external library
  interface ExpressionOpen { T accept<T>(Evaluator<T> eval); }
  record Literal(int n) : ExpressionOpen{
    public T accept<T>(Evaluator<T> eval) => eval.visitLiteral(this);
  }
  record Addition(ExpressionOpen x, ExpressionOpen y) : ExpressionOpen {
    public T accept<T>(Evaluator<T> eval) => eval.visitAddition(this);
  }

  interface Evaluator<T> {
    T visitLiteral(Literal l);
    T visitAddition(Addition a);
  }
  class StringEvaluator : Evaluator<string> {
    public virtual string visitLiteral(Literal l) => l.n.ToString();
    public string visitAddition(Addition a) => a.x.accept(this) + "+" + a.y.accept(this);
  }
  class ExecutionEvaluator : Evaluator<int> {
    public int visitLiteral(Literal l) => l.n;
    public int visitAddition(Addition a) => a.x.accept(this) + a.y.accept(this);
  }
  // ------ END : No longer in some external library

  // ------ BEGIN : My own code

  //note: no need for ComplexEvaluator here
  class HexadecimalEvaluator : StringEvaluator {
    public override string visitLiteral(Literal x) => x.n.ToString("x");

    // thanks to inheritance, the existing expressions can already accept this
    // so no need for an equivalent for extendedEvaluator;
    // we're also inheriting visit(Addition) but that is not fundamental
  }

  interface ComplexExpressionOpen { T accept<T>(ComplexEvaluator<T> eval); }
  record Multiplication(ExpressionOpen x, ExpressionOpen y) : ComplexExpressionOpen {
    public T accept<T> (ComplexEvaluator<T> eval) => eval.visitMultiplication(this);
  }

  //but we do need ComplexEvaluator to go with ComplexExpression
  interface ComplexEvaluator<T> : Evaluator<T> { T visitMultiplication(Multiplication x); }

  class StringEvaluatorExtended : ComplexEvaluator<string> {
    public virtual string visitLiteral(Literal l) => l.n.ToString();
    public string visitAddition(Addition a) => a.x.accept(this) + "+" + a.y.accept(this);
    public string visitMultiplication(Multiplication m) => m.x.accept(this) + " * " + m.y.accept(this);
  }

  class ExecutionEvaluatorExtended : ComplexEvaluator<int> {
    public int visitLiteral(Literal l) => l.n;
    public int visitAddition(Addition a) => a.x.accept(this) + a.y.accept(this);
    public int visitMultiplication(Multiplication m) => m.x.accept(this) * m.y.accept(this);
  }

  class HexadecimalEvaluatorExtended: StringEvaluatorExtended  {
    public string visit(Multiplication m) => m.x.accept(this) + " * " + m.y.accept(this);
    public override string visitLiteral(Literal x) => x.n.ToString("x");
  }

  //we don't need any kind of boilerplate similar to evaluateExtendedExpression.

  // ------ END : Finished with my own code

However, this version has a major problem that neither your version nor my initial one do: I can't represent something like Addition(Multiplication(1,2),3). I don't think this problem is fixable in .NET, since you can't construct generic types that depend on runtime values, such as an ExpressionOpen<V> where V is anything which implements Evaluator<T>. We could probably hack around it with a lot more boilerplate and some convention (an associated Evaluator interface that Evaluator<T> extends, and explicit casts from Evaluator to Evaluator<T> in the accept methods, plus a convention that no one should implement Evaluator except Evaluator<T>).

> // thanks to inheritance...

Ah ha! You're totally right. I totally forgot about using inheritance to deal with overlapping patterns even though I use it all the time in Java (you could do the same overlap with open unions, but there's other ergonomic reasons you don't want to do that if you have exhaustiveness checking.)

> In an OO language, the reason you'd prefer the Visitor pattern to downcasting is that it allows extending behavior by simply declaring new subtypes, instead of having to edit the central evaluator. This is why I keep insisting that the Visitor pattern is about double virtual dispatch, not about discriminated unions.

I don't think that's true. If you give up statically checked combinations I don't think you need double dispatch at all.

Here's a version that does everything with downcasting (and the accompanying error). One class has disappeared altogether because we don't need them anymore (ComplexExpression) and we don't need any classes for the evaluators (other than a container to hold them as a sort of utility class with a bunch of static methods). And we don't have visitors at all.

And it represents Addition(Multiplication(1, 2), 3) just fine.

Am I missing something here?

  // ------ BEGIN : In some external library
  
  interface Expression { // This could be empty
  }
  record Literal(int n) : Expression {}
  record Addition(Expression x, Expression y) : Expression {}

  // Your evaluator doesn't even need to be a true class anymore and can just be
  // a bag of static methods
  // You could if you wanted to, but there's isn't a lot of value
  class BuiltInEvaluators {
    public static string StringEvaluator(Expression expression) {
      switch(expression) {
        case Literal literal: return literal.n.ToString();
        case Addition addition: return $"{StringEvaluator(addition.x)} + {StringEvaluator(addition.y)}";
        // You could remove this boilerplate if you wanted by wrapping this in a true class
        // and then throwing the exception in a wrapper method
        default: throw new ArgumentException("Expression incompatible with evaluator!");
      }
    }
    public static int ExecutionEvaluator(Expression expression) {
      switch(expression) {
        case Literal literal: return literal.n;
        case Addition addition: return ExecutionEvaluator(addition.x) + ExecutionEvaluator(addition.y);
        default: throw new ArgumentException("Expression incompatible with evaluator!");
      }
    }
  }
  // ------- END : No longer in some external library

  // ------- BEGIN : My own code
  record Multiply(Expression x, Expression y) : Expression {}

  class MyOwnEvaluators {
    public static string ExtendedStringEvaluator(Expression expression) {
      switch(expression) {
        case Multiply multiply: return $"{ExtendedStringEvaluator(multiply.x)} * {ExtendedStringEvaluator(multiply.y)}";
        case Expression x: return BuiltInEvaluators.StringEvaluator(x);
        default: throw new ArgumentException("Expression incompatible with evaluator!");
      }
    }
    public static int ExtendedExecutionEvaluator(Expression expression) {
      switch(expression) {
        case Multiply multiply: return ExtendedExecutionEvaluator(multiply.x) * BuiltInEvaluators.ExecutionEvaluator(multiply.y);
        case Expression x: return BuiltInEvaluators.ExecutionEvaluator(x);
        default: throw new ArgumentException("Expression incompatible with evaluator!");
      }
    }
    public static string HexadecimalEvaluator(Expression expression) {
      switch(expression) {
        // Our only interesting case
        case Literal l: return l.n.ToString("x");
        case Expression x: return ExtendedStringEvaluator(x);
        default: throw new ArgumentException("Expression incompatible with evaluator!");
      }
    }
  }
  // ------- END : My own code

Well, one thing that this solution misses is virtual dispatch - HexadecimalEvaluator won't work for Addition(Literal(1),Literal(2)) for example, because ExtendedStringEvaluator doesn't know to call HexadecimalEvaluator back.

Still, this is easily fixable by transforming the static functions into virtual methods on a class:

  //keeping all of the definitions so far the same

  class ExtendedStringEvaluator : BuiltInEvaluators.StringEvaluator {
    public override string visit(Expression expression) {
      
      switch(expression) {
        case Multiply multiply: return $"{ExtendedStringEvaluator(multiply.x)} * {ExtendedStringEvaluator(multiply.y)}";
        default : return super(expression);
      }
    }
  }

  class HexadecimalEvaluator : ExtendedStringEvaluator {    
    public override string visit(Expression expression) {
  
      switch(expression) {
        case Literal literal: return return l.n.ToString("x");
        default : return super(expression);
      }
    }
  }

Still, this switch on the class type is essentially exactly a virtual call on the type of Expression, just controlled by someone other than the original class. That is, the Expression doesn't "know" any more that it is being visited, so you couldn't create a LoggingExpression let's say. On the other hand, the Visitor now has more control over the way it interprets expression subtypes. So even here we are doing double dispatch: we are dispatching manually based on the runtime type of `expression`, and then allowing the language to dispatch based on the runtime type of `this`.

> Well, one thing that this solution misses is virtual dispatch - HexadecimalEvaluator won't work for Addition(Literal(1),Literal(2)) for example, because ExtendedStringEvaluator doesn't know to call HexadecimalEvaluator back.

Ah of course, I accidentally switched back into closed recursion rather than open recursion (that was ExpressionOpen). This indeed is exactly analogous to early binding vs late binding. You can't exactly write the same form of open recursion I wrote earlier because C# doesn't have higher-kinded generics (so for example you can't write

  interface Mappable<L> { 
    public abstract L<B>Map<A>(Func<A, B> f, L<A> x) 
  }

which prevents you from writing

  record Fix<F>(F<Fix<F>> x)

), but that's a language-specific thing. If you had higher-kinded generics you could exactly model virtual dispatch with open recursion because they're the same thing.

> That is, the Expression doesn't "know" any more that it is being visited, so you couldn't create a LoggingExpression let's say. On the other hand, the Visitor now has more control over the way it interprets expression subtypes. So even here we are doing double dispatch: we are dispatching manually based on the runtime type of `expression`, and then allowing the language to dispatch based on the runtime type of `this`.

You can call it visitor, but at this point it really is the same as discriminated unions (indeed this is exactly analogous to the first iteration of code just without the last bit to ensure that combinations matched up). And you can exactly replicate `LoggingExpression` in the evaluator, there is no difference in expressivity.

This shouldn't be surprising; after all the whole thing that kicked off all of this is that there is an isomorphism between discriminated unions and the visitor pattern. It should not be a surprise that it is possible to interpret one in the other.

My point at the beginning of all this was to object to the statement that this was not an isomorphism. Anything you can do with the visitor pattern you can do with discriminated unions and vice versa. The only differences are in ancillary language support (e.g. virtual dispatch and higher-kinded generics). Hence in a very real real sense the visitor pattern is just discriminated unions and vice versa.

The only preference for one or the two comes down to issues of code size and maintenance.

Forgot one other important distinction: if we wanted to add a new expression subtype that can behave like one of the existing types for existing operations, that is not possible with the discriminated union variant. Say we wanted to add an SmallAddition, that stores its two numbers as bytes, but converts them to int when needed. We could add this without modifying any code in the visitors in the C# version (well, technically we may have a problem because we accessed the fields directly, but that's inconsequential). In the pseudo-Haskell version, we would need to modify all 6 evaluation functions just to handle the new Expression subtype.

> In the pseudo-Haskell version, we would need to modify all 6 evaluation functions just to handle the new Expression subtype.

That's not true. I think you may be misunderstanding open unions (or polymorphic variants) here. Tags are types in and of themselves. That's why these are all `type alias`es. You could decide to modify the type alias if you'd like, but you don't have to. You can add the additional tag just to the call site.

So how would a subtype of Addition look here? Who would tell stringEvaluator how to handle this SmallNumberAddition?

There wouldn't be a subtype of Addition because Haskell doesn't have a notion of subtyping in the same way C# does. It would depend on the particulars of what the difference between SmallNumberAddition and Addition look like (in particular which parts are kept with the same functionality and which are different). The answer could be either a simple function or a typeclass depending on that.