Reading 28: Little Languages II

Since our introduction to recursive data types, we’ve been implementing functions on these types using the interpreter pattern:

• declare the operation as an instance method in the interface that defines the datatype, and
• implement the operation in each concrete variant.

In terms of the composite pattern, composite variants (e.g. Concat in Music, Not in Formula) will implement the operation recursively, and primitive variants (e.g. Note in Music, Variable in Formula) will implement base cases.

For example, here’s the Boolean formula recursive datatype again:

Formula = Variable(name:String)
          + Not(formula:Formula)
          + And(left:Formula, right:Formula)
          + Or(left:Formula, right:Formula)

Suppose we want to add an operation variables to Formula that returns the set of all variable names mentioned in the formula. We saw how to do this in a reading exercise in Recursive Data Types (at the time, using immutable set type ImSet).

public interface Formula {
    // ... other operations ...
    /** @return all the variables that appear in this Formula */
    public Set<String> variables();
}

public class Variable implements Formula {
    // ... name field, constructor, other operations ...
    @Override public Set<String> variables() {
        return singletonSet(this.name);
    }
}

public class Not implements Formula {
    // ... formula field, constructor, other operations ...
    @Override public Set<String> variables() {
(missing answer)
    }
}
public class And implements Formula {
    // ... left and right fields, constructor, other operations ...
    @Override public Set<String> variables() {
        return setUnion(this.left.variables(), this.right.variables());
    }
}
public class Or implements Formula {
    // ... left and right fields, constructor, other operations ...
    @Override public Set<String> variables() {
        return setUnion(this.left.variables(), this.right.variables());
    }
}

In this code, singletonSet is an imagined creator for single-element sets, and setUnion constructs the union of its input sets.

/** @return a set containing only elt */
public static <E> Set<E> singletonSet(E elt) {
    return Collections.singleton(elt);
}

/** @return a set containing all the elements in sets */
@SafeVarargs
public static <E> Set<E> setUnion(Set<E>... sets) {
    return Arrays.stream(sets).flatMap(Set::stream)
                              .collect(Collectors.toSet());
}

Let’s consider two disadvantages of this approach:

1. For a complicated operation on a datatype with many variants, the code to implement the operation will be distributed across all the different variant classes. If we want to inspect all the code, refactor it, or fix a bug, we must find it in all those different places: the code is harder to understand. Bugs in our recursive implementations are easier to introduce and harder to find.

2. If we wish to add a new operation to our type, we must modify the interface and every implementing class. If there are many concrete variants that already have substantial code, perhaps written and maintained by other programmers, this change will be more difficult to make than if we could keep the code for the new operation in its own separate location.

Going back to the functional definitions you wrote in the Recursive Data Types reading exercise:

variables(Variable(x)) = set-add(x, set-empty())
variables(Not(f))      = variables(f)
variables(And(f1, f2)) = set-union(variables(f1), variables(f2))
variables(Or(f1, f2))  = set-union(variables(f1), variables(f2))

Translating those cases directly into code on our classes from above, we’d like to write something like this:

public static Set<String> variables(Formula f) {
    switch(f) {
    case Variable var:
        return singletonSet(var.name);
    case Not not:
        return variables(not.formula);
    case And and:
        return setUnion(variables(and.left), variables(and.right));
    case Or or:
        return setUnion(variables(or.left), variables(or.right));
    }
}

Even putting aside the direct access to the rep of each variant, a switch structure like this is not (yet?) valid Java. There are many languages that do support pattern matching of instances to determine their underlying structure, but Java is not one of them.

Bad pattern matching in Java

public static Set<String> variables(Formula f) {
    if (f instanceof Variable) {
        return singletonSet(((Variable)f).name);
    }
    if (f instanceof Not) {
        return variables(((Not)f).formula);
    }
    if (f instanceof And) {
        And and = (And)f;
        return setUnion(variables(and.left), variables(and.right));
    }
    if (f instanceof Or) {
        Or or = (Or)or;
        return setUnion(variables(or.left), variables(or.right));
    }
    throw new IllegalArgumentException("don't know what " + f + " is");
}

To find our way to a different approach, consider what we want: a function on a recursive data type. Instead of thinking about the function as a piece of code, consider representing the function as data.

Our example operation has the type signature:

variables : Formula → Set<‍String>

Can we use the Function type from the standard library, which represents functions of a single argument, and is parameterized by the argument type and the return type?

Function<Formula, Set<String>> variables = f -> {
    // ... did that help?
};

This doesn’t get us anywhere. The body of the lambda still only knows that f is a Formula, and that’s not enough to implement an appropriate recursive or base case of extracting variables.

Let’s try defining a new type instead, specifically to represent functions on Formula:

/**
 * Represents a function on different kinds of Formulas
 * @param <R> the type of the result of the function
 */
interface FormulaFunction<R> {

    /**  @return this applied to var */
    public R onVariable(Variable var);

    /**  @return this applied to not */
    public R onNot(Not not);

    /**  @return this applied to and */
    public R onAnd(And and);

    /**  @return this applied to or */
    public R onOr(Or or);
}

Now define the variables operation as a class that implements FormulaFunction:

class VariablesInFormula implements FormulaFunction<Set<String>> {

    // The base case works nicely:

    @Override public Set<String> onVariable(Variable var) {
        return singletonSet(var.name);
    }

    // But we're not sure how to implement recursive calls:

    @Override public Set<String> onNot(Not not) {
        return ???; // how to apply this to not.formula?
    }

    @Override public Set<String> onAnd(And and) {
        return setUnion(???); // how to apply this to and.left & and.right?
    }

    @Override public Set<String> onOr(Or or) {
        return setUnion(???); // how to apply this to or.left & or.right?
    }
}

The code for onNot, onAnd, and onOr would be easy to write, but only once we understand how to make the recursive calls that apply the function — this — to their arguments’ Formula children. If we can answer that question, we’ll also answer the question of how to use one of these Variables­In­Formula objects in the first place:

Formula f = ...;
VariablesInFormula getVariables = new VariablesInFormula();
Set<String> theVariables = ...; // how to call getVariables on f?

How can we go from calling some method on a Formula object — like f in this code, or like and.left and or.right in the previous code — to calling the variant-specific method we need? The dynamic dispatch of method calls in Java can help us get there. Dynamic dispatch powers the familiar interpreter pattern: if a method is defined in the Formula interface, we can call it on an instance of any concrete type that implements Formula, and the code that runs is that concrete type’s implementation.

In the double dispatch pattern, we will dispatch twice: first to the concrete variant (e.g. And, Or, etc.), then to the recursive function (e.g. VariablesInFormula or another FormulaFunction).

The first method call serves as an entry point for clients to invoke their function by passing it in:

public interface Formula {

    // ... other operations ...

    /**
     * Call a function on this Formula.
     * @param <R> the type of the result
     * @param function the function to call
     * @return function applied to this
     */
    public <R> R callFunction(FormulaFunction<R> function);
}

And we’ll implement this callFunction operation in each variant to make the second function call, with each concrete variant now taking responsibility for revealing its actual type to the Formula­Function:

public class Variable implements Formula {
    // ... fields, other methods, etc. ...
    
    @Override public <R> R callFunction(FormulaFunction<R> function) {
        return function.onVariable(this);
    }
}
public class Not implements Formula {
    // ... fields, other methods, etc. ...
    
    @Override public <R> R callFunction(FormulaFunction<R> function) {
        return function.onNot(this);
    }
}
public class And implements Formula {
    // ... fields, other methods, etc. ...
    
    @Override public <R> R callFunction(FormulaFunction<R> function) {
(missing answer)
    }
}
public class Or implements Formula {
    // ... fields, other methods, etc. ...
    
    @Override public <R> R callFunction(FormulaFunction<R> function) {
(missing answer)
    }
}

So a call to callFunction, as implemented by all four variants, immediately turns around with a call to the appropriate onVariant method of the passed-in function. Now we can complete our recursive implementations in Variables­In­Formula, and call it:

class VariablesInFormula implements FormulaFunction<Set<String>> {
    
    @Override public Set<String> onVariable(Variable var) {
        return singletonSet(var.name);
    }
    
    @Override public Set<String> onNot(Not not) {
(missing answer)
    }
    
    @Override public Set<String> onAnd(And and) {
        return setUnion(and.left.callFunction(this),
                        and.right.callFunction(this));
    }
    
    @Override public Set<String> onOr(Or or) {
        return setUnion(or.left.callFunction(this),
                        or.right.callFunction(this));
    }
}

Formula f = ...;
VariablesInFormula getVariables = new VariablesInFormula();
Set<String> theVariables = ??? ;

Suppose we represent the formula (P ∨ Q) ∧ ¬R. Following the datatype definition, the structure is:

And( Or(Variable("P"), Variable("Q")),
     Not(Variable("R")) )

Let f be that Formula instance.

If we run f.callFunction(getVariables), how will the execution proceed?

• f.callFunction runs VariablesInFormula.onAnd
  • and.left.callFunction runs VariablesInFormula.onOr
    • or.left.callFunction runs VariablesInFormula.onVariable
      • returns the set { "P" }
    • or.right.callFunction runs VariablesInFormula.onVariable
      • returns the set { "Q" }
    • returns setUnion{ "P" }, { "Q" } = { "P", "Q" }
  • and.right.callFunction runs VariablesInFormula.onNot
    • not.formula.callFunction runs VariablesInFormula.onVariable
      • returns the set { "R" }
    • returns the set { "R" } unchanged
  • returns setUnion({ "P", "Q" }, { "R" }) = { "P", "Q", "R" }

What we have created with the FormulaFunction type and callFunction operation is a special kind of iterator, one that walks over the tree structure of a Formula and processes nodes in the tree one at a time. The name for this pattern is visitor:

• A visitor implements switch over types: the code above to implement Variables­In­Formula reads pretty closely to our original goal of a switch statement, but each case is given its own onVariant method.

• A visitor represents a function over a recursive type: we can create an instance of a particular Formula­Function, pass it around, apply it as needed, etc. If the function we’re representing takes additional parameters, we can make them arguments to the constructor and store those values in fields to be referenced as we go.

• A visitor acts as an iterator: the implementor of a function such as variables does not write explicit control structures to pull out each node in the tree one-by-one. Instead, the implementor makes recursive calls to callFunction(this) and relies on the Formula variants to call it back.

We’ll make a few changes to nomenclature to arrive at a final version of our Formula visitor:

• We’ll call it Formula.Visitor instead of FormulaFunction.
• Walking a visitor over an instance is called accepting it, so we’ll rename callFunction to accept.
• In the visitor interface, since the onVariant methods each take a different static type, we can use a single overloaded name for all of them: on.

Finally, let’s stop referring directly to our fields and implement observer operations our visitors can use. We don’t need to add these operations to the Formula interface: they are variant-specific, and will only be used by clients who are aware of what our concrete types are.

public interface Formula {

    public interface Visitor<R> {
        public R on(Variable var);
        public R on(Not not);
        public R on(And and);
        public R on(Or or);
    }

    public <R> R accept(Visitor<R> visitor);

    // ... other operations ...
}

class Variable implements Formula {
    // ...
    public String name() { ... }
    @Override public <R> R accept(Visitor<R> visitor) {
        return visitor.on(this);
    }

}
class Not implements Formula {
    // ...
    public Formula formula() { ... }
    @Override public <R> R accept(Visitor<R> visitor) {
        return visitor.on(this);
    }
}
class And implements Formula {
    // ...
    public Formula left() { ... }
    public Formula right() { ... }
    @Override public <R> R accept(Visitor<R> visitor) {
        return visitor.on(this);
    }
}
class Or implements Formula {
    // ...
    public Formula left() { ... }
    public Formula right() { ... }
    @Override public <R> R accept(Visitor<R> visitor) {
        return visitor.on(this);
    }
}

class VariablesInFormula implements Formula.Visitor<Set<String>> {
    @Override public Set<String> on(Variable var) {
        return singletonSet(var.name());
    }
    @Override public Set<String> on(Not not) {
        return not.formula().accept(this);
    }
    @Override public Set<String> on(And and) {
        return setUnion(and.left().accept(this), and.right().accept(this));
    }
    @Override public Set<String> on(Or or) {
        return setUnion(or.left().accept(this), or.right().accept(this));
    }
}

Calling it with:

Formula f = ...;
Set<String> variables = f.accept(new VariablesInFormula());

Consider all the code that will implement all the operations of Formula:

And compare the interpreter pattern and the visitor pattern for implementing functions over a recursive data type:

Interpreter

Code is grouped by column. Each variant has all the code for that variant, implementing all the operations.

Visitor

Code is grouped by row. All the code for an operation will reside in a visitor class, with code to handle all the variants.

The interpreter pattern makes it easier to add new variants, because we don’t have to change any of our existing code: we just need to implement all the various operations as methods in the new variant class.

The visitor pattern makes it easier to add new operations. Instead of having to modify both the interface and every variant class, we just need to create a new, e.g., Formula.Visitor implementation with all the code for our new operation. There is no change to existing classes or interfaces.

Therefore, if adding new operations is a change we want to be most ready for, we might choose to define a visitor interface and write our functions as visitors.

We will also use the visitor pattern when we intend clients of our type to know and understand its underlying structure and implement their own operations — and we’ve already seen a common and important use case for this: parse trees! Looking at code we’ve written to transform a concrete syntax tree into an abstract syntax tree, notice a smelly pattern:

static IntegerExpression makeAST(ParseTree<IntegerGrammar> parseTree) {
    switch (parseTree.name()) {
    case ROOT:
        // ... call makeAST recursively on child ...
    case SUM:
        // ... iterate over children and create Plus expressions ...
    case PRIMARY:
        // ... call makeAST recursively on child ...
    case NUMBER:
        final int n = Integer.parseInt(parseTree.text());
        return new Number(n);
    default:
        throw new AssertionError("should never get here");
}

There’s that default case throwing an exception again: we don’t have static checking to make sure we’ve covered all the variants. We’re also stuck talking about generic ParseTree nodes. If there are multiple concrete variants of ParseTree, we would need to use unsafe casting to call any helpful operations they define. ParserLib has the advantage of simplicity, but in a more complex parsing package, we are likely to find makeAST implemented with a visitor.

And what works for concrete syntax trees is often very helpful for abstract syntax trees, where the implementor offers AST clients a visitor interface to define their own operations by walking recursively over the tree.

The visitor pattern overcomes a missing feature in the Java language, giving us a type-safe way to represent and implement functions over a recursive data type as self-contained modules. The visitor instances are also functional objects, so we can naturally pass them around our program as values.

If we expect to add new operations to our recursive ADT in the future, using visitor makes our code ready for change without sacrificing the safety from bugs provided by static checking.

If we are building a little language where the ADT is designed for the purpose of allowing clients to define their own operations on our type from outside our implementation classes, visitor makes this possible.