Reading 16: Recursive Data Types

In this reading we’ll look at recursively-defined types, how to specify operations on such types, and how to implement them. Our main example will be immutable lists.

Before we introduce recursive datatypes — which have a recursive structure of both data and computation — take a minute to review recursive computations.

Just as a recursive function is defined in terms of itself, a recursive datatype is defined in terms of itself. We’ll see the same need for base and recursive cases, which will now appear as different variants of the abstract type.

We’ll start with a classic recursive datatype, the immutable list.

Immutability is powerful not just because of its safety, but also because of the potential for sharing. Sharing actually produces performance benefits: less memory consumed, less time spent copying. Here we’re going to look at how to represent list data structures differently than in our familiar array lists or linked lists.

Let’s define a data type for an immutable list, ImList<E>. The data type has four fundamental operations:

These four operations have a long and distinguished pedigree. They are fundamental to the list-processing languages Lisp and Scheme (where for historical reasons they are called nil, cons, car, and cdr, respectively). They are widely used in functional programming, where first and rest are sometimes called head and tail instead.

Before we design TypeScript classes to implement this datatype, let’s get used to the operations a bit, using lists of integers. We’ll write lists with square brackets, like [ 1, 2, 3 ], and we’ll write the operations as if they are simple functions. Once we get to TypeScript, the syntax will look different but the operations will have the same meaning.

empty() → [ ]  
cons(0, empty()) → [ 0 ]  
cons(0, cons(1, cons(2, empty()))) → [ 0, 1, 2 ]

x = cons(0, cons(1, cons(2, empty())))  → [ 0, 1, 2 ]  
first(x) → 0  
rest(x) → [ 1, 2 ]

first(rest(x)) → 1  
rest(rest(x)) → [ 2 ]  
first(rest(rest(x)) → 2  
rest(rest(rest(x))) → [ ]

The fundamental relationship between first, rest, and cons is:

first(cons(x, y)) = x  
rest(cons(x, y)) = y

What cons puts together, first and rest peel back apart.

Immutable lists in TypeScript

To implement this datatype in TypeScript, we’ll use an interface:

// todo: empty() returning ImList<E>
interface ImList<E> {
    cons(first: E): ImList<E>;
    readonly first: E;
    readonly rest: ImList<E>;
}

This interface declares a generic type ImList<E> that can be instantiated for any type E: ImList<number>, ImList<string>, etc. The E in these declarations is a placeholder that the compiler will fill in when it checks our code for type safety.

And we’ll write two classes that implement this interface:

• Empty represents the result of the empty operation (an empty list)
• Cons represents the result of a cons operation (an element glued together with another list)

class Empty<E> implements ImList<E> {
  public constructor() {
  }
  public cons(first: E): ImList<E> {
    return new Cons<E>(first, this);
  }
  public get first(): E {
    throw new Error("unsupported operation");
  }
  public get rest(): ImList<E> {
    throw new Error("unsupported operation");
  }
}

Note the use of getter methods for the first and rest properties, which allow these operations to fail fast when called on an empty list.

class Cons<E> implements ImList<E> {
  public readonly first: E;
  public readonly rest: ImList<E>;

  public constructor(first: E, rest: ImList<E>) {
    this.first = first;
    this.rest = rest;
  }
  public cons(first: E): ImList<E> {
    return new Cons<E>(first, this);
  }
}

So we have operations for cons, first, and rest, but where is the fourth operation of our datatype, empty?

One way to implement empty is to have clients call the Empty class constructor to obtain empty lists. This sacrifices representation independence — clients have to know about the Empty class!

As we saw in an earlier reading, a better way to do it is as a factory function that takes no arguments and produces an instance of Empty:

function emptyList<E>(): ImList<E> {
  return new Empty<E>();
}

Now that we have all the operations, here’s some actual TypeScript code that parallels the abstract examples we wrote earlier. Hover or tap on each row in the table to update the snapshot diagram:

TypeScript syntax	Functional syntax	Result
const nil:ImList<number> = emptyList();	nil = empty()	[ ]
nil.cons(0)	cons(0, nil)	[ 0 ]
nil.cons(2).cons(1).cons(0)	cons(0, cons(1, cons(2, nil)))	[ 0, 1, 2 ]
const x = nil.cons(2).cons(1).cons(0);	x = cons(0, cons(1, cons(2, nil)))	[ 0, 1, 2 ]
x.first	first(x)	0
x.rest	rest(x)	[ 1, 2 ]
x.rest.first	first(rest(x))	1
x.rest.rest	rest(rest(x))	[ 2 ]
x.rest.rest.first	first(rest(rest(x)))	2
x.rest.rest.rest	rest(rest(rest(x)))	[ ]
const y = x.rest.cons(4);	y = cons(4, rest(x))	[ 4, 1, 2 ]

The key thing to note here is the sharing of structure that immutable list provides. In the last example above, x and y are sharing their representation of the sublist [ 1, 2 ]. Only one copy of this sublist exists in memory, and both x and y point to it, but this aliasing is perfectly safe because the list is immutable.

Two classes implementing one interface

class Empty<E> implements ImList<E> {
  public constructor() { }
  public cons(first: E): ImList<E> { return new Cons<E>(first, this); }
  public get first(): E { throw new Error("unsupported operation"); }
  public get rest(): ImList<E> { throw new Error("unsupported operation"); }
}

class Cons<E> implements ImList<E> {
  public readonly first: E;
  public readonly rest: ImList<E>;
  public constructor(first: E, rest: ImList<E>) {
    this.first = first;
    this.rest = rest;
  }
  public cons(first: E): ImList<E> { return new Cons<E>(first, this); }
}

const airports = empty().cons("SFO").cons("IAD").cons("BOS");

TypeScript shortcut: parameter properties

class Cons<E> implements ImList<E> {
  public readonly first: E;
  public readonly rest: ImList<E>;

  public constructor(first: E, rest: ImList<E>) {
    this.first = first;
    this.rest = rest;
  }
  public cons(first: E): ImList<E> {
    return new Cons<E>(first, this);
  }
}

class Cons<E> implements ImList<E> {
  public readonly first: E;
  public readonly rest: ImList<E>;

  public constructor(
    public readonly first: E,
    public readonly rest: ImList<E>
  ) {
    this.first = first;
    this.rest = rest;
  }
  public cons(first: E): ImList<E> {
    return new Cons<E>(first, this);
  }
}

/**
 * Represents an immutable tweet from Twitter.
 */
class Tweet {
    public constructor(
      public readonly author:string, 
      public readonly text:string,
      public readonly timestamp:Date) {
    }
}

The abstract data type ImList, and its two concrete classes Empty and Cons, form a recursive data type. Cons is an implementation of ImList, but it also uses ImList inside its own rep (for the rest field), so it recursively requires an implementation of ImList in order to successfully implement its contract.

To make this fact clearly visible, we’ll write a datatype definition:

ImList<E> = Empty + Cons(first:E, rest:ImList<E>)

This is a recursive definition of ImList as a set of values. Here’s the high-level meaning: the set ImList consists of values represented in two ways: either by an Empty object (which has no fields), or by a Cons object whose fields are an element first and an ImList rest.

A more detailed reading: ImList<E> is a generic type where for any type E, the set ImList<E> consists of the values represented either by an Empty object, or by a Cons object, where a Cons object has fields consisting of first of type E and rest of type ImList<E>.

The recursive nature of the datatype becomes far more visible when written in a datatype definition.

We can write any ImList value as a term or expression using this definition, e.g. the list [ 0, 1, 2 ] is written as

Cons(0, Cons(1, Cons(2, Empty)))

And we can build up the infinite set of all ImList values by starting from the base case that is Empty and recursively applying Cons. For example, with ImList<boolean>:

ImList<boolean>

= Empty

+ Cons(true, Empty)
+ Cons(false, Empty)

+ Cons(true, Cons(true, Empty))
+ Cons(true, Cons(false, Empty))
+ Cons(false, Cons(true, Empty))
+ Cons(false, Cons(false, Empty))

+ Cons(true, ...)
+ Cons(true, ...)
+ Cons(true, ...)
+ Cons(true, ...)
+ Cons(false, ...)
+ Cons(false, ...)
+ Cons(false, ...)
+ Cons(false, ...)

+ ...

Formally, a datatype definition has:

• an abstract datatype on the left, defined by its representation (or concrete datatype) on the right
• the representation consists of variants of the datatype combined by a union operator, which we’ll write as +
• each variant is a class name with zero or more fields, written with name and type separated by a colon (:).

A recursive datatype definition is a datatype definition where the abstract type (on the left) appears in its own definition (as the type of a field on the right). Another example is a binary tree:

Tree<E> = Empty + Node(e:E, left:Tree<E>, right:Tree<E>)

The Tree type is defined as the set of values formed either by Empty or by Node, which takes an element e (of type E) and left and right subtrees.

We’ll see more examples below.

When you write a recursive datatype, document the recursive datatype definition as a comment in the interface:

interface ImList<E> {

    // Datatype definition:
    //   ImList<E> = Empty + Cons(first:E, rest:ImList<E>)

    // ...

Geology = Core(a:number, b:number, c:number, d:number)
          + Planet(core:Core, a:number, b:number, c:number, d:number)
          + System(geology:Geology, a:number, b:number)

This way of thinking about datatypes — as a recursive definition of an abstract datatype with concrete variants — is appealing not only because it can handle recursive and unbounded structures like lists and trees, but also because it provides a convenient way to describe operations over the datatype, as functions with one case per variant.

First, notice how the datatype definition maps to the abstract interface and concrete variants that we’ve already seen:

ImList<E>

interface ImList<E> { ... }

 =

Empty

class Empty<E> implements ImList<E> {
    public constructor() { }
}

 +

Cons(first:E, rest:ImList<E>)

class Cons<E> implements ImList<E> {
    private readonly first: E;
    private readonly rest: ImList<E>;
    public constructor(first: E, rest: ImList<E>) {
        this.first = first;
        this.rest = rest;
    }
}

Now: let’s add a new operation by thinking about it as a recursive function with one case per variant: the size of the list, which is certainly an operation we’ll want in ImList. We can define it like this:

size : ImList → number  // returns the size of the list

and then fully specify its meaning by defining size for each variant of ImList:

size(Empty) = 0  
size(Cons(first:E, rest:ImList)) = 1 + size(rest)

This function is recursive. We can think about the execution of size on a particular list as a series of reduction steps:

size(Cons(0, Cons(1, Empty)))  
   = 1 + size(Cons(1, Empty))  
   = 1 + (1 + size(Empty))  
   = 1 + (1 + 0)  
   = 1 + 1  
   = 2

And the cases from the definition can be translated directly into TypeScript as methods in ImList, Empty, and Cons:

interface ImList<E> {
    // ...
    size(): number;
}

class Empty<E> implements ImList<E> {
    // ...
    public size(): number { return 0; }
}

class Cons<E> implements ImList<E> {
    // ...
    public size(): number { return 1 + this.rest.size(); }
}

This pattern of implementing an operation over a recursive datatype by

• declaring the operation in the abstract datatype interface
• implementing the operation (recursively) in each concrete variant

is a very common and practical design pattern. It sometimes goes by the unhelpful name interpreter pattern.

Let’s try a few more examples:

isEmpty : ImList → boolean

isEmpty(Empty) = true  
isEmpty(Cons(first:E, rest:ImList)) = false

contains : ImList × E → boolean

contains(Empty, e:E) = false  
contains(Cons(first:E, rest:ImList), e:E) = (first=e) or contains(rest, e)

get: ImList × number → E

get(Empty, n:number) = undefined  
get(Cons(first:E, rest:ImList), n:number) = if n=0 then first else get(rest, n-1)

append: ImList × ImList → ImList

append(Empty, list2:ImList) = list2  
append(Cons(first:E, rest:ImList), list2:ImList) = cons(first, append(rest, list2))

reverse: ImList → ImList

reverse(Empty) = empty()  
reverse(Cons(first:E, rest:ImList)) = append(reverse(rest), cons(first, empty()))

For reverse, it turns out that the recursive definition produces a pretty bad implementation in TypeScript, with performance that’s quadratic in the length of the list you’re reversing. We could rewrite it using an iterative approach if needed.

interface ImList<‍E> {
    // ...
    /**
     * @returns true iff this list is empty
     */
(missing answer)
}

class Empty<‍E> implements ImList<‍E> {
    // ...
    
(missing answer)

    {
(missing answer)
    }
}

class Cons<‍E> implements ImList<‍E> {
    private readonly first: E;
    private readonly rest: ImList<‍E>;
    // ...
    
(missing answer)

    {
(missing answer)
    }
}

interface ImList<‍E> {
    // ...
    /**
     * @param other list to append to this list
     * @returns list with the elements of this followed by the elements of other
     */
(missing answer)
}

class Empty<‍E> implements ImList<‍E> {
    // ...
    
(missing answer)

    {
(missing answer)
    }
}

class Cons<‍E> implements ImList<‍E> {
    private readonly first: E;
    private readonly rest: ImList<‍E>;
    // ...
    
(missing answer)

    {

    }
}

Tuning the rep

class Cons<E> implements ImList<E> {
    private readonly first: E;
    private readonly rest: ImList<E>;
    private cachedSize = 0;
    // rep invariant:
    //   first != null, rest != null, cachedSize >= 0
    //   cachedSize > 0 implies cachedSize == 1+rest.size()

    // ...
    public size(): number { 
        if (this.cachedSize === 0) this.cachedSize = 1 + this.rest.size();
        return this.cachedSize;
    }
}

Does our TypeScript implementation of ImList still have rep independence? We’ve concealed the Empty contructor behind the function empty(), and clients should never need to use the Empty or Cons constructors directly.

We have a great deal of freedom to change our implementation — indeed, we just added a size field to the internal rep of Cons. We could even have an extra array in there to make get() run fast! This might get expensive in space, however, but we are free to make those tradeoffs.

What about an operation like isEmpty above: does that operation break rep independence by revealing the concrete variant to clients? No! The specification of isEmpty is not, e.g., “return true iff this is an instance of the Empty concrete class.” Instead, like any operation of an abstract type, we specify isEmpty abstractly, e.g., “return true iff this list contains no elements.” Any implementation of immutable lists, whether a recursive Empty-plus-Cons implementation, an array-backed list, a linked list, or something else, can support an isEmpty operation with that spec.

Any time you write an ADT, its specs must not talk about the rep. The concrete variants of a recursive ADT are its rep, so the specs must not mention them.

Finally, is there rep exposure because Cons.rest() returns a reference to its internal list? Could a clumsy client add elements to the rest of the list? If so, this would threaten two of Cons’s invariants: that it’s immutable, and that the cached size is always correct. But there’s no risk of rep exposure, because the internal list is immutable. Nobody can threaten the rep invariant of Cons.

It might be tempting to get rid of the Empty class and just use null instead. Resist that temptation.

Using an object, rather than a null reference, to signal the base case or endpoint of a data structure is an example of a design pattern called sentinel objects. The enormous advantage that a sentinel object provides is that it acts like an object in the datatype, so you can call methods on it. So we can call the size() method even on an empty list. If empty lists were represented by null, then we wouldn’t be able to do that, and as a consequence our code would be full of tests like:

if (list !== null) n = list.size();

which clutter the code, obscure its meaning, and are easy to forget. Better the much simpler

n = list.size();

which will always work, including when an empty list refers to an Empty object.

Keep null values out of your data structures, and your life will be happier.

Now that we’re using interfaces and classes, it’s worth taking a moment to reinforce an important point about how TypeScript’s type-checking works. In fact every statically-checked object-oriented language works this way.

There are two worlds in type checking: compile time before the program runs, and run time when the program is executing. We saw this distinction when we discussed dynamic dispatch.

At compile time, every variable has a static type, also called its declared type or compile-time type, which is either stated in its declaration or inferred from its initializer. The compiler uses the static types of variables (and method return values) to deduce static types for every expression in the program. For example, after a variable declaration s: string, the expression s has static type string, s[0] has static type string, and s.length has static type number. Note that TypeScript allows a variable’s static type to be inferred from its initializer: so, for example, in let n = 5, the static type of n is number.

At run time, every object has a dynamic type, also called its actual type or runtime type, imbued in it by the constructor that created the object. For example, new String() makes an object whose dynamic type is String. new Empty() makes an object whose dynamic type is Empty. new ImList() is forbidden by TypeScript, because ImList is an interface — it has no object values of its own, and no constructors.

let hello = "Hello";

let words1:Array<string> = [];
words1.push("hello");

let words2:ImList<string> = empty();
words2.cons("hello");

Another useful sort of recursive datatype in computer science is for Boolean formulas. For instance, here’s a formula of propositional logic:

(P ∨ Q) ∧ (¬P ∨ R)

which means “either P or Q is true, and either P is false or R is true.”

We can give a datatype definition suitable for representing all formulas of propositional logic.

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

(P ∨ Q) ∧ (¬P ∨ R) would be

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

ImSet<E> = ImList<E>

A key operation for Boolean formulas is testing whether they are satisfiable, that is, whether some assignment of true/false values to the variables leads the formula to evaluate to true. There is a simple but slow algorithm for checking satisfiability:

1. Extract the set of variables from the formula.
   We’ve already implemented this with the variables operation.

2. Try all possible assignments of true/false values to those variables.
   We can represent an assignment with an Environment: a list of variables and their values. We could use ImList to implement the Environment, or develop an immutable map type.

3. Evaluate the formula for each environment.
   For this, we’ll define evaluate : Formula × Environment → boolean.

4. Return the first environment in which the formula evaluates to true.

Defining these pieces and putting them together into a satisfiable : Formula → boolean function is an exercise for another time.

We started out this reading with immutable lists, which are a representation that permits a lot of sharing between different list instances. Sharing of a particular kind, though: only the ends of lists can actually be shared. If two lists are identical at the beginning but then diverge from each other, they have to be stored separately. (Why?)

It turns out that backtracking search is a great application for these lists, and here’s why. A search through a space (like the space of assignments to a set of Boolean variables) generally proceeds by making one choice after another, and when a choice leads to a dead end, you backtrack.

Mutable data structures are typically not a good approach for backtracking. If you use a mutable Map, say, to keep track of the current variable bindings you’re trying, then you have to undo those bindings every time you backtrack. That’s error-prone and painful compared to what you do with immutable maps — when you backtrack, you just throw the map away!

But immutable data structures with no sharing aren’t a great idea either, because the space you need to keep track of where you are (in the case of the satisfiability problem, the environment) will grow quadratically if you have to make a complete copy every time you take a new step. You need to hold on to all the previous environments on your path, in case you need to back up.

Immutable lists have the nice property that each step taken on the path can share all the information from the previous steps, just by adding to the front of the list. When you have to backtrack, you stop using the current step’s state — but you still have references to the previous step’s state.

Finally, a search that uses immutable data structures is immediately ready to be parallelized. You can delegate multiple processors to search multiple paths at once, without having to deal with the problem that they’ll step on each other in a shared mutable data structure. We’ll talk about this more when we get to concurrency.

In addition to the big idea of recursive datatypes, we saw in this reading:

• datatype definitions: a powerful way to think about abstract types, particularly recursive ones
• functions over recursive datatypes: declared in the specification for the type, and implemented with one case per concrete variant
• immutable lists: a classic, canonical example of an immutable datatype

As always, we ask how these ideas make our code safer from bugs, easier to understand, and more ready for change. Look again at the definition and implementation of size() in ImList. The definition is little more than the mathematical definition of size. The code is little more than the definition, with some syntactic changes to placate the compiler.

If we examine the definitions for further methods — isEmpty, contains, etc. — in each case we see a safe, easy-to-read implementation waiting to be coded. Since we’ve taken the time to specify these operations, if we avoid rep exposure and maintain rep independence, we know our code is ready for change: different clients will be able to reuse our datatype, and we will be able to update the implementation without breaking them.