Reading 11: 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 data types — which have a recursive structure of both data and computation — take a minute to review what you know about recursive functions. You should have already practiced this heavily in 6.101 (formerly called 6.009). Here are some 6.101 readings to review:

• Recursion discusses base cases and recursive cases, recursive helper functions, and shows how to implement several functions recursively.
• Recursion and Iteration discusses the duality of recursion (a function calling itself) and iteration (loops) as ways to express repeated computation. It introduces three common patterns for recursive computation: list-like, tree-like, and graph-like. List-like and tree-like patterns will be very important in the rest of this reading.

If you are at all uncertain about what these terms mean and how to use them, take some time to review this 6.101 material before going on.

Just as a recursive function is defined in terms of itself, a recursive data type 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. Some of our recursive types will be list-like and some will be tree-like.

function factorial(n: number): number {
    if (n === 0) {
        return 1; // this is called the base case
    } else {
        return n * factorial(n-1); // this is the recursive step
    }
}

function fibonacci(n: number): number {
    if (n === 0 || n === 1) {
        return 1; // base cases
    } else {
        return fibonacci(n-1) + fibonacci(n-2); // recursive step
    }
}

/**
 * @param array must be nonempty
 * @returns the maximum integer in array
 */
function maxArray(array: Array<number>): number {
  return maxOfRange(array, 0, array.length-1);
}

// helper function -- finds the max of array[start]...array[end]
function maxOfRange(array: Array<number>, start: number, end: number): number {
  if (start === end) {
    const elem = array[start];
    array.splice(start, 1);
    return elem;
  } else {
    const midpoint = Math.floor((start + end + 1) / 2);
    return Math.max(maxOfRange(array, start, midpoint), maxOfRange(array, midpoint+1, end));
  }
}

We’ll start with a classic recursive data type, 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 data type, 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 data type 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 data type, 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 empty<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> = empty();	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
  ) {}
}

/**
  * Represents an immutable tweet from Twitter.
  */
class Tweet {
  private readonly _timestamp: Date;

  public constructor(
    public readonly author: string, 
    public readonly text: string,
    timestamp: Date
  ) {
    this._timestamp = timestamp;
  }
  public get timestamp(): Date {
    return new Date(this._timestamp);
  }
}

const author = "6.102";
const text = "SFB, RFC, ETU";
const timestamp = new Date(); // current time

const tweet = new Tweet(author, text, timestamp);

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 data type 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 data type becomes far more visible when written in a data type 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 data type definition has:

• an abstract data type on the left, defined by its representation (or concrete data type) on the right
• the representation consists of variants of the data type 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 data type definition is a data type 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.

Data type definitions are called algebraic data types in functional programming. The syntax we are using here is designed for object-oriented programming languages like TypeScript, Python, and Java, in which classes and interfaces are used to implement the recursive data type. Functional programming languages like Haskell and ML use a different syntax, but the idea is the same: the type consists of a union of variants, some of which may use the type recursively.

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

interface ImList<E> {

    // Data type 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)

T = A(x: number) + B(a1: A, a2: A)
U = D(x: number, y: number)
V = F(z: number, v: V) + G(z: number, v: V)
W = K(n: number) + L(n: number, m: W)
X = M + N(here: number, there: X)

V = F(z: number, v: V) + G(z: number, v: V)

W = K(n: number) + L(n: number, m: W)
X = M + N(here: number, there: X)

This way of thinking about data types — as a recursive definition of an abstract data type 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 data type, as functions with one case per variant.

First, notice how the data type 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 in ImList, Empty, and Cons:

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

class Empty<E> implements ImList<E> {
    // ...
    public readonly size = 0;
}

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

This pattern of implementing an operation over a recursive data type by

• declaring the operation in the abstract data type 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: number|undefined = undefined;
    // rep invariant:
    //   cachedSize is either a positive integer or undefined

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

Does our TypeScript implementation of ImList still have rep independence? We’ve concealed the Empty constructor 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 cachedSize 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 or undefined 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 data type, so you can call methods on it. So we can call the size operation even on an empty list. If empty lists were represented by null or undefined, 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(), on the other hand, is forbidden by TypeScript, because ImList is an interface — it has no object values of its own, and no constructors.

In short, variables at compile time have static types, whereas objects at run time have dynamic types.

const hello = "Hello";

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

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

Dynamic dispatch makes it possible to run different code depending on the dynamic type of this, and we’re using it to run the variant-specific implementations when clients call the operations of our recursive type.

You may find that you are tempted to inspect the runtime type of an object, especially with recursive types, as you implement an algorithm. This is a bad idea, an anti-pattern that will quickly get us mired in weeds.

For example, suppose we want to implement:

last: ImList → E
// returns the last element of the list, requires the list to be nonempty

Do not do it this way:

class Cons<E> implements ImList<E> {
    // ...
    public last(): E {
        if (this.rest instanceof Empty) { return this.first; } // do not do this
        return this.rest.last();
    }
}

The instanceof operator tests whether an object is an instance of a particular type. It is dynamic type checking, and it is both less safe from bugs and less ready for change than static type checking. The fundamental reason is that new variants might be added to the type. For operations implemented as instance methods of the interface, like first and rest, those new variants will be required to make sure that the operations are correctly implemented, or the code won’t even compile. (Static checking!) But for operations implemented like this – using instanceof – the compiler won’t help you find the places that you need to change to handle the new variants, and some of those places might quietly return wrong answers instead of throwing an error.

For example, suppose someone were to later on introduce a CachingList variant that cached the results of some otherwise expensive operations, like size. This is a different way of doing the caching we did above, but now using a common design pattern called a wrapper (also known as a decorator):

class CachingList<E> implements ImList<E> {
    private readonly wrappedList: ImList<E>;
    private readonly cachedSize: number;
    // rep invariant:
    //   cachedSize === wrappedList.size

    // ... constructor and other operations omitted

    public get first(): E {
      return wrappedList.first;
    }
    public get rest(): ImList<E> {
      return wrappedList.rest;
    }
    public get size(): number {
      return cachedSize;
    }
}

With CachingList added to the system, there are now multiple ways to represent an empty list: an instance of Empty, or an instance of CachingList wrapped around an Empty. In general, any instanceof checks that assumed that Cons and Empty were the only ways to make lists, are now broken, without any static type error. Do not use instanceof.

How, then, can we implement the last operation on ImList? Any time the thought of instanceof seems appealing, take a step back and rethink the problem.

In trying to decide whether its first element is, in fact, also the last element, Cons doesn’t care about the representation of rest, it only cares about its abstract value. If our type doesn’t provide enough operations for clients (in this case, we are both implementers of ImList and, recursively, clients of it) to do what they want, we can make the type more powerful by adding operations:

class Cons<E> implements ImList<E> {
    // ...
    public last(): E {
        if (this.rest instanceof Empty ________________) { return this.first; }
        return this.rest.last();
    }
}

One place where using instanceof may be the least-bad option is defining an equalValue() operation for immutable types.

In Equality when we defined equalValue(..) for our immutable TypeScript types, we required that the type of argument that match the type of this. And on the surface, that makes sense.

But in languages where equality is defined as part of the language, as we saw with Python’s __eq__, the spec of the equality operation may allow clients to call it with any other object as an input. Since you can’t make any useful determination about equality without knowing the type, you must first use instanceof (in Python, isinstance).

And a similar problem exists for a recursive type. Suppose we want to define equalValue(..) for ImList: what will be the type of the argument? We might wish to define only equalValue(Empty<E>) in Empty, and only equalValue(Cons<E>) in Cons. But clients only know about the type ImList, so we need a signature like equalValue(ImList<E>) in the ImList interface that both concrete variants will implement.

There are two implementation options:

1. Define observers that allow us to make the equality determination without knowing the concrete variant of that. For example, if we have both size and get(..) as defined above:

   interface ImList<E> {
      // ... includes size and get(..) ...
      equalValue(that: ImList<E>): boolean;
   }
   class Empty<E> implements ImList<E> {
      public equalValue(that: ImList<E>): boolean {
          return that.size === 0;
      }
   }
   class Cons<E> implements ImList<E> {
      public equalValue(that: ImList<E>): boolean {
          if (this.size !== that.size) { return false; }
          for (let ii = 0; ii < this.size; ii++) {
              if (this.get(ii) !== that.get(ii)) { return false; }
          }
          return true;
      }
   }

   This approach is quite clunky (harder to understand), but effective, and keeps everything above the abstraction barrier (safer from bugs, if all those operations are well-tested).

2. Check the runtime type:

   interface ImList<E> {
      // ...
      equalValue(that: ImList<E>): boolean;
   }
   class Empty<E> implements ImList<E> {
      public equalValue(that: ImList<E>): boolean {
          return that instanceof Empty;
      }
   }
   class Cons<E> implements ImList<E> {
      public equalValue(that: ImList<E>): boolean {
          if ( ! (that instanceof Cons)) { return false; }
          return this.first === that.first && this.rest.equalValue(that.rest);
      }
   }

   This approach has the frisson of runtime type inspection (less safe from bugs), but is elegant and with a recursive structure that matches the data (easier to understand). It’s also far less likely that introducing new variants will change the checking of equality in existing variants.

   Note that this implementation relies on the fact that a given abstract immutable list can ony be represented by exactly one rep value: if there are alternative reps, with a different structure of concrete variant instances, we need to correctly account for those alternatives.

Another useful sort of recursive data type 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 data type 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 finding a satisfying assignment if it exists:

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, or fail if it’s not possible.

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

SAT Solver enthusiasts may be quick to point out that a faster algorithm exists which makes use of backtracking. Can our Formula recursive data type handle this improvement? As you’ll see in the next section, the answer is it can, and quite well!

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.

Since we’re talking about immutable types here, this is a good moment to discuss performance as well. One common reason for choosing mutability rather than immutability is performance optimization. But it’s not correct to conclude that a mutable type is always more efficient than an immutable type.

The key question to think about is how much sharing is possible, and how much copying is required. An immutable value may be safely shared by many different parts of a program, where a mutable value in the same context would have to be defensively copied over and over, at a cost of both time and space. On the other hand, if a value needs to be edited, in the sense of having many small mutations made to it, then a mutable value may be more efficient, because it doesn’t require the whole value to be copied on every edit.

But even when an immutable value needs to be heavily edited, it may still be possible to design it in a way that exploits sharing in order to reduce copying. For example, if you have an immutable string that is a million characters long, editing a single character in the middle of the string seems to require copying all the unchanged characters too. But a clever string implementation might internally share the unchanged regions of characters before and after the edit, so that even though you get a new string object as a result of the edit, it actually occupies very little additional memory space. We will see an example of this kind of implementation in a few classes, when we talk about abstract data types. This kind of internal sharing is only possible because of immutability.

The git object graph is another example of the benefit of sharing. Because commits are immutable, other commits can point to them without fear that they will become broken because the parent commit is modified (e.g. reverted or undone). Without immutability, this kind of structural sharing is impossible.

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

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

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 data type, and we will be able to update the implementation without breaking them.