Chapter 3. Functional data structures

We said in the introduction that functional programs don't update variables or modify mutable data structures (except in iterative loops, of course). So what kinds of data structures can we use? Functional data structures, of course! This chapter talks about how we define data types in functional programming and a related logical branching technique called pattern matching. It has a ton of exercises, too, to provide practice writing and generalizing pure functions. In general, the book is trying to lead us to higher level functional abstractions - things like monoids, functors, monads, and other structures with weird names borrowed from category theory.

Defining functional data structures

A functional data structure is only operated on with pure functions. Pure functions cannot modify data in place or perform side effects. Thus, by definition, functional data structures are immutable.

Let's start by talking about everyone's favorite functional data structure - a singly linked list. Of course, we're going to dive into some new TypeScript syntax, as well.

/**
 * list.ts - a functional singly linked list implementation using a
 * discriminated union, a.k.a. algebraic data type (ADT). Based on
 * guidance at
 * https://www.typescriptlang.org/docs/handbook/advanced-types.html
 */

/**
 * The "root" definition of our List data type, with one type parameter, `A`.
 * Note that List is defined entirely by its constituent types, using
 * TypeScript's type union syntax, `|`.
 */
export type List<A> = Cons<A> | Nil;

/**
 * Nil represents the empty list
 */
export interface Nil {
  tag: "nil";
}

/**
 * Our "data constructor" for Nil, which has only one instance
 */
export const Nil: Nil = { tag: "nil" };

/**
 * A data constructor representing a link in the list, containing a value in
 * `head` and a pointer to the remainder of the list in `tail`. Every tail is
 * a complete List in its own right, with Nil signifying the "end" of the
 * list.
 */
export class Cons<A> {
  tag: "cons" = "cons";

  constructor(readonly head: A, readonly tail: List<A>) { }
}

/**
 * Creates a List from a variable number of arguments.
 */
export const List = <A>(...vals: A[]): List<A> => {
  if (vals.length === 0)
    return Nil;
  else
    return new Cons(vals[0], List(...vals.slice(1)));
};

/**
 * Uses recursion and pattern matching to add up a list of numbers
 */
export const sum = (ns: List<number>): number => {
  switch (ns.tag) {
    // the sum of the empty list is 0
    case "nil": return 0;

    // the sum of a list is the value in the head plus the sum of the tail
    case "cons": return ns.head + sum(ns.tail);
  }
};

/**
 * Multiplies a list of numbers together, using the same technique as `sum`
 */
export const product = (ns: List<number>): number => {
  switch (ns.tag) {
    case "nil": return 1.0;
    case "cons":
      if (ns.head === 0.0)
        return 0.0;
      else
        return ns.head * product(ns.tail);
  }
};
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One bit of new syntax shows up twice in the List function: the ... operator. It's actually two different operators that happen to look the same and do related things. In const List = <A>(...vals: A[]), the ... is the rest-parameter operator. Other languages call this variadic function syntax. Either way, the result is a function that accepts a variable number of arguments of type A, which are represented inside the function as an array.

Later, in List(...vals.slice(1)), the ... is called the spread operator. If we think of a rest parameter as "collapsing" a parameter list into an array, then the spread operator does the opposite. It spreads an array into a parameter list.

A second bit of new syntax is in the constructor of Cons. Recall that the Charge class of Chapter 1 defines a couple properties and initializes them in its constructor.

class Charge {
  readonly cc: CreditCard;
  readonly amount: number;

  constructor(cc: CreditCard, amount: number) {
    this.cc = cc;
    this.amount = amount;
  }
}
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This is a pretty common thing to need to do, so TypeScript provides a shortcut. The following code achieves the exact same result, but is much less verbose.

class Charge {
  constructor(readonly cc: CreditCard, readonly amount: number) { }
}
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The Cons constructor, along with the constructor functions of many data types throughout this book, uses this shortcut.

If you're familiar with object-oriented programming, and someone asks you to create a data type called List with two implementations, you probably immediately reach for some kind of subtyping, maybe with List as the parent interface or abstract class and two implementors or subclasses, Nil and Cons. We don't see that in the code above, so what exactly is going on here?

In FP, we often use a pattern called an Algebraic Data Type (ADT), also called a discriminated union, to represent data structures. This kind of structure ends up being more natural to work with than traditional OOP class hierarchies. It also enables a technique called pattern matching, which allows us to get some help from the compiler when writing functions for ADTs.

TypeScript, being both a "lightweight" language and something of a hybrid beast between OOP and FP, does not directly support either ADTs or pattern matching, but does have some lower-level features that allow us to "build up" to those techniques. Strap in, we've got some reading to do.

Representing algebraic data types in TypeScript

Our goal is to define an ADT that represents a singly linked list. Lists of this kind are useful tools for exploring FP because they can be defined and constructed recursively: each subpart of the list is itself a list. Therefore, we need two variants of our List type: one to represent a link in the list and one to represent the empty list. We call the variants data constructors, and they are Cons and Nil. The List type is defined as the union of Cons and Nil.

Now, we could have chosen to create a parent class or interface for List and still achieved roughly the same thing. We didn't do that for a couple reasons. There is no common functionality between Cons and Nil, so a class isn't really appropriate.

The reasoning behind not using an interface is a bit more complex. Unlike traditional class hierarchies, an ADT is not meant to be extended. The "algebraic" descriptor means that our data type is really a formulation of other, known data types, which cannot be true if there is the possibility of someone else adding a variant to our ADT! So we want a way to define a closed set of types that allows us to write functions that accept any member of the set and prevents the addition of new members. Enter TypeScript's union type:

type List<A> = Cons<A> | Nil;
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This declares a type alias named List, with one type parameter A, that can be either a Cons<A> or Nil. We cannot add types to List after it has been declared, but we can write functions that accept any List value. In the snippet below, ns will be either a Cons<number> or Nil.

const sum = (ns: List<number>): number => ...
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The union type is the first TypeScript feature that we're using to implement an ADT. The second is the literal type. You may have noticed that both Nil and Cons have a property called tag, each with an unusual type annotation: "nil" and "cons", respectively. The code tag: "nil" declares the tag property to be of the literal type "nil", which is a kind of constrained string type that can only ever hold one value: "nil". Attempting to assign any other value to it results in a compiler error. Because literal types can only hold one value (i.e. have only one inhabitant), they are also called singleton types. ADTs, or discriminated unions, are also called tagged unions, hence the property name tag. We could have called it discriminator, but that's kind of a long word and we're lazy.

To understand why the tags matter, we need to talk about a third TypeScript feature called type guards. A type guard is a conditional that guarantees a value has a specific type. Inside the conditional, we can then safely treat the value as that type. For example:

const printHead = <A>(ls: List<A>) => {
  if (ls.tag === "cons") {
    console.log(`The head is ${ls.head}`);
  } else {
    console.log("We're headless!");
  }
};
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This code compiles because ls.tag === "cons" is a type guard, allowing the TypeScript compiler to know that, inside the if block, ls must be a Cons, so that the ls.head property access is legal. This is type inferencing at work, and it hinges on TypeScript knowing that every variant of List has a tag property, but only one variant has a tag property that can possibly hold a value of "cons" (thanks, literal types!), so that if ls.tag === "cons" is true, then ls must be a Cons.

There is one more feature of ADTs often supported by FP languages that we have not yet achieved, which is exhaustiveness checking. It would be helpful for the compiler to warn us if we haven't handled every variant of an ADT, in order to avoid undefined behavior. Fortunately, once again due to TypeScript's type inferencing, we don't have to do anything special. Consider this function:

const getHead = <A>(ls: List<A>): A => {
  if (ls.tag === "cons") {
    return ls.head;
  }
};
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Attempting to compile it leads to this error:

[eval].ts(13,35): error TS2366: Function lacks ending return statement and
return type does not include 'undefined'.
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This is a bit of a subtle message, but makes more sense if you keep in mind that a function lacking an explicit return value returns undefined. In strict mode, TypeScript does not allow us to have undefined or null values unless we explicitly allow it, which we won't get into now.

We'll often use type guards and exhaustiveness checking with switch statements, which gives us something close to the pattern matching capability offered by other FP languages. Hopefully, the sum function for List makes complete sense to you now:

const sum = (ns: List<number>): number => {
  switch (ns.tag) {
    case "nil": return 0;
    case "cons": return ns.head + sum(ns.tail);
  }
};
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Exercise 3.1. Type inference

Given these declarations:

interface Nil { tag: "nil"; }

interface SingleCons<A> {
  tag: "cons";
  head: A;
  tail: SingleList<A>;
}
type SingleList<A> = SingleCons<A> | Nil;

interface DoubleCons<A> {
  tag: "cons";
  head: A;
  tail: DoubleList<A>;
  prev: DoubleList<A>;
}
type DoubleList<A> = DoubleCons<A> | Nil;

type AnyList<A> = SingleList<A> | DoubleList<A>;
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What is the result of compiling the following snippets?

const foo = <A>(l: SingleList<A>) => {
  if (l.tag === "cons")
    console.log(l.head);
};
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Answer

This snippet compiles. The function argument l is constrained to be of type SingleList<A>. Within that type, only SingleCons<A> has a tag property that can contain the value "cons", so the compiler has enough information to lookup the head property inside the if statement.

const bar = <A>(l: AnyList<A>) => {
  if (l.tag === "cons")
    console.log(l.head);
};
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Answer

This snippet compiles. The function argument l is no longer constrained to a SingleList<A>; now it can be either that or a DoubleList<A>. Within those types, both SingleCons<A> and DoubleCons<A> have a tag of "cons". So the compiler can't narrow l down to only one data constructor inside the if statement. But, it doesn't matter, because both possibilities support the head property, and it has the same type for each.

const baz = <A>(l: AnyList<A>) => {
  if (l.tag === "cons")
    console.log(l.prev);
};
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Answer

This snippet does not compile. Again, inside the if statement, the type of l has been inferred to be either SingleCons<A> or DoubleCons<A>. But we try to access the prev property, which only exists on DoubleCons. Attempting to compile this leads to an error message similar to:

TSError: ⨯ Unable to compile TypeScript:
test.ts(34,19): error TS2339: Property 'prev' does not exist on type 'SingleCons<A> | DoubleCons<A>'.
  Property 'prev' does not exist on type 'SingleCons<A>'.
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Data sharing in functional data structures

How do we write functions that actually do things with immutable data structures? For instance, how can we add or remove elements from a List? To add a 1 to the front of a list named xs, we just create a new Cons with the old list as the tail: new Cons(1, xs). Since List is immutable, we dont' have to worry about anyone changing a list out from under us, so it's safe to reuse in new lists, without pessimistically copying it. According to the book:

...in the large, FP can often achieve greater efficiency than approaches that rely on side effects, due to much greater sharing of data and computation.

Similarly, to remove an element from the front of a list, we just return its tail.

                       Data Sharing

┌─────┬─────┐  ┌─────┬─────┐  ┌─────┬─────┐  ┌─────┬─────┐
│ "a" │  ●──┼─>│ "b" │  ●──┼─>│ "c" │  ●──┼─>│ "d" │  ●  │
└─────┴─────┘  └─────┴─────┘  └─────┴─────┘  └─────┴─────┘
   ↑              ↑
   |              ╰──────────────────╮
   |                                 |
   ╰─ List("a", "b", "c", "d")       │
      |                              |
      |         ╭───────╮            │
      ╰─────────┤ .tail ├─────────> List("b", "c", "d")
                ╰───────╯
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And now... some exercises for manipulating lists.

Note

As we complete these exercises, we'll be creating a reusable List module. You can find the code for List in the code repo at /fpbookclub/data_structures/list.ts.

Exercise 3.2. getTail

Implement the getTail function for removing the first element of a list. Notice that this function takes constant time.

const getTail = <A>(l: List<A>): List<A> => ...
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Answer
// This version throws an exception if the caller passes an empty list, but
// there are several equally "correct" ways to handle this scenario, such as
// returning `Nil`. It's up to you as the library designer to choose a method!
const getTail = <A>(l: List<A>): List<A> => {
  if (l.tag === "nil")
    throw new Error("Attempt to take tail of empty list");

  return l.tail;
};

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Exercise 3.3. setHead

Now implement setHead, which replaces the first element of a list with a different value. For example, setHead(List(1, 2, 3), 4) should return List(4, 2, 3).

const setHead = <A>(l: List<A>, head: A): List<A> => ...
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Answer
const setHead = <A>(l: List<A>, a: A): List<A> => {
  if (l.tag === "nil")
    throw new Error("Attempt to set head of empty list");

  return new Cons(a, l.tail);
};
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The following exercises demonstrate the efficiency of data sharing.

Exercise 3.4. drop

Generalize tail to drop, which removes the first n elements from a list. Note that this function takes time proportional to n, and we do not have to make any copies.

const drop = <A>(l: List<A>, n: number): List<A> => ...
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Answer
const drop = <A>(l: List<A>, n: number): List<A> => {
  if (l.tag === "nil")
      throw new Error("Attempt to drop from empty list");

  if (n <= 0) return l;
  else return drop(l.tail, n - 1);
};
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Exercise 3.5. dropWhile

Implement dropWhile, which removes elements from the front of a list as long as they match a predicate.

const dropWhile = <A>(l: List<A>, f: (a: A) => boolean): List<A> => ...
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Answer
const dropWhile = <A>(l: List<A>, p: (a: A) => boolean): List<A> => {
  if (l.tag === "nil")
      throw new Error("Attempt to drop from empty list");

  if (p(l.head))
    switch (l.tail.tag) {
      case "nil": return l.tail;
      default: return dropWhile(l.tail, p);
    }

  return l;
};
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A "more surprising" example of data sharing, this append implementation's runtime and memory usage are determined entirely by the first list. The second is just tacked onto the end, so to speak.

const append = <A>(a1: List<A>, a2: List<A>): List<A> => {
  switch (a1.tag) {
    case "nil":
      return a2;
    case "cons":
      return new Cons(a1.head, append(a1.tail, a2));
  }
};
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Exercise 3.6. init

Implement a function, init, that returns a List consisting of all but the last element of another List. Given List(1, 2, 3, 4) your function should return List(1, 2, 3). Why can't this function be implemented in constant time, like tail?

const init = <A>(l: List<A>): List<A> => ...
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Answer
const init = <A>(l: List<A>): List<A> => {
  if (l.tag === "nil")
    throw new Error("Attempt to get init of empty list");

  if (l.tail === Nil)
    return Nil;

  return new Cons(l.head, init(l.tail));
};
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Due to the structure and immutability of a Cons, any time we want to replace the tail of a list, we have to create a new Cons element with the new tail value and the old head value. If that particular Cons was itself the tail of a different Cons, that element now needs to be copied as well, and so on until we reach the head of the list. According to the book:

Writing purely functional data structures that support different operations efficiently is all about finding clever ways to exploit data sharing. As an example of what’s possible, in the Scala standard library there’s a purely functional sequence implementation, Vector (documentation here), with constant-time random access, updates, head, tail, init, and constant-time additions to either the front or rear of the sequence. See the chapter notes for links to further reading about how to design such data structures.

Recursion over lists and generalizing to higher-order functions

Take another look at sum and product. These versions are simplified a bit compared to what you saw before.

const sum = (ns: List<number>): number => {
  switch (ns.tag) {
    case "nil":  return 0;
    case "cons": return ns.head + sum(ns.tail);
  }
};

const product = (ns: List<number>): number => {
  switch (ns.tag) {
    case "nil":  return 1.0;
    case "cons": return ns.head * product(ns.tail);
  }
};
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Notice how similar they are? They both return a constant value in the "nil" case. In the "cons" case, they both call themselves recursively and combine the result with head using a binary operator. When you see this level of structural similarity, you can often write one generalized polymoprhic function with the same shape that takes function parameters for the differing bits of logic. In this example, our general function needs a normal single-valued parameter for the "nil" case and a function parameter for the operation to apply in the "cons" case.

const foldRight = <A, B>(l: List<A>,
                         z: B,
                         f: (a: A, b: B) => B): B => {
  switch (l.tag) {
    case "nil": return z;
    case "cons": return f(l.head, foldRight(l.tail, z, f));
  }
};

const sum = (ns: List<number>): number =>
  foldRight(ns, 0, (a, b) => a + b);

const product = (ns: List<number>): number =>
  foldRight(ns, 1.0, (a, b) => a * b);
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foldRight can operate on lists of any element type, and, it turns out, can return any type of value, not just the type contained in the list! It just so happens that the types of the results of sum and product match the element types of their List arguments (that is, number). In a sense, foldRight replaces the data constructors of the list, Cons and Nil, with f and z, respectively:

Cons(1, Cons(2, Nil))
f   (1, f   (2, z  ))
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Here's a more complete example, tracing the evaluation of foldRight(List(1, 2, 3), 0, (x, y) => x + y) by repeatedly substituting the definition of foldRight for its usages.

foldRight(Cons(1, Cons(2, Cons(3, Nil))), 0, (x, y) => x + y)
1 + foldRight(Cons(2, Cons(3, Nil)), 0, (x, y) => x + y)
1 + (2 + foldRight(Cons(3, Nil), 0, (x, y) => x + y))
1 + (2 + (3 + foldRight(Nil, 0, (x, y) => x + y)))
1 + (2 + (3 + (0)))
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You can see that foldRight has to traverse the list all the way to the end (using stack frames for each recursive call as it goes) before it can fully evaluate the result. In other words, it is not tail recursive.

Exercise 3.7. foldRight and short-circuiting

In our simplified version of product, above, we left out the "short-circuit" behavior of immediately returning 0 when any list element is 0. Can we regain this behavior, now that product is implemented in terms of foldRight? Why or why not?

Answer

We cannot, because the way foldRight iterates through the list is hidden as an implementation detail. There is nothing in foldRight's signature that would let us specify a short-circuiting behavior.

Exercise 3.8. Relationship between foldRight and List data constructors

See what happens when you pass Nil and Cons themselves to foldRight, like this: foldRight(List(1, 2, 3), Nil, (h, t) => new Cons(h, t)). What does this say about the relationship between foldRight and the data constructors of List?

Answer

You get a copy of the original list. You could say that foldRight replaces the Nil constructor with the z value and the Cons constructor with the result of applying the function f.

Exercise 3.9. length via foldRight

Compute the length of a list using foldRight.

const length = <A>(l: List<A>): number => ...
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Answer
const length = <A>(l: List<A>): number =>
  foldRight(l, 0, (a, b) => b + 1);
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Exercise 3.10. foldLeft

Our implementation of foldRight is not tail-recursive and therefore not stack-safe. Write another function, foldLeft, that is stack-safe. Recall that even tail recursion isn't stack-safe in TypeScript. Because our List library will be reused in future exercises, we should probably spend some time making it scale well. Therefore, this is one of the few times that we'll sacrifice our functional ideals for practical usability gains. Your function should use an iterative loop and local state mutations, rather than recursion, to achieve stack safety. It may be helpful to start by writing a tail-recursive version of foldLeft and then transform it into an iterative version.

const foldLeft = <A, B>(l: List<A>, z: B, f: (b: B, a: A) => B): B => ...
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Answer
// tail-recursive solution
const foldLeft = <A, B>(l: List<A>, z: B, f: (b: B, a: A) => B): B => {
  const go = (la: List<A>, acc: B): B => {
    if (la.tag === "nil")
      return acc;
    else
      return go(la.tail, f(acc, la.head));
  };

  return go(l, z);
};

// iterative solution
const foldLeft = <A, B>(l: List<A>, z: B, f: (b: B, a: A) => B): B => {
  var state = z;
  while (l.tag !== "nil") {
    state = f(state, l.head);
    l = l.tail;
  }
  return state;
};
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Exercise 3.11. Refactor sum, product, and length

Rewrite sum, product, and length in terms of foldLeft.

Answer
const sum = (ns: List<number>): number =>
  foldLeft(ns, 0, (tot, n) => n + tot);

const product = (ns: List<number>): number =>
  foldLeft(ns, 1.0, (prod, n) => n * prod);

const length = <A>(l: List<A>): number =>
  foldLeft(l, 0, (len, a) => len + 1);
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Exercise 3.12. reverse

Write a function that returns the reverse of a list. See if you can do it using a fold.

const reverse = <A>(l: List<A>): List<A> => ...
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Answer
const reverse = <A>(l: List<A>): List<A> =>
  foldLeft(l, List(), (rev, a) => new Cons(a, rev));
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Exercise 3.13. foldRight and foldLeft in terms of each other

Can you write foldLeft in terms of foldRight? How about the other way around? If we can express foldRight in terms of foldLeft, we gain the ability to fold a list right-wise in a stack-safe way.

Answer

To implement foldLeft in terms of foldRight, we can't just reverse the list and call foldRight, because reverse calls foldLeft, which would lead to an infinite loop. We could rewrite reverse, but that's no fun!

Thinking about reverse leads us to a useful insight: we need to somehow "reverse" the usual order of execution of foldRight. One strategy is to delay evaluation of the combiner function, f. So rather than immediately computing a result, we can use foldRight to build up a function that, when evaluated, applies the combiner function to the list elements in first-to-last order, instead of foldRight's usual last-to-first order.

const foldLeft = <A, B>(l: List<A>, z: B, f: (b: B, a: A) => B): B => {
  // This is the result type for our inner call to `foldRight`, a function
  // that transforms a value of type `B` to a result also of type `B`. Such a
  // function is called an "endomorphism", hence the name of our type alias:
  // `END_B`.
  type END_B = (b: B) => B;

  // This is the `z` value for the call to `foldRight`, which is a simple
  // identity function.
  const id: END_B = b => b;

  // This is the combiner function for the call to `foldRight`. It takes a
  // list element, `a`, and an `END_B` function, `g`, and returns another
  // `END_B` function that, when applied, calls our original combiner
  // function, `f`, with the provided `B` value and the captured `A` value,
  // and then passes the result to the captured `END_B` function.
  const deferred = (a: A, g: END_B) => (b: B) => g(f(b, a));

  // We use `foldRight` to build up our reversing function, call it with the
  // original `z` value, and return the result.
  return foldRight<A, END_B>(l, id, deferred)(z);
};
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Implementing foldRight in terms of foldLeft is much more straightforward. We just reverse the list and pass it to foldLeft. Since this gives us a nice, stack-safe foldRight implementation, we'll keep it.

const foldRight = <A, B>(l: List<A>, z: B, f: (a: A, b: B) => B): B =>
  foldLeft(reverse(l), z, (b, a) => f(a, b));
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Exercise 3.14. Refactor append

Rewrite append using either fold.

Answer
const append = <A>(a1: List<A>, a2: List<A>): List<A> =>
  foldRight(a1, a2, (a, b) => new Cons(a, b));
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Exercise 3.15 concat

Write a function that concatenates a list of lists into a single list, using functions we've already defined. The runtime of concat should be proportional to the total length of all lists.

const concat = <A>(ll: List<List<A>>): List<A> => ...
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Answer
const concat = <A>(ll: List<List<A>>): List<A> =>
  foldRight(ll, List(), (a, b) => append(a, b));
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More functions for working with lists

The next set of exercises introduces a few more useful list functions. For each one, we follow the pattern of trying to accomplish some specific tasks, noticing the commonality between them, writing the general function, and then refactoring the tasks using our new tool. The purpose of doing these exercises is not to commit to memory every list function and when to use it, but to begin to develop an intuition for detecting patterns in working with lists and functional data structures in general. As we proceed through the book, we'll see that these patterns apply to a variety data structures beyond the humble list, and that there are opportunities for extracting these patterns into highly abstract functions that we can use in any domain.

Exercise 3.16. Add 1 to each element

Write a function that transforms a list of integers by adding 1 to each element. (Reminder: this should be a pure function that returns a new List!)

Answer
const addOne = (l: List<number>): List<number> =>
  foldRight(l, List(), (a, b) => new Cons(a + 1, b));
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Exercise 3.17. Convert number to string

Write a function that turns each value in a List<number> into a string. You can use the expression n.toString() to convert some n: number to a string.

Answer
const toString = (l: List<number>): List<string> =>
  foldRight(l, List(), (a, b) => new Cons(a.toString(), b));
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Exercise 3.18. map

Write a function map that generalizes modifying each element in a list while maintaining the structure of the list. Then, refactor your answers to the previous two exercises using map. Here is its signature:

const map = <A, B>(l: List<A>, f: (a: A) => B): List<B> => ...
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Answer
const map = <A, B>(la: List<A>, f: (a: A) => B): List<B> =>
  foldRight(la, List(), (a, b) => new Cons(f(a), b));
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Exercise 3.19. filter

Write filter, a function that removes elements from a list unless they satisfy a predicate. Then practice using it by removing all odd numbers from a List<number>. Is it possible to implement filter in terms of map? Why, or why not?

const filter = <A>(l: List<A>, p: (a: A) => boolean): List<A> => ...
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Answer
const filter = <A>(l: List<A>, p: (a: A) => boolean): List<A> =>
  foldRight(l, List(), (a, acc) => p(a) ? new Cons(a, acc) : acc);
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Exercise 3.20. flatMap

Write a function named flatMap that works similarly to map, except the provided function returns a List instead of a raw value. That List should be inserted into the resulting List. For example, flatMap(List(1, 2, 3), i => List(i, i)) should return List(1, 1, 2, 2, 3, 3).

const flatMap = <A, B>(l: List<A>, f: (a: A) => List<B>): List<B> => ...
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Answer
const flatMap = <A, B>(l: List<A>, f: (a: A) => List<B>): List<B> =>
  foldRight(l, List(), (a, acc) => append(f(a), acc));
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Exercise 3.21. filter in terms of flatMap

Re-implement filter using flatMap. As a bonus, can you implement map in terms of flatMap?

Answer
const filter = <A>(l: List<A>, p: (a: A) => boolean): List<A> =>
  flatMap(l, a => p(a) ? List(a) : List());

const map = <A, B>(la: List<A>, f: (a: A) => B): List<B> =>
  flatMap(la, a => List(f(a)));
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Exercise 3.22. Add corresponding elements

Write a function that, given two lists, returns a new list consisting of the sums of corresponding elements in the argument lists. For example, given List(1, 2, 3) and List(4, 5, 6), it should return List(5, 7, 9).

Answer
const addCorresponding = (a1: List<number>, a2: List<number>): List<number> => {
  return foldRight(
    a1,
    [List() as List<number>, reverse(a2)],
    (a, [acc, other]) => {
      if (other.tag === "cons")
        return [new Cons(a + other.head, acc), other.tail];
      else
        return [acc, other];
    }
  )[0];
};
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Exercise 3.23. zipWith

Generalize the previous function so that it's not specific to numbers or addition. Call it zipWith, and then refactor the previous function to use zipWith.

Answer
const zipWith = <A, B, C>(
    la: List<A>,
    lb: List<B>,
    f: (a: A, b: B) => C): List<C> => {
  if (la.tag === "nil" || lb.tag === "nil")
    return Nil;
  else
    return new Cons(f(la.head, lb.head), zipWith(la.tail, lb.tail, f));
};

// or, using a fold:
const zipWith = <A, B, C>(
    la: List<A>,
    lb: List<B>,
    f: (a: A, b: B) => C): List<C> => {
  return reverse(foldLeft<A, [List<C>, List<B>]>(
    la,
    [Nil, lb],
    ([acc, rem], a) => {
      if (rem.tag === "cons")
        return [new Cons(f(a, rem.head), acc), rem.tail];
      else
        return [acc, rem];
    }
  )[0]);
};

// and the refactored `addCorresponding`:
const addCorresponding = (a1: List<number>, a2: List<number>): List<number> => {
  return zipWith(a1, a2, (n1, n2) => n1 + n2);
};
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Loss of efficiency when assembling list functions from simpler components

Sometimes, when we express List operations in terms of general-purpose functions, we end up with inefficient implementations, even though the code ends up being very concise and readable. We may wind up making several passes over input lists, or having to write explicit loops to allow for early termination.

Exercise 3.24. hasSubsequence

Write a function, called hasSubsequence, to check whether a List contains another List as a subsequence. For example, List("a", "b", "r", "a") has subsequences List("b", "r") and List("a"), among others. This is meant to illustrate the previous point about efficiency, so you will probably have some difficulty expressing this in a purely functional, efficient manner. We'll return to this problem, and hopefully find a better answer, in Chapter 5.

const hasSubsequence = <A>(sup: List<A>, sub: List<A>): boolean => ...
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Answer
// This is just one of many possible implementations, none of which can be
// both succinct and efficient. This version of the function, for example,
// is terse, but must process every element of `sup`, even if the subsequence
// has already been found. Our `List` API just doesn't give us the tools we
// need to accomplish elegantly this task.
const hasSubsequence = <A>(sup: List<A>, sub: List<A>): boolean => {
  const folded = foldLeft(sup, sub, (rem, cur) => {
    if (rem.tag === "nil")
      return Nil;
    else if (cur === rem.head)
      return rem.tail;
    else
      return sub;
  });
  return folded === Nil;
};
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Trees

List is just one example of an algebraic data type (ADT), which we discussed earlier in the chapter. In this section we'll introduce the Tree, which is a recursive, hierarchical data structure.

Tuple types in TypeScript

We briefly touched on tuples in Chapter 1. A tuple is a bit like a List, except its size and the types of all its elements, which need not be the same type, are known at compile time. A tuple of "arity", or size, 2 is often referred to simply as a pair. Tuples are themselves ADTs. TypeScript borrows JavaScript's array syntax to define both tuple types and tuple values.

> const p: [string, number] = ["Bob", 42];
> p[0];
'Bob'
> p[0] = 41;
[eval].ts(5,1): error TS2322: Type '2' is not assignable to type 'string'.
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In fact, TypeScript tuples are just JavaScript arrays with some additional type information. TypeScript keeps track of the type of each element of a tuple, and throws errors like the one above if you attempt to assign a value to a position in a tuple with an incompatible type.

Here's a simple binary tree data structure. You should recognize the tagged union technique we introduced earlier with List.

type Tree<A> = Leaf<A> | Branch<A>;

class Leaf<A> {
  tag: "leaf" = "leaf";
  readonly value: A;

  constructor(value: A) {
    this.value = value;
  }
}

class Branch<A> {
  tag: "branch" = "branch";
  readonly left: Tree<A>;
  readonly right: Tree<A>;

  constructor(left: Tree<A>, right: Tree<A>) {
    this.left = left;
    this.right = right;
  }
}
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                          A Tree

                          Branch
                      ┌─────┬─────┐
             ┌────────┼──●  │  ●──┼───────┐
             │        └─────┴─────┘       │
             ↓         left  right        ↓
           Branch                       Branch
       ┌─────┬─────┐                ┌─────┬─────┐
   ┌───┼──●  │  ●──┼───┐        ┌───┼──●  │  ●──┼───┐
   │   └─────┴─────┘   │        │   └─────┴─────┘   │
   ↓    left  right    ↓        ↓    left  right    ↓
  Leaf                Leaf     Leaf                Leaf
┌─────┐             ┌─────┐  ┌─────┐             ┌─────┐
│ "a" │             │ "b" │  │ "c" │             │ "d" │
└─────┘             └─────┘  └─────┘             └─────┘
 value               value    value               value
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Exercise 3.25. size

Write a function named size that counts the number of leaves and branches (collectively called "nodes") in a tree.

const size = (ta: Tree<unknown>): number => ...
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The unknown type

The unknown type signifies to TypeScript that we do not know what type of values the passed-in tree contains, and also that we do not care. Our size function needs to know nothing about the values to do its job. Attempting to reference the values in the Leaf nodes would cause a compile error. The unknown type can help enforce the constraint that a function maintain a strict ignorance about values contained in the structures it deals with, and therefore prevent accidentally coupling the function to a type.

Answer
const size = (t: Tree<unknown>): number => {
  if (t.tag === "leaf")
    return 1;
  else
    return 1 + size(t.left) + size(t.right);
};
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Exercise 3.26. maximum

Write a function maximum that returns the maximum value contained in a tree of numbers. You can use TypeScript's built-in Math.max(x, y) function to calculate the maximum of two numbers x and y.

const maximum = (tn: Tree<number>): number => ...
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Answer
const maximum = (t: Tree<number>): number => {
  if (t.tag === "leaf")
    return t.value;
  else
    return Math.max(maximum(t.left), maximum(t.right));
};
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Exercise 3.27. depth

Write a function depth that returns the maximum path length from the "root", or top, of a tree to any leaf.

const depth = (ta: Tree<unknown>): number => ...
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Answer
const depth = (t: Tree<unknown>): number => {
  if (t.tag === "leaf")
    return 1;
  else
    return Math.max(depth(t.left), depth(t.right)) + 1;
};
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Exercise 3.28. map

Write the Tree version of map. As with the List version, it should return a Tree with the same shape as the input Tree, but with elements transformed by a provided function.

const map = <A, B>(ta: Tree<A>, f: (a: A) => B): Tree<B> => ...
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Answer
const map = <A, B>(t: Tree<A>, f: (a: A) => B): Tree<B> => {
  if (t.tag === "leaf")
    return new Leaf(f(t.value));
  else
    return new Branch(map(t.left, f), map(t.right, f));
};
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ADTs and encapsulation

As you've worked through the exercises in this chapter, you may have caught yourself thinking, "Wow, it sure seems like we're exposing an awful lot of the internals of our data structures to the users of our API." To quote the book:

In FP, we approach concerns about encapsulation differently—we don’t typically have delicate mutable state which could lead to bugs or violation of invariants if exposed publicly. Exposing the data constructors of a type is often fine, and the decision to do so is approached much like any other decision about what the public API of a data type should be.

ADTs are typically used in scenarios where the set of possible data constructors is known to be fixed, or closed. Since we do not expect to have to support adding new cases to our types very often, the advantages of techniques like pattern matching and directly exposing the structure of data types to API consumers outweigh the disadvantages, such as the amount of code that would have to be touched to add a new case.

Exercise 3.29

Write a new function, called fold, that abstracts over the similarities between size, maximum, depth, and map. Then, reimplement those functions in terms of fold. Can you describe the connection between this fold and the left and right folds of List?

Answer
const fold = <A, B>(t: Tree<A>,
                    f: (a: A) => B,
                    g: (l: B, r: B) => B): B => {
  if (t.tag === "leaf")
    return f(t.value);
  else
    return g(fold(t.left, f, g), fold(t.right, f, g));
};

const size = (t: Tree<unknown>): number =>
  fold(t, x => 1, (l, r) => l + r + 1);

const maximum = (t: Tree<number>): number =>
  fold(t, n => n, (l, r) => Math.max(l, r));

const depth = (t: Tree<unknown>): number =>
  fold(t, a => 1, (l, r) => Math.max(l, r) + 1);

const map = <A, B>(t: Tree<A>, f: (a: A) => B): Tree<B> => {
  return fold(
    t,
    a => new Leaf(f(a)) as Tree<B>,
    (l, r) => new Branch(l, r),
  );
};
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Folding over a tree and folding over a list do the same thing, conceptually: collapse a data structure into a value (which may itself be a data structure) by incrementally applying functions to each node of the structure, carrying along an intermediate value in an "accumulator". Our fold implementation for trees is similar to foldRight for lists in that the caller needs to specify how to deal with each data constructor. Recall foldRight:

const foldRight = <A, B>(l: List<A>, z: B, f: (a: A) => B): B => ...
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The function f handles the Cons data constructor, while Nil is simply replaced with the "zero" value, z. For trees, we need to specify two functions, f for Leaf nodes and g for Branch nodes. Analagously to the List case, passing the data constructor functions for f and g copies the tree: fold(t, a => new Leaf(a), (l, r) => new Branch(l, r)).

Summary

Breathe a sigh of relief, you made it through Chapter 3! Hopefully, you've become more comfortable with writing and and generalizing pure functions in TypeScript. The rabbit hole only gets deeper from here. Next up: how to handle errors while adhering to functional principals. In other words, without exceptions.

Last Updated: 6/19/2019, 10:33:36 PM