Last updated on Friday, December 13, 2024

Laziness

Welcome to week 7 of CS-214 — Software Construction!

As usual, ⭐️ indicates the most important exercises and questions and 🔥 indicates the most challenging ones.

You do not need to complete all exercises to succeed in this class, and you do not need to do all exercises in the order they are written.

We strongly encourage you to solve the exercises on paper first, in groups. After completing a first draft on paper, you may want to check your solutions on your computer. To do so, you can download the scaffold code (ZIP).

“I’ll think about it tomorrow” - Introduction

In this exercise sheet we’ll explore lazy computation by experimenting with lazy lists, where procrastination meets programming.

Lazy lists (as LazyLists in Scala) are called “lazy” because they compute elements only when they are needed:

Elements are memoized; that is, the value of each element is computed at most once.
Elements are computed in-order and are never skipped. In other words, accessing the tail causes the head to be computed first.

This allows for powerful new constructs, such as infinite data structures.

Remember, in this session, procrastination isn’t a bad thing… it’s the entire point! Let’s dive together into the world of LazyLists, at our own pace, of course. 🐢💤

Why call right now when you can call later?

Quick recap

In this course, we have studied three evaluation strategies:

Call-by-value: function arguments are fully evaluated before the function.
Call-by-name: function arguments are evaluated every time they are used, and not before then.
Call-by-need: similar to call-by-name, but the result of the argument evaluation is memoized (stored) for future uses.

In Scala, call-by-need variables or fields are denoted by a lazy keyword before the definition. This means that once a lazy variable (or field) is evaluated, the result is memoized (stored) for future uses. Lazy lists are a great example of call-by-need evaluation, the elements are computed on-demand and also stored for future use.

Now that we’ve refreshed our mind, let’s dive into to the first exercise.

Understanding evaluation strategies ⭐️

First, let’s warm up on code you’ve seen during the lecture. What will the following code print?

val x = { print("x"); 1 }
lazy val y = { print("y"); 2 }
def z = { print("z"); 3 }
println(z + y + x + z + y + x)

Take the time to discuss your answer with your classmates before reading the solution.

Who evaluates what and when? ⭐️

Think of lazy lists as a box that only shows items when requested. After accessing an item, you don’t revisit it in the box. This “on-demand” behavior is deeply connected to the concept of call-by-need evaluation.

Now, let’s consider the following implementation of List1, List2 and List3:

class List1[+A](val state: List1State[A])

enum List1State[+A]:
  case Cons(head: A, tail: List1[A])
  case Nil

class List2[+A](val init: => List2State[A]):
  lazy val state: List2State[A] = init

enum List2State[+A]:
  case Cons(head: A, tail: List2[A])
  case Nil

class List3[+A](val state: () => List3State[A])

enum List3State[+A]:
  case Cons(head: () => A, tail: () => List3[A])
  case Nil

Which evaluation strategy do each of these lists use: by-value, by-name, or by-need?

“Infinity and beyond!” - Implementing infinite lazy lists

During the lecture, you were introduced to the concept of infinite lists, made possible through lazy evaluation in programming.

Lazy lists only compute elements as they are needed, allowing for the definition of potentially infinite data structures.

Before you proceed, we highly recommend that you disable any programming assistant during the exercise and solve the problems on paper first.

MyLazyList implementation

This week we’ll use MyLazyList, our own implementation of lazy lists in Scala.

The MyLazyList class has a lazy field pointing to a state, which can either be a LZCons (representing a non-empty list) or LZNil (representing an empty list). The class is described below:

class MyLazyList[+A](init: () => MyLazyListState[A]):
  lazy val state: MyLazyListState[A] = init()

enum MyLazyListState[+A]:
  case LZCons(elem: A, tail: MyLazyList[A])
  case LZNil

src/main/scala/laziness/MyLazyList.scala

We’ve provided utility methods to help you create instances of MyLazyList:

def cons[A](elem: => A, tail: => MyLazyList[A]): MyLazyList[A] =
  MyLazyList(() => LZCons(elem, tail))

def empty: MyLazyList[Nothing] = MyLazyList(() => LZNil)

src/main/scala/laziness/MyLazyList.scala

Also, here are the provided methods to interact with MyLazyList:

isEmpty: Check if the list is empty.
head: Retrieve the first element.
tail: Get the rest of the list after the first element.
size: Get the size of the list.
foreach: Iterate over each element of the list. This method takes a function and applies it to each item in the list (without modifying it).
contains: Check if the list contains a given element.
get: Get the element at a given index.

def isEmpty: Boolean = self.state match
  case LZNil        => true
  case LZCons(_, _) => false

def head: A = self.state match
  case LZNil        => throw RuntimeException("head of empty list")
  case LZCons(x, _) => x

def tail: MyLazyList[A] = self.state match
  case LZNil         => throw RuntimeException("tail of empty list")
  case LZCons(_, xs) => xs

def size: Int = self.state match
  case LZNil         => 0
  case LZCons(_, xs) => 1 + xs.size

def foreach(f: A => Unit): Unit = self.state match
  case LZNil => ()
  case LZCons(x, xs) =>
    f(x)
    xs.foreach(f)

def contains(elem: A): Boolean = self.state match
  case LZNil => false
  case LZCons(x, xs) =>
    if x == elem then true
    else xs.contains(elem)

def get(i: Int): A =
  if i < 0 then throw RuntimeException("index out of bounds")
  else
    self.state match
      case LZNil => throw RuntimeException("index out of bounds")
      case LZCons(x, xs) =>
        if i == 0 then x
        else xs.get(i - 1)

src/main/scala/laziness/MyLazyList.scala

Note

These operations will execute delayed computation through the lazy list. size, foreach, and contains ensure the whole lazy list is evaluated; hence, they do not terminate on infinite lazy lists.

Before we continue, take a moment to familiarize yourself with their implementation.

This week, we’ll implement other operations for lazy lists, and they should take constant time (O(1)) regardless of the size of the list or the content. They should delay computation and not execute immediately.

Basic lazy list Operations - Part 1 ⭐️

Your task is to implement the following methods:

take(n) returns a lazy list containing the first n elements of the original lazy list. If the original lazy list has fewer than n elements, it will return a lazy list with the same elements as the original.
drop(n) returns a lazy list with the first n elements removed. If the original lazy list has fewer than n elements, it will return an empty lazy list.

Before you start, take a look at the “wrong” implementation of take:

def wrongTake(n: Int): MyLazyList[A] =
  if n <= 0 then empty
  else
    self.state match
      case LZNil         => empty
      case LZCons(x, xs) => cons(x, xs.wrongTake(n - 1))

src/main/scala/laziness/MyLazyList.scala

wrongTake seems to work fine, but it has a problem: forcing the computing too early. The first element of the lazy list is evaluated immediately when wrongTake is called. If we call wrongTake on an infinite lazy list with complicated references to the first element, it may never terminate.

We don’t need to look at the first element until we need it, so we can wrap the logic of take inside the closure of MyLazyList class. Remember, these operations should delay computation as much as possible.

Here are the definition of these methods:

def take(n: Int): MyLazyList[A] =
  if n <= 0 then empty
  else
    MyLazyList(() =>
      self.state match
        case LZNil => ???
        case LZCons(x, xs) => ???
    )

src/main/scala/laziness/MyLazyList.scala

def drop(n: Int): MyLazyList[A] =
  ???

src/main/scala/laziness/MyLazyList.scala

Test these methods on infinite lazy lists and check whether they terminate.
How does the behavior and result differ compared to a normal list?

Building infinite lazy lists using recursive functions ⭐️

Infinite lazy lists can be defined using recursive functions, often referencing themselves. Such definitions do not cause infinite loops by themselves due to the nature of the lazy list’s lazy computation.

Reimplement the from and range methods, as mentioned in the lecture.

Remember,

from(x) generates an infinite lazy list starting from the given number x and increments by 1 for each subsequent element.
range(x, y) generates a lazy list starting from the number x and going up to (but not including) the number y, incrementing by 1 for each subsequent element.

def from(x: Int): MyLazyList[Int] =
  ???

src/main/scala/laziness/IntLazyLists.scala

def range(x: Int, y: Int): MyLazyList[Int] =
  ???

src/main/scala/laziness/IntLazyLists.scala

Hint

You can use the operations you just implemented.

Basic lazy list operations - Part 2 ⭐️

Next, your task is to implement the following methods:

map(f) applies a function f to each element of the lazy list and returns a new lazy list with the results.
filter(f) returns a new lazy list containing only the elements of the original lazy list for which the given predicate f returns true.
zip(that) returns a lazy list of pairs of corresponding elements of the original lazy list and that. If one of the two lazy lists is longer than the other, its remaining elements are ignored.

def map[B](f: A => B): MyLazyList[B] =
  ???

src/main/scala/laziness/MyLazyList.scala

def filter(p: A => Boolean): MyLazyList[A] =
  ???

src/main/scala/laziness/MyLazyList.scala

def zip[B](that: MyLazyList[B]): MyLazyList[(A, B)] =
  ???

src/main/scala/laziness/MyLazyList.scala

Building infinite lazy lists using recursive data ⭐️

Let’s practice more infinite lazy lists using the operations you just implemented. In this section, we focus on building infinite lazy lists using recursive data.

Simple recursion lazy list

Consider the following example:

val anonymList: MyLazyList[Int] = cons(1, anonymList)

What does this list corresponds to?
Does the definition of anonymList cause an infinite loop?
Implement an infinite list of 2s using anonymList and map.

lazy val infiniteTwoes: MyLazyList[Int] =
  ???

src/main/scala/laziness/IntLazyLists.scala

We can easily construct the lazy list of all natural numbers using from.

val naturalNumbers1: MyLazyList[Int] = from(0)

Try to implement the same lazy list using itself and map.

lazy val naturalNumbers2: MyLazyList[Int] =
  cons(0, ???)

src/main/scala/laziness/IntLazyLists.scala

Fibonacci’s magic show! 🔥

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. The first few values in the sequence are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.

Implement the Fibonacci sequence by filling the hole using zip and map.

lazy val fib: MyLazyList[Int] =
  cons(0, cons(1, ???))

src/main/scala/laziness/IntLazyLists.scala

Be careful, if your implementation of the operations does not follow our rule strictly, you may get an infinite loop when you try to access the elements of fib.

Basic lazy list operations - Part 3 ⭐️

Last but not least, your task is to implement the following methods:

append returns a new lazy list consisting of the elements of the first lazy list followed by the elements of the second lazy list.
flatMap applies a function f to each element of the lazy list and returns a new lazy list with the results. The resulting lazy list will be a concatenation of all the lazy lists returned by f.

def append(that: MyLazyList[A]): MyLazyList[A] =
  ???

src/main/scala/laziness/MyLazyList.scala

def flatMap[B](f: A => MyLazyList[B]): MyLazyList[B] =
  ???

src/main/scala/laziness/MyLazyList.scala

After you finish the implementation of append, you will notice how “efficient” it is compared to ++ on regular lists. ++ from regular lists has to traverse the first list to the end and build a new list. append on lazy lists only needs to traverse the first list when needed, spreading the cost over the entire process.

String lazy list 🔥

Write a lazy list of all non-empty strings using the characters “0” and “1” and the concatenation operation +. In other words, every non-empty string composed of “0” and “1” should be reached at some point.

lazy val codes: MyLazyList[String] =
  ???

src/main/scala/laziness/OtherPractice.scala

Using codes, write a lazy list of all possible non-empty palindromes of “0” and “1”. You may use the .reverse function defined on strings.

lazy val palCodes: MyLazyList[String] =
  ???

src/main/scala/laziness/OtherPractice.scala

Computing the reverse of large number of strings and comparing them are expensive. Can you achieve the same result without filtering? The palindromes need not to be in the same order.

Note for each palindrome, there is a unique code s which is a prefix of the palindrome and s.reverse is a suffix of the palindrome.

s + s.reverse
s + "0" + s.reverse
s + "1" + s.reverse

Based on this observation, palCodes can be implemented using middle:

val middle: MyLazyList[String] = cons("", cons("0", cons("1", empty)))

(middle does not need to be a lazy list. We define it this way to make the list methods compatible with our implementation.)

Now, we can implement palCodes2 without filtering:

lazy val palCodes2: MyLazyList[String] =
  // need to add base cases at the beginning
  cons("0", cons("1", empty)).append(???)

src/main/scala/laziness/OtherPractice.scala

More practices on infinite lazy lists

Having explored the foundations of infinite lazy lists and their recursive nature, let’s transition to understanding specific sequences that can be represented using lazy lists.

Look And Say ⭐️

Consider the following series:

1
1 1
2 1
1 2 1 1
1 1 1 2 2 1
3 1 2 2 1 1
...........

Can you guess the next element in the …?
Now, let us encode an element of the sequence above as a List[Int]. Write a function to compute the next line, based on the current line.

def nextLine(currentLine: List[Int]): List[Int] =
  ???

src/main/scala/laziness/OtherPractice.scala

Implement a lazy list funSeq which constructs this sequence.

lazy val funSeq: MyLazyList[List[Int]] =
  ???

src/main/scala/laziness/OtherPractice.scala

Sieve of Erathostenes ⭐️

Sieve of Eratosthenes is an ancient technique to calculate prime numbers:

Start with all integers from 2, the first prime number.
Eliminate all multiples of 2.
The first element of the resulting list is 3, a prime number.
Eliminate all multiples of 3.
Iterate forever. At each step, the first number in the list is a prime number and we eliminate all its multiples.

Your task is to implement the Sieve of Eratosthenes using lazy lists.

lazy val primeNumbers: MyLazyList[Int] =
  import MyLazyListState.*
  def sieve(s: MyLazyList[Int]): MyLazyList[Int] =
    ???
  sieve(???)

src/main/scala/laziness/IntLazyLists.scala

Thinking About Lazy Lists

Lazy list is a clever way to represent delayed computation and infinite data structures. However, it is not a silver bullet. Abusing lazy lists may lead to performance issues, due to the overhead of closures and memoization.