Homework 10

Due by 11:59pm on Friday, 8/7

Instructions

Download hw10.zip. Inside the archive, you will find a file called hw10.scm, along with a copy of the OK autograder.

Submission: When you are done, submit with python3 ok --submit. You may submit more than once before the deadline; only the final submission will be scored. See Lab 1 for instructions on submitting assignments.

Using OK: If you have any questions about using OK, please refer to this guide.

Readings: You might find the following references useful:

Required questions

Question 1: Filter

Write a function filter, which takes in a predicate and a Scheme list and returns a Scheme list whose elements are the elements from the inputted Scheme list that satisfy the predicate.

Make sure filter is tail recursive!

Hint: Try writing the function iteratively in Python, then convert into a tail recursive Scheme function.

You will also find the built-in append function useful. append concatenates two Scheme lists, much like the + operator in Python.

(define (filter pred lst)
  'YOUR-CODE-HERE
)

Use OK to test your code:

python3 ok -q filter

This question uses the following tail-recursive version of int-list (which creates a list of integers from 1 to n) and equal? (which checks if two lists s0 and s1 contain the same elements):

(define (int-list n total)
    (if (= n 0)
        total
        (int-list (- n 1) (cons n total))))

(define (equal? s0 s1)
  (cond ((and (null? s0) (null? s1)) True)
        ((or (null? s0) (null? s1)) False)
        (else (and (= (car s0) (car s1))
                   (equal? (cdr s0) (cdr s1))))))

Stream utilities

For the following questions use the functions integers and stream-to-list in their tests:

(define (integers first)
  (cons-stream first (integers (+ first 1))))

(define (stream-to-list s num-elements)
  (if (or (null? s) (= num-elements 0))
    nil
    (cons (car s) (stream-to-list (stream-cdr s) (- num-elements 1)))))

Question 2

Implement the function scale-stream, which returns a stream over each element of an input stream, scaled by k:

(define (scale-stream s k)
  'YOUR-CODE-HERE
)

Use OK to test your code:

python3 ok -q scale-stream

Question 3

Implement merge, which takes two streams s1 and s2 whose elements are ordered. merge returns a stream that contains elements from s1 and s2 in sorted order, elimnating repetition. You may assume s0 and s1 themselves do not contain repeats. s1 and s2 may or may not be infinite streams.

(define (merge s0 s1)
  (cond ((null? s0) s1)
        ((null? s1) s0)
        ; YOUR CODE HERE
  ))

Use OK to test your code:

python3 ok -q merge

Question 4

A famous problem, first raised by Richard Hamming, is to enumerate, in ascending order with no repetitions, all positive integers with no prime factors other than 2, 3, or 5. These are called regular numbers. One obvious way to do this is to simply test each integer in turn to see whether it has any factors other than 2, 3, and 5. But this is very inefficient, since, as the integers get larger, fewer and fewer of them fit the requirement.

As an alternative, we can write a function that returns an infinite stream of such numbers. Let us call the stream of numbers s and notice the following facts about it.

  • s begins with 1.
  • The elements of (scale-stream s 2) are also elements of s.
  • The same is true for (scale-stream s 3) and (scale-stream s 5).
  • These are all of the elements of s.

Now all we have to do is combine elements from these sources. Use the merge function you defined previously to fill in the definition of make-s:

(define (make-s)
  (cons-stream 1
      'YOUR-CODE-HERE
  ))

Use OK to test your code:

python3 ok -q make-s