Homework 6: Scheme

Due by 11:59pm on Wednesday, August 4

Instructions

Download hw06.zip. Inside the archive, you will find a file called hw06.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. Check that you have successfully submitted your code on okpy.org. See Lab 0 for more 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:

Grading: Homework is graded based on correctness. Each incorrect problem will decrease the total score by one point. There is a homework recovery policy as stated in the syllabus. This homework is out of 3 points.

You may find it useful to try code.cs61a.org/scheme when working through problems, as it can draw environment and box-and-pointer diagrams and it lets you walk your code step-by-step (similar to Python Tutor). Don't forget to submit your code through Ok though!

Scheme Editor

As you're writing your code, you can debug using the Scheme Editor. In your scheme folder you will find a new editor. To run this editor, run python3 editor. This should pop up a window in your browser; if it does not, please navigate to localhost:31415 and you should see it.

Make sure to run python3 ok in a separate tab or window so that the editor keeps running.

If you find that your code works in the online editor but not in your own interpreter, it's possible you have a bug in code from an earlier part that you'll have to track down. Every once in a while there's a bug that our tests don't catch, and if you find one you should let us know!

Required Questions

Scheme

Assignment Hint Videos

Q1: Thane of Cadr

Define the procedures cadr and caddr, which return the second and third elements of a list, respectively:

; Question 1
;
(define (cddr s)
  (cdr (cdr s)))

(define (cadr s)
  'YOUR-CODE-HERE
)

(define (caddr s)
  'YOUR-CODE-HERE
)

Use Ok to unlock and test your code:

python3 ok -q cadr-caddr -u
python3 ok -q cadr-caddr

Q2: Pow

Implement a procedure pow for raising the number base to the power of a nonnegative integer exp for which the number of operations grows logarithmically, rather than linearly (the number of recursive calls should be much smaller than the input exp). For example, for (pow 2 32) should take 5 recursive calls rather than 32 recursive calls. Similarly, (pow 2 64) should take 6 recursive calls.

Hint: Consider the following observations:

  1. x2y = (xy)2
  2. x2y+1 = x(xy)2

You may use the built-in predicates even? and odd?. Scheme doesn't support iteration in the same manner as Python, so consider another way to solve this problem.

; Question 2
;
(define (square x) (* x x))

(define (pow base exp)
  'YOUR-CODE-HERE
)

Use Ok to unlock and test your code:

python3 ok -q pow -u
python3 ok -q pow

Scheme Lists

Assignment Hint Videos

Q3: Filter Lst

Write a procedure filter-lst, which takes a predicate func and a list lst, and returns a new list containing only elements of the list that satisfy the predicate. The output should contain the elements in the same order that they appeared in the original list.

Note: Make sure that you are not just calling the built-in filter function in Scheme - we are asking you to re-implement this!

; Question 3
;
(define (filter-lst func lst)
  'YOUR-CODE-HERE
)

;;; Tests
(define (even? x)
  (= (modulo x 2) 0))
(filter-lst even? '(0 1 1 2 3 5 8))
; expect (0 2 8)

Use Ok to unlock and test your code:

python3 ok -q filter_lst -u
python3 ok -q filter_lst

Q4: Accumulate

Fill in the definition for the procedure accumulate, which merges the first n natural numbers according to the following parameters:

  1. merger: a function of two arguments
  2. start: a number with which to start merging
  3. n: the number of natural numbers to merge
  4. term: a function of one argument that computes the nth term of a sequence

For example, we can find the product of all the numbers from 1 to 5 by using the multiplication operator as the merger, and starting our product at 1:

scm> (define (identity x) x)
scm> (accumulate * 1 5 identity)  ; 1 * 1 * 2 * 3 * 4 * 5
120

We can also find the sum of the squares of the same numbers by using the addition operator as the merger and square as the term:

scm> (define (square x) (* x x))
scm> (accumulate + 0 5 square)  ; 0 + 1^2 + 2^2 + 3^2 + 4^2 + 5^2
55
scm> (accumulate + 5 5 square)  ; 5 + 1^2 + 2^2 + 3^2 + 4^2 + 5^2
60

You may assume that the merger will always be commutative: i.e. the order of arguments do not matter.

Hint: You may find it useful to refer to the recursive implementation of accumulate we implemented in Python in HW 2.

; Question 4
;
(define (accumulate merger start n term)
  'YOUR-CODE-HERE
)

Use Ok to test your code:

python3 ok -q accumulate

Q5: Without Duplicates

Implement without-duplicates, which takes a list of numbers lst as input and returns a list that has all of the unique elements of lst in the order that they first appear, but without duplicates. For example, (without-duplicates (list 5 4 5 4 2 2)) evaluates to (5 4 2).

Hints: To test if two numbers are equal, use the = procedure. To test if two numbers are not equal, use the not procedure in combination with =. You may find it helpful to use the filter-lst procedure with a helper lambda function to use as a filter.

; Question 5
;
(define (without-duplicates lst)
  'YOUR-CODE-HERE
)

Use Ok to unlock and test your code:

python3 ok -q without_duplicates -u
python3 ok -q without_duplicates

More Scheme, Tail Recursion

Assignment Hint Videos

Q6: WWSD: Quasiquote

Use Ok to test your knowledge with the following "What Would Scheme Display?" questions:

python3 ok -q wwsd-quasiquote -u
scm> '(1 x 3)

______
(1 x 3)

scm> (define x 2)

______
x

scm> `(1 x 3)

______
(1 x 3)

scm> `(1 ,x 3)

______
(1 2 3)

scm> '(1 ,x 3)

______
(1 (unquote x) 3)

scm> `(,1 x 3)

______
(1 x 3)

scm> `,(+ 1 x 3)

______
6

scm> `(1 (,x) 3)

______
(1 (2) 3)

scm> `(1 ,(+ x 2) 3)

______
(1 4 3)

scm> (define y 3)

______
y

scm> `(x ,(* y x) y)

______
(x 6 y)

scm> `(1 ,(cons x (list y 4)) 5)

______
(1 (2 3 4) 5)

Q7: Tail Recursive Accumulate

In a previous part of the homework, you implemented accumulate in scheme. As a reminder, accumulate merges the first n natural numbers according to the parameters merger, start, n, and term.

You can refer to your implementation of accumulate as a reminder of what the function does and a refresher of its implementation.

Update your implementation of accumulate to be tail recursive. It should still pass all the tests for "regular" accumulate!

You may assume that the input merger and term procedures are properly tail recursive.

If your implementation for accumulate in the previous question is already tail recursive, you may simply copy over that solution (replacing accumulate with accumulate-tail as appropriate).

If you're running into an recursion depth exceeded error and you're using the staff interpreter, it's very likely your solution is not properly tail recursive.

We test that your solution is tail recursive by calling accumulate-tail with a very large input. If your solution is not tail recursive and does not use a constant number of frames, it will not be able to successfully run.

; Question 7
;
(define (accumulate-tail merger start n term)
  'YOUR-CODE-HERE
)

Use Ok to test your code:

python3 ok -q accumulate-tail

Symbolic Differentiation - Optional Question

The following problems develop a system for symbolic differentiation of algebraic expressions. The derive Scheme procedure takes an algebraic expression and a variable and returns the derivative of the expression with respect to the variable. Symbolic differentiation is of special historical significance in Lisp. It was one of the motivating examples behind the development of the language. Differentiating is a recursive process that applies different rules to different kinds of expressions.

; derive returns the derivative of EXPR with respect to VAR
(define (derive expr var)
  (cond ((number? expr) 0)
        ((variable? expr) (if (same-variable? expr var) 1 0))
        ((sum? expr) (derive-sum expr var))
        ((product? expr) (derive-product expr var))
        ((exp? expr) (derive-exp expr var))
        (else 'Error)))

To implement the system, we will use the following data abstraction. Sums and products are lists, and they are simplified on construction:

; Variables are represented as symbols
(define (variable? x) (symbol? x))
(define (same-variable? v1 v2)
  (and (variable? v1) (variable? v2) (eqv? v1 v2)))

; Numbers are compared with =
(define (=number? expr num)
  (and (number? expr) (= expr num)))

; Sums are represented as lists that start with +.
(define (make-sum a1 a2)
  (cond ((=number? a1 0) a2)
        ((=number? a2 0) a1)
        ((and (number? a1) (number? a2)) (+ a1 a2))
        (else (list '+ a1 a2))))
(define (sum? x)
  (and (list? x) (eqv? (car x) '+)))
(define (first-operand s) (cadr s))
(define (second-operand s) (caddr s))

; Products are represented as lists that start with *.
(define (make-product m1 m2)
  (cond ((or (=number? m1 0) (=number? m2 0)) 0)
        ((=number? m1 1) m2)
        ((=number? m2 1) m1)
        ((and (number? m1) (number? m2)) (* m1 m2))
        (else (list '* m1 m2))))
(define (product? x)
  (and (list? x) (eqv? (car x) '*)))
; You can access the operands from the expressions with
; first-operand and second-operand
(define (first-operand p) (cadr p))
(define (second-operand p) (caddr p))

Note that we will not test whether your solutions to this question correctly apply the chain rule. For more info, check out the extensions section.

Q8: Derive Sum

Implement derive-sum, a procedure that differentiates a sum by summing the derivatives of the first-operand and second-operand. Use data abstraction for a sum.

Note: the formula for the derivative of a sum is (f(x) + g(x))' = f'(x) + g'(x)

(define (derive-sum expr var)
  'YOUR-CODE-HERE
)

The tests for this section aren't exhaustive, but tests for later parts will fully test it.

Before you start, check your understanding by running

python3 ok -q derive-sum -u

To test your code, if you are in the local Scheme editor, hit Test. You can click on a case, press Run, and then use the Debug and Environments features to figure out why your code is not functioning correctly.

You can also test your code from the terminal by running

python3 ok -q derive-sum

Q9: Derive Product

Note: the formula for the derivative of a product is (f(x) g(x))' = f'(x) g(x) + f(x) g'(x)

Implement derive-product, which applies the product rule to differentiate products. This means taking the first-operand and second-operand, and then summing the result of multiplying one by the derivative of the other.

The ok tests expect the terms of the result in a particular order. First, multiply the derivative of the first-operand by the second-operand. Then, multiply the first-operand by the derivative of the second-operand. Sum these two terms to form the derivative of the original product. In other words, f' g + f g', not some other ordering.

(define (derive-product expr var)
  'YOUR-CODE-HERE
)

Before you start, check your understanding by running

python3 ok -q derive-product -u

To test your code, if you are in the local Scheme editor, hit Test. You can click on a case, press Run, and then use the Debug and Environments features to figure out why your code is not functioning correctly.

You can also test your code from the terminal by running

python3 ok -q derive-product

Q10: Make Exp

Implement a data abstraction for exponentiation: a base raised to the power of an exponent. The base can be any expression, but assume that the exponent is a non-negative integer. You can simplify the cases when exponent is 0 or 1, or when base is a number, by returning numbers from the constructor make-exp. In other cases, you can represent the exp as a triple (^ base exponent).

You may want to use the built-in procedure expt, which takes two number arguments and raises the first to the power of the second.

; Exponentiations are represented as lists that start with ^.
(define (make-exp base exponent)
  'YOUR-CODE-HERE
)

(define (exp? exp)
  'YOUR-CODE-HERE
)

(define x^2 (make-exp 'x 2))
(define x^3 (make-exp 'x 3))

Before you start, check your understanding by running

python3 ok -q make-exp -u

To test your code, if you are in the local Scheme editor, hit Test. You can click on a case, press Run, and then use the Debug and Environments features to figure out why your code is not functioning correctly.

You can also test your code from the terminal by running

python3 ok -q make-exp

Q11: Derive Exp

Implement derive-exp, which uses the power rule to derive exponents. Reduce the power of the exponent by one, and multiply the entire expression by the original exponent.

Note: the formula for the derivative of an exponent is [f(x)^(g(x))]' = f(x)^(g(x) - 1) * g(x), if we ignore the chain rule, which we do for this problem

(define (derive-exp exp var)
  'YOUR-CODE-HERE
)

Before you start, check your understanding by running

python3 ok -q derive-exp -u

To test your code, if you are in the local Scheme editor, hit Test. You can click on a case, press Run, and then use the Debug and Environments features to figure out why your code is not functioning correctly.

You can also test your code from the terminal by running

python3 ok -q derive-exp

Extensions

There are many ways to extend this symbolic differentiation system. For example, you could simplify nested exponentiation expression such as (^ (^ x 3) 2), products of exponents such as (* (^ x 2) (^ x 3)), and sums of products such as (+ (* 2 x) (* 3 x)). You could apply the chain rule when deriving exponents, so that expressions like (derive '(^ (^ x y) 3) 'x) are handled correctly. Enjoy!

Submit

Make sure to submit this assignment by running:

python3 ok --submit