Due by 11:59pm on Wednesday, 11/4

Instructions

Download hw08.zip. Inside the archive, you will find a file called hw08.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:

Question 1

Write a procedure substitute that takes three arguments: a list s, an old word, and a new word. It returns a list with the elements of s, but with every occurrence of old replaced by new, even within sub-lists.

(define (substitute s old new)
  'YOUR-CODE-HERE
)

Use OK to unlock and test your code:

python3 ok -q substitute -u
python3 ok -q substitute

Question 2

Write sub-all, which takes a list s, a list of old words, and a list of new words; the last two lists must be the same length. It returns a list with the elements of s, but with each word that occurs in the second argument replaced by the corresponding word of the third argument.

(define (sub-all s olds news)
  'YOUR-CODE-HERE
)

Use OK to unlock and test your code:

python3 ok -q sub-all -u
python3 ok -q sub-all

Differentiation

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 exp 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) (eq? 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) (eq? (car x) '+)))
(define (addend s) (cadr s))
(define (augend 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) (eq? (car x) '*)))
(define (multiplier p) (cadr p))
(define (multiplicand p) (caddr p))

Question 3

Implement derive-sum, a procedure that differentiates a sum by summing the derivatives of the addend and augend. Use data abstraction for a sum:

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

Use OK to unlock and test your code:

python3 ok -q derive-sum -u
python3 ok -q derive-sum

Question 4

Implement derive-product, which applies the product rule to differentiate products:

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

Use OK to unlock and test your code:

python3 ok -q derive-product -u
python3 ok -q derive-product

Question 5

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).

Hint: The built-in procedure expt 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 (base exp)
  'YOUR-CODE-HERE
  )

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

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

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

Use OK to unlock and test your code:

python3 ok -q make-exp -u
python3 ok -q make-exp

Question 6

Implement derive-exp, which uses the power rule to derive exps:

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

Use OK to test your code:

python3 ok -q derive-exp