Solution Files

You can find the solutions in hw08.scm.

Scheme Editor

How to launch

In your hw08 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.

Features

The hw08.scm file should already be open. You can edit this file and then run Run to run the code and get an interactive terminal or Test to run the ok tests.

Environments will help you diagram your code, and Debug works with environments so you can see where you are in it. We encourage you to try out all these features.

Reformat is incredibly useful for determining whether you have parenthesis based bugs in your code. You should be able to see after formatting if your code looks weird where the issue is.

By default, the interpreter uses Lisp-style formatting, where the parens are all put on the end of the last line

(define (f x)
    (if (> x 0)
        x
        (- x)))

However, if you would prefer the close parens to be on their own lines as so

(define (f x)
    (if (> x 0)
        x
        (- x)
    )
)

you can go to Settings and select the second option.

Required Questions

Q1: 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)

Q2: WWSD: Eval and Apply

How many calls to scheme_eval and scheme_apply would it take to evaluate each of these Scheme expressions? Use OkPy to test your knowledge by writing the number of calls needed to evaluate each expression:

python3 ok -q wwsd-eval_apply -u

scm> (+ 2 4 6 8) ; number of calls to scheme_eval
______
6
scm> (+ 2 4 6 8) ; number of calls to scheme_apply
______
1
scm> (+ 2 (* 4 (- 6 8))) ; number of calls to scheme_eval
______
10
scm> (+ 2 (* 4 (- 6 8))) ; number of calls to scheme_apply
______
3
scm> (if #f (+ 2 3) (+ 1 2)) ; number of calls to scheme_eval
______
5
scm> (if #f (+ 2 3) (+ 1 2)) ; number of calls to scheme_apply
______
1
scm> (define (cube a) (* a a a)) ; number of calls to scheme_eval
______
1
scm> (define (cube a) (* a a a)) ; number of calls to scheme_apply
______
0
scm> (cube 3) ; number of calls to scheme_eval
______
8
scm> (cube 3) ; number of calls to scheme_apply
______
2

Symbolic 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 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) (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 (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) (eq? (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.

Q3: 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)
  (make-sum (derive (first-operand expr) var)
            (derive (second-operand expr) var))
)

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

  scm> (derive (make-sum x^3 x^2) 'x)
  (+ (* 3 (^ x 2)) (* 2 x))

Q4: 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)
  (make-sum
    (make-product (derive (first-operand expr) var)
                  (second-operand expr))
    (make-product (first-operand expr)
                  (derive (second-operand expr) var)))
)

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

Optional Questions

Q5: 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)
  (cond ((= exponent 0) 1)
        ((= exponent 1) base)
        ((and (number? base) (integer? exponent)) (expt base exponent))
        (else (list '^ base exponent)))
)

(define (exp? exp)
  (and (list? exp) (eq? (car exp) '^))
)

(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

Q6: 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)
  (let ((b (first-operand exp))
        (n (second-operand exp)))
    (if (number? n)
      (let ((first (make-product n (make-exp b (- n 1)))))
        (make-product first (derive b var))) ;; Note: Chain rule isn't tested by ok.
        'unknown))
)

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

(make-product n (make-exp b (- n 1)))

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