# 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:

- x
^{2y}= (x^{y})^{2}- x
^{2y+1}= x(x^{y})^{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:

`merger`

: a function of two arguments`start`

: a number with which to start merging`n`

: the number of natural numbers to merge`term`

: a function of one argument that computes the*n*th 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`

`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`