# Homework 7:

*Due by 11:59pm on Thursday, 10/25*

## Instructions

Download hw07.zip.

Our course uses a custom version of Scheme (which you will build for Project 4) included in the starter ZIP archive. To start the interpreter, type

`python3 scheme`

. To run a Scheme program interactively, type`python3 scheme -i <file.scm>`

. To exit the Scheme interpreter, type`(exit)`

.

**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 effort, not
correctness. However, there is no partial credit; you must show substantial
effort on every problem to receive any points.

### Q0: Survey

Before you get started writing code, please fill out the midterm survey.

#### Important Submission Note

You're not done yet!Add the passphrase you receive at the end of the survey to passphrase at the top of the homework. For example, if the passphrase was`CS61A`

(it isn't ðŸ™‚), then the first line of your file should read:

`passphrase = 'CS61A'`

Instead of:

`passphrase = '*** PASSPHRASE HERE ***'`

Use Ok to test your code:

`python3 ok -q survey`

### Q1: Cadr and Caddr

Define the procedures `cadr`

and `caddr`

, which return the second
and third elements of a list, respectively:

```
(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
```

### Conditional expressions

The `cond`

special form is a general conditional expression, similar to
a multi-clause conditional statement in Python. The general form of a
conditional expression is:

```
(cond
(<p1> <el1>)
(<p2> <el2>)
...
(<pn> <eln>)
(else <else-expressions>))
```

This consists of the symbol `cond`

followed by sequences of expressions ```
(<p>
<el>)
```

called *clauses*.

The first expression in each pair is a *predicate*: an expression whose value is
interpreted as either being true or false.

In Scheme, *all* values except the special boolean value `False`

(historically
called `#f`

) are interpreted as true values. There is also a boolean value
`True`

(historically called `#t`

). In the 61A Scheme interpreter, `False`

and
`#f`

can be used interchangeably.

Conditional expressions are evaluated as follows:

- The predicates
`<p1>`

,`<p2>`

, ...,`<pn>`

are evaluated in that order until one of them evaluates to a true value (anything but`False`

). - If some predicate, such as
`<p2>`

, evaluates to a true value, then the following sequence of consequent expressions, such as`<el2>`

, is evaluated, and its final expression provides the value of the whole`cond`

expression. - Otherwise, if an
`else`

clause is present, then the sequence of`else-expressions`

is evaluated, and its final expression provides the value of the whole`cond`

expression. - If no clause has a true predicate and there is no
`else`

clause, then the value of the`cond`

is unspecified and should not be used.

### Q2: Sign

Using a `cond`

expression, define a procedure `sign`

that takes in one
parameter `x`

and returns -1 if `x`

is negative, 0 if `x`

is zero, and 1 if `x`

is positive.

```
(define (sign x)
'YOUR-CODE-HERE
)
```

Use Ok to unlock and test your code:

```
python3 ok -q sign -u
python3 ok -q sign
```

### Q3: Pow

Implement a procedure `pow`

for raising the number `b`

to the power of a
nonnegative integer `n`

that runs in Î˜(log n) time.

Hint:Consider the following observations:

- b
^{2k}= (b^{k})^{2}- b
^{2k+1}= b(b^{k})^{2}You may use the built-in predicates

`even?`

and`odd?`

.

```
(define (square x) (* x x))
(define (pow b n)
'YOUR-CODE-HERE
)
```

Use Ok to unlock and test your code:

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

### Q4: Ordered

Implement a procedure called `ordered?`

, which takes a list of numbers and
returns `True`

if the numbers are in nondescending order, and `False`

otherwise. Numbers are considered nondescending if each subsequent number is
either larger or equal to the previous, that is:

`1 2 3 3 4`

Is nondescending, but:

`1 2 3 3 2`

Is not.

Hint: The built-in`null?`

function returns whether its argument is`nil`

.

```
(define (ordered? s)
'YOUR-CODE-HERE
)
```

Use Ok to unlock and test your code:

```
python3 ok -q ordered -u
python3 ok -q ordered
```

## Sets as Ordered Lists

A set is a type of collection that stores unique elements. The main operation associated with sets is checking whether a given value is in a set.

There is no such built-in set data type in Scheme, but one way to represent a set is by using an ordered list, where the ordering is used to make union/intersection functions more convenient. The following few questions explore this idea. Specifically, we will represent sets using Scheme lists ordered from least to greatest with no repeated elements.

### Q5: Add

First, define `add`

, which takes a set `s`

and a value `v`

as arguments. It
returns a representation of a set containing the values in `s`

and the value
`v`

. There should be no repeated elements in the return value.

We've provided a function `empty?`

which returns `True`

if the given set `s`

has
no elements.

```
(define (empty? s) (null? s))
(define (add s v)
'YOUR-CODE-HERE
)
```

Use Ok to unlock and test your code:

```
python3 ok -q add -u
python3 ok -q add
```

### Q6: Contains

Next, define `contains?`

, which returns whether a set `s`

contains value `v`

.

```
; Sets as sorted lists
(define (contains? s v)
'YOUR-CODE-HERE
)
```

Use Ok to unlock and test your code:

```
python3 ok -q contains -u
python3 ok -q contains
```

### Q7: Intersect and Union

Finally, define `intersect`

, which returns a set containing only values that
appear in *both* sets `s`

and `t`

, and `union`

, which returns a set containing
all values that appear in *either* set `s`

or `t`

.

Your implementation for both functions should run in linear time in the length of the input sets.

```
(define (intersect s t)
'YOUR-CODE-HERE
)
(define (union s t)
'YOUR-CODE-HERE
)
```

Use Ok to unlock and test your code:

```
python3 ok -q intersect -u
python3 ok -q intersect
python3 ok -q union -u
python3 ok -q union
```