# Homework 7

*Due by 11:59pm on Thursday, 10/29*

## Instructions

Download hw07.zip. Inside the archive, you will find a file called hw07.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:

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 exit the Scheme interpreter, type`(exit)`

.

### Question 1

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

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> <e1>)
(<p2> <e2>)
...
(<pn> <en>)
(else <else-expression>))
```

consisting of the symbol `cond`

followed by pairs of expressions ```
(<p>
<e>)
```

called clauses. The first expression in each pair is a
*predicate*: an expression whose value is interpreted as either `true`

or `false`

.

Conditional expressions are evaluated as follows. The predicate `<p1>`

is evaluated first. If its value is `false`

, then `<p2>`

is evaluated.
If `<p2>`

's value is also `false`

, then `<p3>`

is evaluated. This
process continues until a predicate is found whose value is `true`

, in
which case the interpreter returns the value of the corresponding
consequent expression `<e>`

of the clause as the value of the
conditional expression. If none of the `<p>`

's is found to be `true`

,
the interpreter returns the value of the `else`

expression.

The word "predicate" is used for procedures that return `true`

or
`false`

, as well as for expressions that evaluate to `true`

or `false`

.

### Question 2

Using `cond`

, define a procedure `sign`

that returns `-1`

for negative
arguments, `0`

for zero, and `1`

for positive arguments:

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

Use OK to unlock and test your code:

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

### Question 3

Implement a procedure `pow`

for raising the number `b`

to the power of integer
`n`

that runs in Θ(log n) time.

Hint:Using the built-in predicates`even?`

and`odd?`

, implement the fast exponentiation procedure from the lecture about orders of growth:

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

Use OK to unlock and test your code:

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

### Question 4

Implement a function called `ordered?`

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

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

otherwise.

Hint: The built-in`null?`

function returns whether its argument is`nil`

.

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

Use OK to unlock and test your code:

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

### Question 5

Implement the procedure `nodots`

, which takes a nested list of numbers that
may not be well-formed and returns a nested list with the same content and
structure, but which does not have any dots when displayed.

```
(define (nodots s)
'YOUR-CODE-HERE
nil
)
```

You can unlock and test using OK:

```
python3 ok -q nodots -u
python3 ok -q nodots
```

### Sets as Ordered Lists

The lecture on sets described one representation of a set using an ordered list, where the ordering was used to speed up set intersection. The following few questions explore this idea, assuming a "set" is a Scheme list with no repeated elements that is already ordered from least to greatest.

### Question 6

Define `contains?`

, which returns whether a set `s`

contains value `v`

. The
Python implementation of this procedure is provided for your reference.

```
; Sets as sorted lists
(define (empty? s) (null? s))
(define (contains? s v)
(cond ((empty? s) false)
'YOUR-CODE-HERE
(else nil) ; replace this line
))
; Equivalent Python code, for your reference:
;
; def empty(s):
; return len(s) == 0
;
; def contains(s, v):
; if empty(s):
; return False
; elif s.first > v:
; return False
; elif s.first == v:
; return True
; else:
; return contains(s.rest, v)
```

You can unlock and test using OK:

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

### Question 7

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.

```
(define (add s v)
(cond ((empty? s) (list v))
'YOUR-CODE-HERE
(else nil) ; replace this line
))
```

You can unlock and test using OK:

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

### Question 8

Define `intersect`

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

and `t`

. Your implementation should run in linear time in the
length of the input sets. The Python implementation of this procedure is
provided for your reference.

Also, define `union`

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

or `t`

.

```
(define (intersect s t)
(cond ((or (empty? s) (empty? t)) nil)
'YOUR-CODE-HERE
(else nil) ; replace this line
))
; Equivalent Python code, for your reference:
;
; def intersect(set1, set2):
; if empty(set1) or empty(set2):
; return Link.empty
; else:
; e1, e2 = set1.first, set2.first
; if e1 == e2:
; return Link(e1, intersect(set1.rest, set2.rest))
; elif e1 < e2:
; return intersect(set1.rest, set2)
; elif e2 < e1:
; return intersect(set1, set2.rest)
(define (union s t)
(cond ((empty? s) t)
((empty? t) s)
'YOUR-CODE-HERE
(else nil) ; replace this line
))
```

You can unlock and test using OK:

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