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?
andodd?
, 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 isnil
.
(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