Homework 10
Due by 11:59pm on Friday, 11/3
Instructions
Download hw10.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, typepython3 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:
Q1: How Many Dots?
Implement how-many-dots
, which takes in a nested scheme list s
and returns
the number of dots that appear when it is displayed by the interpreter (not
counting decimal points). You may assume that s
is a nested list that
contains only numbers.
Hints: A dot appears when the second element of a pair is not a well formed
list. The procedures pair?
, null?
, and number?
test whether a value is a
pair, nil
, or a number, respectively.
(define (how-many-dots s)
'YOUR-CODE-HERE
)
Use Ok to unlock and test your code:
python3 ok -q how-many-dots -u
python3 ok -q how-many-dots
Note: Q2 and Q3 (Substitute, Sub All) were extra lab questions from Lab 9. You may check the solutions if you are stuck, but we highly recommend you work through the problem on your own for practice.
Q2: Substitute
Write a procedure substitute
that takes three arguments: a list s
, an old
word, and a new
word. It returns a list with the elements of s
, but with
every occurrence of old
replaced by new
, even within sub-lists.
Hint: The built-in pair?
predicate returns True if its argument is a cons
pair.
Hint: The =
operator will only let you compare numbers, but using equal?
or eq?
will let you compare symbols as well as numbers. For more information,
check out the
Scheme Primitives Reference.
Use Ok to unlock and test your code:
python3 ok -q substitute -u
python3 ok -q substitute
(define (substitute s old new)
'YOUR-CODE-HERE
)
Q3: Sub All
Write sub-all
, which takes a list s
, a list of old
words, and a
list of new
words; the last two lists must be the same length. It returns a
list with the elements of s
, but with each word that occurs in the second
argument replaced by the corresponding word of the third argument. You may use
substitute
in your solution.
(define (sub-all s olds news)
'YOUR-CODE-HERE
)
Use Ok to unlock and test your code:
python3 ok -q sub-all -u
python3 ok -q sub-all
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 (addend s) (cadr s))
(define (augend 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) '*)))
(define (multiplier p) (cadr p))
(define (multiplicand 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.
Q4: Derive Sum
Implement derive-sum
, a procedure that differentiates a sum by
summing the derivatives of the addend
and augend
. Use data abstraction
for a sum.
(define (derive-sum expr var)
'YOUR-CODE-HERE
)
Use Ok to unlock and test your code:
python3 ok -q derive-sum -u
python3 ok -q derive-sum
Q5: Derive Product
Implement derive-product
, which applies the product
rule to differentiate
products.
The
ok
tests expect the terms of the result in a particular order. First, multiply the derivative of the multiplier by the multiplicand. Then, multiply the multiplier by the derivative of the multiplicand. Sum these two terms to form the derivative of the original product.
(define (derive-product expr var)
'YOUR-CODE-HERE
)
Use Ok to unlock and test your code:
python3 ok -q derive-product -u
python3 ok -q derive-product
Q6: 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 (base exp)
'YOUR-CODE-HERE
)
(define (exponent exp)
'YOUR-CODE-HERE
)
(define (exp? exp)
'YOUR-CODE-HERE
)
(define x^2 (make-exp 'x 2))
(define x^3 (make-exp 'x 3))
Use Ok to unlock and test your code:
python3 ok -q make-exp -u
python3 ok -q make-exp
Q7: Derive Exp
Implement derive-exp
, which uses the power
rule to derive exponents.
(define (derive-exp exp var)
'YOUR-CODE-HERE
)
Use Ok to unlock and test your code:
python3 ok -q derive-exp -u
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!