Due by 11:59pm on Wednesday, 7/27

Instructions

Download hw08.zip. Inside the archive, you will find a file called hw08.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 0 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 run a Scheme program interactively, type python3 scheme -i <file.scm>. To exit the Scheme interpreter, type (exit).

Hinting

Homework 8 includes a hinting sytem that generates hints specifically for your submitted code. If you ever find yourself stuck, you can ask for hints by using adding the --hint flag to the ok command.

$ python3 ok -q pow --hint
...
Hint: There are extra parentheses on line ...

The system tries to generate hints specific to your program by applying multiple techniques, including:

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

Question 2

Implement a procedure called ordered?, which takes a list of numbers and returns True if the numbers are in nondescending 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 3

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. Lists are not well-formed if they do not properly terminate in a null list. In such cases, the list will print with a dot before the final item to indicate that its last two items are contained in a single pair. For example,

(cons 1 (cons 2 3))

would print as

(1 2 . 3)

for which nodots should substitute

(1 2 3)

You may find the built-in null? and pair? predicates useful.

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

Use OK to unlock and test your code:

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

Sets as Ordered Lists

One way to represent a set is by using an ordered list, where the ordering is used to speed up search (although only by a constant factor). 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 4

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 s is Link.empty
;
; 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)

Use OK to unlock and test your code:

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

Question 5

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
          ))

Use OK to unlock and test your code:

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

Question 6

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. A 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
          ))

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

Question 7

Another way to represent a set is to store the elements in a binary search tree. A binary search tree's branches are either empty or binary search trees. Also, the entry at the root of the tree must be greater than all entries in the left branch and less than all entries in the right branch.

The following procedures define a data abstraction for a binary search tree.

; A data abstraction for binary trees where nil represents the empty tree
(define (tree label left right) (list label left right))
(define (label t) (car t))
(define (left t) (cadr t))
(define (right t) (caddr t))
(define (empty? s) (null? s))
(define (leaf label) (tree label nil nil))

Define the procedure in?, which returns whether a binary search tree t contains an element v. Your implementation should run in logarithmic time in the number of entries in a balanced input tree. The Python implementation of this procedure is provided for your reference.

(define (in? t v)
    (cond ((empty? t) false)
          'YOUR-CODE-HERE
          (else nil)
          ))

; Equivalent Python code, for your reference:
;
; def contains(t, v):
;     if t is BST.empty:
;         return False
;     elif t.entry == v:
;         return True
;     elif v < t.entry:
;         return contains(t.left, v)
;     elif v > t.entry:
;         return contains(t.right, v)

You can unlock and test using OK:

Use OK to unlock and test your code:

python3 ok -q in -u
python3 ok -q in

Question 8

Define as-list, which takes a set (represented with a binary search tree) and returns a sorted list containing the entries of the tree. For an (optional) extra challenge, define the procedure so that it runs in linear time in the length of the resulting list. Converting to and from the sorted list representation in linear time allows binary search trees to be intersected in linear time.

(define (as-list t)
    'YOUR-CODE-HERE
    (else nil)
    )

Use OK to unlock and test your code:

python3 ok -q as-list -u
python3 ok -q as-list

Extra Questions: Symbolic Differentiation

Extra questions are not worth extra credit and are entirely optional. They are designed to challenge you to think creatively!

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))

Question 9

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

Question 10

Implement derive-product, which applies the product rule to differentiate products:

(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

Question 11

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).

Hint: The built-in procedure expt 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

Question 12

Implement derive-exp, which uses the power rule to derive exps:

(define (derive-exp exp var)
  'YOUR-CODE-HERE
  )

Use OK to test your code:

python3 ok -q derive-exp