1 Getting started

You should be using the same lab partners as the previous Scheme lab from last week.

Start (as always) by going to your git repo at si413/scheme, and run these commands to update from the starter code repo:

roche@ubuntu$ git pull starter main
roche@ubuntu$ git pull

You should see a new folder lab02 with a file lab02.scm. That is where you will do your work for this lab!

(See git instructions here as a reminder, and don't ever hesitate to ask your instructor or a classmate for help dealing with git things. It should be a useful tool, not an annoying chore!)

2 Symbols

Recall that a symbol is just a quoted identifier, such as 'si413, or 'wombats. We can use the function symbol? to find out whether something is a symbol, and eqv? to compare two symbols.

Exercises

For these exercises, use the following exchange rates:

  1 US dollar = 0.76 Euros (euro)
  1 US dollar = 98.18 Japanese Yen (yen)
  1 US Dollar = 1109.85 South Korean Won (won)

1. Conversion to US dollars
   Create a function (to-usdollar amt cur) that takes an amount of money amt in some foreign currency cur and returns that amount in US dollars. The parameter cur will be one of the symbols 'euro, 'yen, 'won, or 'usd.
   So for instance (to-usdollar 500 'yen) produces \(500/98.18\), which comes out to \(5.0927\ldots\).
2. Conversion from US dollars
   Create a function (from-usdollar amt cur) that does the opposite: takes an amount in US Dollars and converts it to the named currency.
   If you're clever, you can write a helper function for this problem and the previous one that will avoid your having to enter the conversion rates twice.
3. Any kind of conversion!
   Create a function (convert amt fromCur toCur) that takes an amount in currency fromCur and converts it to currency toCur. Use your functions from parts 1 and 2 and life will be easy!

3 Lists

Recall "big 4" of list processing:

• '(): the name for an empty list.
• (cons a L): Take an item a and a list L and return the new list with a inserted in the front of L.
• (car L): Returns the first item in L.
• (cdr L): Returns the list containing all the items in L after the first item.

If you're faced with nested lists, you sometimes need cars of cdrs, and cdrs of cars, and so forth. The shortcut for a bunch of these in a row is cXXXr, where each X is either a or d, corresponding to car and cdr. For example, calling (caddr lst) is exactly equivalent to calling (car (cdr (cdr lst))).

There are two more extremely useful shortcuts:

• list: Takes an arbitrary number of items and makes a single list containing them.
• append: Takes an arbitrary number of lists and makes a single list containing all their items.

Make sure you understand the difference between these two!

In class we looked at the general pattern for a recursive function on a list. When that function is also producing a list, it extends the pattern by (usually) returning '() in the base case, and (usually) returning a cons in the general case. For example, here is a simple recursive function that multiplies every number in a list by 2:

; alon must be a list of numbers.
(define (mulby2 alon)
  (if (null? alon)
      '()
      (cons (* 2 (car alon))
            (mulby2 (cdr alon)))))

See what's happening? Make sure you can identify the base case condition, the base case return value, and the recursive call. Notice what gets multiplied by 2, and where it happens. If it's confusing, now is the time to ask your able and handsome instructor!

Exercises

4. List of Squares
   Write a function squares that takes integers i and j and returns list of the squares \(i^2, (i+1)^2, \ldots, (j-1)^2, j^2\).

   > (squares 2 12)
   (4 9 16 25 36 49 64 81 100 121 144)

5. Length Comparison

   In class we saw that there is a built-in Scheme function length which returns the length of a list. Using this, we could write a function longer? to test whether the first list is longer than the second:

   ; L1 and L2 must both be lists
   (define (longer? L1 L2)
     (> (length L1) (length L2)))

   Now I want you to write your own version of longer? that doesn't use length calculation as a subroutine. Instead, you will have to recursively process both L1 and L2 at the same time until one of them is empty.

   (I really mean it - no length, even if you write it yourself! I will test you code using an infinite list as one of the arguments. Yes, those exist in Scheme.

6. Count cash!
   Write a function called sum-cash that returns the value in US dollars of a collection of amounts of different currencies (same currencies as above). The amounts will be given in a list L, such that each element of L is itself a list, whose first element is an amount and whose second element is a currency name. So, for example,

   '((12.20 usd) (340 yen) (850 won))

   as an argument to sum-cash would mean adding 12.20 dollars, 340 yen and 850 won, and giving the total in dollars. Here's an example:

   > (sum-cash '((12.20 usd) (340 yen) (850 won)))
   16.4289...

7. Top-down design in Scheme: standard deviation
   Write a function called std-dev that takes a list of numbers and returns their standard deviation. (Recall: the standard deviation of \(x_1, x_2, \ldots, x_n\) is \[\sqrt{\frac{(x_1-\mu)^2 + (x_2-\mu)^2 + \cdots + (x_n-\mu)^2}{n-1}},\] ... where \(\mu\) is the average of \(x_1,x_2,\ldots,x_n\).) When you write this function, use top-down design. In scheme this means writing the std-dev function using functions you wish were already defined, then going back and defining them later. It can be useful to quote these function calls before you implement them, so you can even do top-down testing!
   Example:

   > (std-dev '(34 18 25 23 29 11 28 24 27 29))
   6.460134157533676

8. Uniquefication (OPTIONAL)
   Write a function uniquefy that takes a list of numbers, and returns a list with all the distinct numbers in the original, i.e., with all the duplicate numbers removed. For example:

   > (uniquefy '(1 2 4 1 2 2 3 15))
   (1 2 4 3 15)

   At the very least, you will probably want to also define at least one nice helper function. Try thinking about the efficiency of your function. Could you make it faster?

4 Let

The let construct in Scheme allows you to give a name to a common subexpression. For example, consider the expression

\[\frac{33 \cdot (501 - 33)}{1 - 33\cdot(501-33)}\]

The natural way to code this in Scheme is probably

(/ (* 33 (- 501 33)) (- 1 (* 33 (- 501 33))))

But you could say "let a = 33 * (501 - 33), and return a / (1 - a)". That's what let allows you to do:

(let ((a (* 33 (- 501 33))))
  (/ a (- 1 a)))

Essentially, let is the Scheme way of getting local variable functionality. What you've got is the let keyword, followed by a list of variable-name/value pairs, followed by an expression (presumably using the new names) that provides the value of the whole let expression. For example, the following code produces 12 as a result:

(let ((a 2)
      (b 4)
      (c 6))
  (+ a b c))

There is power in using let in functions. Suppose I want to define a function called shifted-cube, which computes (x + 1)^3 for a value x.

(define (shifted-cube x)
  (let ((a (+ x 1)))
    (* a a a)))

Then running (shifted-cube 2) produces the value 27.

Exercises:

9. Let for a common sub-expression
   Using what you just learned, write a function called test-sin that takes any number x and computes \[\frac{1}{(\sin x)^2 + 1} + \sqrt{(\sin x)^2 + 1} - \left((\sin x)^2 + 1\right)^2\]
10. Let for readability
    Write a function dist that computes the absolute value of the difference (in inches) between two lengths (given in feet and inches). For example, to compute the distance between 3'7" and 2'11":

    > (dist 3 7 2 11)
    8

    Write this function using a let expression to create values L1 and L2 for the lengths in total number of inches for each of the input feet-and-inches values.

5 Functions as parameters to other functions

As you have perhaps noticed, you never actually tell scheme what the type of a function parameter is. Thus, if I define some function (f x y), there's nothing to stop me from calling it like this: (f sqrt 4). In other words, there's nothing to stop me from passing f a function as one of its arguments. This is something very special about functional programming languages: functions can be used like any other kind of data. Take a moment to allow this to sink in.

Now check out this example. f uses its first argument, x, as a function. So x should be a function!

> (define (f x y)
    (* y (x y)))
> (f sqrt 4)
8

What happened here? Well since x is sqrt and y is 4, the evaluation produces (* 4 (sqrt 4)) which is 8. So f is the "apply function x to argument y and then multiply by y" function. Passing functions to functions like this is very powerful. This is part of what we mean when we say that "functions are first class objects" in a functional language.

Exercises:

11. Finite difference.
    Given a function \(g(x)\), the finite difference of \(g(x)\) at \(x=n\) is \(g(n + 1) - g(n)\). Define a function (fd-at g n) that takes a function g and a value n and returns the finite difference of g at n. For example:

    > (define (f x) (* x x))
    > (fd-at f 3)
    7

6 Functions and lists: map and apply

map is a really useful function that takes functions as arguments. The expression (map f L) applies the function f to each element of the list L and puts the results together in a new list. For example:

> (map abs '(-4 12 -3 -8 11 0))
(4 12 3 8 11 0)

If you have a function with k arguments, then you give map k lists, and it will take the first argument from list1, the second from list2, etc.

> (map * '(2 3 4) '(6 5 4))
(12 15 16)

Another useful function of this type is apply. The expression (apply f L) calls the function f with arguments the elements of L. Here are some examples:

> (apply max '(4 6 2)) ; same as (max 4 6 2)
6
> (apply - '(3 7))     ; same as (- 3 7)
-4

Combining map and apply can be very interesting.

> (define (sqr x) (* x x))
> (map sqr '(1 2 3 4 5 6 7 8 9 10))
(1 4 9 16 25 36 49 64 81 100)
> (apply + (map sqr '(1 2 3 4 5 6 7 8 9 10)))
385

Yet another useful function-that-takes-a-function is filter. The expression (filter f? L) applies the predicate function f? to each element in the list L, and returns the list containing only the elements for which the predicate returns true. For example:

> (filter number? '(a b 2 #t + 4))
(2 4)

Exercises

For these, you will probably want to use this helper function. Just copy its definition into your code.

; Returns a list containing integers a, a+1, ..., b.
(define (range a b)
  (if (> a b)
      '()
      (cons a (range (+ a 1) b))))

12. Product of square roots
    Create a function sqrt-prod that takes a number n and computes the product of the square roots of the integers from 1 up to n. For example:

    > (sqrt-prod 10)
    1904.94...

13. Find the special numbers
    Define a function special-nums that takes an integer n and computes a list of all integers between 1 and n that are both divisible by 3 and are perfect squares. For example:

    > (special-nums 100)
    (9 36 81)

    Hint: you should make some helper functions that are predicates.
14. Polygonal transformation (OPTIONAL)
    Make the following definition for P1:

    (define P1 '((3 5) (9 2) (11 6) (8 8) (4 6)))

    P1 represents a polygon whose vertex coordinates are given by the pairs in P1. Write a function (translate p d) that takes a polygon (in the sense of P1, i.e., a list of coordinate pairs) and an offset d (represented by a pair, the x offset and the y offset) and translates the given polygon by the given offset.

    > (define P1 '((3 5) (9 2) (11 6) (8 8) (4 6)))
    > (translate P1 '(1 2))
    ((4 7) (10 4) (12 8) (9 10) (5 8))

    Depending on your approach, the following function might be useful to you:

    ;Creates a list of k copies of x
    (define (repeat x k)
      (if (= k 0)
          '()
          (cons x (repeat x (- k 1)))))

    For full credit, your function should also work in higher dimensions. By this I mean, if each point has x, y, and z coordinates, for example, everything should still work, by providing a list of 3 offsets to the translate function.