Homework 4 (Due Friday 2/12/2010 at 10:00am)

Use the Intermediate Student with lambda language level. You can choose to use lambda, as appropriate, in any of the assigned problems. [For 2011: You may not use local to define functions; all program functions (including helpers) must be defined at the top level.]

Book Problems:

18.2.2 (10 pts)

In this problem, simply annotate each definition by attaching identifying subscripts (written in line such as x1 for "x sub 1" and y4 for "y sub 4") to _all_variable occurrences so that all of the uses of each defining occurrence are identified. A good way to choose a subscript value for a variable is to use the lexical nesting level of the binding occurrence. For example,

(define (my-max lon) 
  (cond [(empty? lon) (error 'my-max "applied to no arguments")]
        [(empty? (rest lon)) (first lon)]
        [else 
         (local [(define head (first lon))
                 (define max-tail (my-max (rest lon)))]
           (if (>= head max-tail) head max-tail))]))

becomes

(define (my-max1 lon1) 
  (cond [(empty? lon1) (error 'my-max "applied to no arguments")]
        [(empty? (rest lon1)) (first lon1)]
        [else 
         (local [(define head2 (first lon1))
                 (define max-tail2 (my-max1 (rest lon1)))]
           (if (>= head2 max-tail2) head2 max-tail2))]))

20.1.1 (10 pts)
21.1.2 (10 pts)
21.2.1 (10 pts)
21.2.3 (10 pts)
- "Lists of names" just means "lists of symbols."
22.2.2 (10 pts) [For 2011: 12 pts]
- Name your function make-sort .
- make-sort produces functions. Test these results by applying them to a few different arguments.
23.3.9 (10 pts) [For 2011: 8 pts]
- The series represented by greg is (4, -4/3, 4/5, -4/7, ...).
24.0.8 (5 pts)
- Instead of drawing arrows from the underlined occurrences, simply annotate each expression by attaching subscripts (written in line such as x1 for "x sub 1" and y4 for "y sub 4") to all variable occurrences so that all of the uses of each defining occurrence are identified. For example,
  (lambda (f g x) (f x (lambda (x) (g (g x)))))
  becomes
  (lambda (f1 g1 x1) (f1 x1 (lambda (x2) (g1 (g1 x2)))))

Problems not in the book:*

(5 pts) Write the most general (parametric) contract for the following function:

; compose : ?
; Purpose: (compose f g) returns the result of composing functions f and g: x |-> f(g(x))
  (define (compose f g)
    (lambda (x) (f (g x))))

(20 pts) Write the same mergesort function as described in exercise 26.1.2, except decompose the problem "top-down" rather than "bottom-up". You will need to define a function split: (list-of number) -> (list-of (list-of number)) that partitions its input into two lists of approximately (+/- 1) the same length in O(n) time where n is the length of the input list. (split l) returns a list containing two lists of numbers. After splitting the list in half, mergesort recursively sorts each half and then merges them together.