Homework 4 (Due Monday 2/14/2011 at 10:00am)
Use the {{ Wiki Markup 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,
becomesCode Block (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))]))
Code Block (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))]))
- 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,
- 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\]Wiki Markup - Name your function
make-sort
. make-sort
produces functions. Test these results by applying them to a few different arguments.
- Name your function
- 23.3.9 (10 pts) \ [For 2011: 8 pts\]
- The series represented by
greg
is (4, -4/3, 4/5, -4/7, ...).
- The series represented by
- 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,
becomesCode Block (lambda (f g x) (f x (lambda (x) (g (g x)))))
Code Block (lambda (f1 g1 x1) (f1 x1 (lambda (x2) (g1 (g1 x2)))))
- 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,
...
- (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 functionsplit: (list-of number) -> 2-element-structure-of-lists-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 a a 2-element structure containing two lists of numbers. See below for hints in defining a 2-element structure. After splitting the list in half,mergesort
recursively sorts each half and then merges them together. Be careful about attempting to split a one-element list!
Hints for writing the "split" function:
...
b) Use Scheme's built-in functions. First, you must explicitly define that you are going to use a list of exactly two elements, i.e. (list a b)
. Scheme already provides functions to access the first and second elements of such a list, called, oddly enough, called "first" and "second":
Code Block |
---|
(first (list a b)) ==> a (second (list a b)) ==> b |
Again, you will still need to define exactly what types the first and second elements are supposed to be. DO NOT USE (first (rest lon))
as this an encapsulation violation of an indefinite-length list!
Think simple!
Follow these important rules whenever you write function on a list using structural recursion (note that the generative recursion that mergesort uses follows slightly different rules):
...