Natural Numbers & List Abbreviations

Instructions for students & labbies: Students use DrScheme, following the design recipe, working on the exercises at their own pace, while labbies wander among the students, answering questions, bringing the more important ones to the lab's attention. Students should feel free to skip the challenge exercises.

Scheme's Built-in Naturals

We already know Scheme has lots of numbers built-in, like 3, 17.83, and -14/3. It is often convenient to limit our attention to a subset of these such as the naturals: 0, 1, 2, 3, ... . We can define the naturals and its template as follows:

; A natural (*N*) is either:
; - 0
; - (add1 n)
; where n is a natural

; Template
; nat-f : natural -> ...
(define (f ... n ... ) 
  (cond [(zero? n) ...] 
        [(positive? n)
         ... (f ... (sub1 n) ... ) ...]))

Of course, we already know what the example data looks like: 0, 1, 2, 3, ...

Unlike most data definitions, we are not defining new Scheme values here (i.e., there's no

), but
we are defining (identifying)
a subset of Scheme numbers. The definition and template use some built-in Scheme functions (

,

,

) that may be unfamiliar, but which mean just what their names suggest.

If we ignore that Scheme has a built-in function {+}, we could define it ourselves as follows:

; add: natural natural \-> natural
; Purpose: (add x y) returns the sum of x and y .

(define (add n m) 
  (cond [(zero? n) m]
        [(positive? n) (add1 (add (sub1 n) m))]))

Optional Exercise

• Use the stepper on
  Unknown macro: {(add-nat 2 2)}
  to see how it works.

Example functions

Exercises

Write each of the functions on N.

1. The factorial function
   Unknown macro: {!}
   , which is defined by the equations:

, and

.

1. The function
   Unknown macro: {down}
   that takes an input
   Unknown macro: {n}

   in N and returns the list of N

   Unknown macro: {(n ... 1 0)}

   .

   1. The function
      Unknown macro: {up}
      that takes an input
   in N and returns the list of N
   Unknown macro: {(0 1 ... n)}
   . Hint: define an auxiliary function
   Unknown macro: {upfrom}
   such that
   Unknown macro: {(upfrom m n)}
   returns {(m (add1 m) ... n). Assume that
   Unknown macro: {m}
   is less than or equal to
   Unknown macro: {n}
   .

List Abbreviations

Chapter 13 of the book introduces some new, compact methods for representing lists, which have already been mentioned in lecture. The following exercises simply let you explore how this notation behaves.

Finger Exercises on list abbreviations

1. Evaluate he following in the DrScheme interactions pane. You can cut and paste to save time if you want.

   (list 1 2 3)
   (cons 1 (cons 2 (cons 3 empty)))
   (list 1 2 3 empty)
   (cons 1 (cons 2 (cons 3 (cons empty empty)))
   

1. Rewrite the following using
   Unknown macro: {list}
   .

   (cons (cons 1 empty) empty) (cons 1 (cons (cons 2 (cons 3 empty)) (cons 4 (cons (cons 5 empty) empty))))
   (cons (cons (cons 'bozo empty) empty) empty)
   

List constants

Using

produces strange results for embedded references to

,

,

, and empty

'(1 2 3 4)
(list 1 2 3 4)
'(rabbit bunny)
(list 'rabbit 'bunny)
'(rabbit (2) (3 4 5))
(list 'rabbit (list 2) (list 3 4 5))
'(true)
'(empty)
'(())
(list empty)
(list ())
(list 'empty)
(list '())

'(1 2 3 4) => (list 1 2 3 4) => (cons 1 (cons 2 (cons 3 (cons 4 empty)))) '(rabbit bunny) => (list 'rabbit 'bunny) => (cons 'rabbit (cons 'bunny empty)) '(rabbit (2) (3 4 5)) => (list 'rabbit (list 2) (list 3 4 5)) (cons 'rabbit (cons (cons 2 empty) (cons (cons 3 (cons 4 (cons 5 empty))) empty)))

Exercise

Re-write the lists from above using
'
notation.

We can think of the
'
as distributing over the elements. We apply this rule recursively (Yes! Recursion strikes again!) until there are no more'(
s left.

'(rabbit (2) (3 4 5)) => (list 'rabbit '(2) '(3 4 5)) (list 'rabbit (list '2) (list '3 '4 '5)) => ... => (cons 'rabbit (cons (cons 2 empty) (cons (cons 3 (cons 4 (cons 5 empty))) empty)))

NB:

'1
reduces to1
. In general,'<a number>
evaluates to<a number>
.

Exercise

What do we get in these cases?

'((make-posn 1 2)) '(1 (+ 1 1) (+ 1 1 1))

You cannot use list abbreviation one nested in another. For example
'(1 '(+ 1 1)) (will be treated in DrScheme as (list 1 (list 'quote (list '+ 1 1)))

If we want to apply functions, we have to use eitherconsorlist. (Not exactly true. There is another abbreviation,quasiquote, that we won't talk about in this course.)

Trees & Mutually Recursive Data Definitions

Instructions for students & labbies:

Students use DrScheme, following the design recipe,
working on the exercises at their own pace,
while labbies wander among the students, answering questions,
bringing the more important ones to the lab's attention.
Students should feel free to skip the challenge exercises.

Trees

In class, we used ancestor family trees as an example form of trees.
In ancestor family trees, each person (a
make-child
structure)
has 0, 1, or 2 ancestors (alsomake-child
structures).
Here, we'll use a similar, but slightly different, form of trees for
more experience.

In mathematics, we can model arithmetic expressions as trees. For
example,

5+(1-8)×(7+1)

or equivalently, the Scheme code

(+ 5 (* (- 1 8) (+ 7 1)))

is pictorially

+ / \ 5 × / \ - + / \ / \ 1 8 7 1

This tree form has some advantages. To understand the more familiar
linear form, you must know about the order of operator precedence,
whereas that is unnecessary in the tree form. The tree also eliminates
the need for parentheses. The tree also gets us away from the relatively
minor concerns of the precise details of mathematical or Scheme
notation, like infix vs. prefix operators.

Consider if you were developing a computer program like DrScheme
(or, similarly, a "compiler," if you know what that is).
Such a program would take the linear form, which is convenient for
a person to type in, but then convert or

parse

it to the tree form
for internal use.
Since parsing is beyond the scope of this course, let's just skip straight
to the tree form.

We'll require that each addition, subtraction, multiplication, and
division has exactly two subexpressions. Of course, recursively,
each subexpression can be another addition, subtraction, multiplication,
or division. As a base case, an expression can also be a number.

(define-struct add (m n)) (define-struct sub (m n)) (define-struct mul (m n)) (define-struct div (m n)) ; An Arithmetic-Expression (AExp) is one of ; - a number ; - (make-add <var>m</var> <var>n</var>) ; where <var>m</var>,<var>n</var> are AExps ; - (make-sub <var>m</var> <var>n</var>) ; where <var>m</var>,<var>n</var> are AExps ; - (make-mul <var>m</var> <var>n</var>) ; where <var>m</var>,<var>n</var> are AExps ; - (make-div <var>m</var> <var>n</var>) ; where <var>m</var>,<var>n</var> are AExps

With this data definition, the above tree is modeled by the structure

(define AExp1 (make-add 5 (make-mul (make-sub 1 8) (make-add 7 1))))

Another sample AExp is

(define AExp2 16)

As always, we distinguish between the information (the mathematical
expression or its corresponding tree) and its data representation
(this AExp).
Just writing this piece of data doesn't mean we can do anything with it
yet, such as compute the intended result.

Exercise

Make more example data.

Develop the function
evaluate
which takes an
AExp as input and returns the number that the expression
mathematically computes. For example,
(evaluate AExp1)
should result in -51, and
(evaluate AExp2)
should result in 16.

Challenge exercise:

Let's say we hadmanybasic operations, not just
these four. We would want to have one structure for any
binary operation, using a separate data definition enumerating
all of our operations. Rewrite the data definitions, examples,
andevaluate
for this.
As a further challenge, also allow unary operations.

Files and Directories

The following are data definitions for one possible (simplified) representation
of files and directories (a.k.a. folders). These definitions follow the Windows convention of attaching a name to a file. They also collapse the definition of the directory type into a clause of the definition of for file, which makes the definitions more compact but makes it less clear how to write functions that process directories (instead of files). For this reason, none of the following exercises uses a directory as the primary input to a function.

Observe the mutual recursion between files and list-of-files.

(define-struct dir (name contents))

; A file is one of
; - a symbol
; representing a "simple" file's name
; - a directory
; (make-dir name contents)
; where name is a symbol, and contents is a l-o-f.

; A list-of-files (l-o-f) is one of
; - empty
; - (cons f lofd)
; where f is a file, and lofd is a l-o-f

This is very similar to the descendant trees data structure seen in class.

Tree-based data structures are very common!

Directory exercises

Create some sample data for the above types.

Write the templates for the above types.

Develop a function

; find? : symbol file -> boolean
; Returns whether the filename is anywhere in the
; tree of files represented by the file. This includes both
; simple file names and directory names.

Aside

Note that this is a vast simplification of
find
, the
mother-of-all everything-but-the-kitchen-sink UNIX directory
traversing command. If you're curious, logon to a UNIX machine at a UNIX shell prompt, enter
man findto see what it can do.

Use DrScheme's stepper to step through an example use
offind?.
Following the templates leads to an overall strategy known as

depth-first search

. I.e., it explores each tree branch to the
end before moving on to the next branch.

Develop the following function:

; any-duplicate-names? : file -> boolean
; Returns whether any (sub)directory directly or indirectly contains
; another directory or file of the same name. It does NOT check
; for duplicated names in separate branches of the tree.

There's a straightforward way that just follows the template.

Challenge:

Develop a program to check for duplicated names among
alldirectories and files in the given tree, not just
subdirectories.
Here's a hint.

Develop the following function:

; flatten-dir-once : symbol file -> (file or l-o-f)
; Returns a structure like the original file, except that any
; (sub)directory with that name is removed and its contents
; moved up one level.

Here are two pictorial examples, in both cases removing the directory
namedto-remove. These illustrate why this function can
return either a file or a list of files.

Input

Output

Example 1:

foo
/ | \
bar baz to-remove
/ \
one two

foo
/ / \ \
bar baz one two

Example 2:

to-remove
/ | \
foo bar baz

foo bar baz

Follow the templates and think about a single
case at a time.

If you do that, it shouldn't be too difficult. If you don't, you'll
probably have real trouble.

Sample solutions.

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