Comp 211 Laboratory 2

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 define-struct), but
we are defining (identifying)
a subset of Scheme numbers. The definition and template use some built-in Scheme functions (add1, sub1, zero?) 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 (add 2 2) to see how it works.

Example functions

Exercises

Write each of the functions on N.

The factorial function !, which is defined by the equations:

(! 0) = 1 , and
(! (add1 n)) = (* (add1 n) (! n)).

The function down that takes an input n in N and returns the list of N
(n ... 1 0).

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

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)))

Rewrite the following using 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 ' notation we can abbreviate constant lists even more.

Finger Exercises on list constants

Evaluate he following in the DrScheme interactions pane. You can cut and paste to save time if you want. Note that ' produces strange results for embedded references to true, false, (), 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 '())
'((cons x y) (1 (+ 1 1) (+ 1 1 1)))

Notice that no expressions within the scope of the ' operator are evaluated.
We can think of the ' operator as distributing over the elements. We apply this rule recursively until there are no more ' operators left. This simple rule makes embedded references to true, false, and empty behave strangely because 'true, 'false, and 'empty reduce to themselves as symbols, not to true, false, and empty. In contrast, 'n for some number n reduces to n.

Trees and Mutually Recursive Data Definitions

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 of inductively defined tree data.
In ancestor family trees, each person (a make-child structure)
has two ancestors (also make-child structures) which may be empty.
In this lab, we'll use a similar, but slightly different, form of tree as an example.

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

5+(1-8)×(7+1)

or equivalently, the Scheme code

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

which encodes expressions as lists revealing their nesting structure.

The string representation for expressions is particularly unattractive for computational purposes because we have to
parse the string to understand its structure. The parsing process must understand which symbols are variables, operatorsj incorporate the precedence of infix operators.

We can define a tree formulation of simple Scheme expressions which avoids representing them as list and encodes far more information about their structure. Parsers build tree representations for programs.

To simplify the formulation of Scheme expressions as trees,
we will limit each addition, subtraction, multiplication, and
division operation to exactly two subexpressions. We will limit the
atomic elements of expressions to numbers.

;; Given

(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 either:
;; - 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; or
;; - (make-div <var>m</var> <var>n</var>) where <var>m</var>,<var>n</var> are AExps,

Usinbg this data definition, the arithmetic expression above corresponds to the structure ae1
defined by

(define ae1 (make-add 5 (make-mul (make-sub 1 8) (make-add 7 1))))

A trival AExp is ae2 defined by

(define ae2 16)

Exercises on Arithmetic Expressions

Develop the function eval: AExp -> N where (eval ae) returns the number denoted by the expression
ae. For example,
(eval ae1)
should return -51, and
(eval ae2)
should return 16.

[Challenge] Assume that our expression language includes many basic operations, not just
the four supported by AExp. We would want a single representation for the application
of a binary operator to arguments and use a separate data definition enumerating
all of our operations. Rewrite the preceding data definitions, examples,
and the function
Unknown macro: {eval}
using
for this. As a further challenge, extend your data definition to accommodate unary operations including
negation and absolute value as unary operators.

Files and Directories

The following are data definitions are idealized (for the sake of simplicity) representations
of files and directories (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 in the definition of a file, which makes the set f definitions more compact but obfuscates 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 either:
; - a symbol (representing a "simple" file's name) or
; - a directory (make-dir name contents) where name is a symbol, and contents is a lof.


; A list-of-files (lof) is one of
; - empty or
; - (cons f lofd) where f is a file and lofd is a lof

This set of definitions is very similar to the descendant trees data structure discussed 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.

Note that this function is a vast simplification of
find, the
mother-of-all everything-but-the-kitchen-sink UNIX directory
traversing command. If open a terminal window and enter

man find

to see what it can do.

Use DrScheme's stepper to step through an example use of find?.
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 is a straightforward way to write this function that just follows the template.

Challenge: develop a program to check for duplicated names among
all directories and files in the given tree, not just subdirectories.
Here's a hint. Develop the following function:
; flatten-dir-once : symbol file -> (file or lof)
; Purpose: returns a structure like the original file, except that any (sub)directory with that name is removed and its contents are promoted up one level in the tree.

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.

Example 1:

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

becomes

foo
/ / \ \
bar baz one two

Example 2:

to-remove
/ \ \
foo bar baz

becomes

foo bar baz

Follow the templates and think about a single
case at a time. If you do that, this exercise is not too difficult. If you don't follow the templates, you
are likely to run into difficulty.

Child pages

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