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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.

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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 following functions on N.

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

   Code Block
   (! 0) = 1

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1. (! (add1 n)) = (* (add1 n) (! n))

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2. The function down that takes an input n in N and returns the list of N
   (n ... 1 0).
3. 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.

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Finger Exercises on List Abbreviations

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

   Code Block
   (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)))

2. Rewrite the following using list.

   Code Block
   (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)

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Using ' notation we can abbreviate constant lists even more concisely.

Finger Exercises on list constants

1. Evaluate he the 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.

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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

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bringing the more important ones to the lab's attention.
Students should feel free to skip the challenge exercises.

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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 use formalized arithmetic expressions as trees. For
example,

Code Block
5+(1-8)×(7+1)

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Code Block
(+ 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

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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, operators 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

parse

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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 anddivision operation to 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.. We will limit the atomic elements of expressions to numbers.

Code Block
;; Given (define-struct add (

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left right)) (define-struct sub (

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left right)) (define-struct mul (

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left right)) (define-struct div (

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left right))

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;; an Arithmetic-Expression (AExp)

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is either: ;; - a number ; ;; - (make-add

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l r) where l,r are AExps; ;; - (make-sub l r) where l,r are AExps; ;; - (make-mul l r) where l,r are AExps; or ;; - (make-div l r) where l,r are AExps,

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

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

A trival AExp is ae2 defined by

Code Block
(define ae2 16)

Exercises on Arithmetic Expressions

1. 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.
2. [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

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.

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Exercise

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Make more example data.

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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.

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Challenge exercise:

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1. all of our operations. Rewrite the preceding data definitions, examples,

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1. and the function eval using for this.

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1. As a further challenge,

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1. extend your data definition to accommodate unary operations

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1. including negation and absolute value as unary operators.

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Files and Directories

The following are data definitions for one possible (simplified) representation
are idealized (for the sake of simplicity) representations 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 in the definition of for a file, which makes the set f definitions more compact but makes it less clear 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))

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; A file

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 is either:
; - a symbol (representing a "simple" file's name

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) or
; - a directory

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 (make-dir name contents)

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 where name is a symbol, and contents is

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 a lof.


; A list-of-files (

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lof) is one of

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; - empty

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 or
; - (cons f lofd)

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 where f is a file

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 and lofd is

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 a lof

This set of definitions

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

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Tree-based data structures are very common!

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Directory exercises

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1. Create some sample data for the above types.

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2. Write the templates for the above types.

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2. Develop a function

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   Code Block
   ; find? : symbol file -> boolean

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1. ; Returns whether the filename is anywhere in the

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1. ; tree of files represented by the file. This includes both

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1. ; simple file names and directory names.

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Aside

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1. Note that this function is a vast simplification of

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1. {{find

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1. }}, the

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1. mother-of-all everything-but-the-kitchen-sink UNIX directory

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1. traversing command. If

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1. open a terminal window and enter

   Code Block
   man find

   to see what it can do.
   

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1. Use DrScheme's stepper to step through an example use

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1. of find?.

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1. Following the templates leads to an overall strategy known as

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1. depth-first search

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1. , i.e., it explores each tree branch to the

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1. end before moving on to the next branch.

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1. Develop the following function:

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   Code Block
   ; any-duplicate-names? : file -> boolean

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1. ; Returns whether any (sub)directory directly or indirectly contains

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1. ; another directory or file of the same name. It does NOT check

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1. ; for duplicated names in separate branches of the tree.

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1. There

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1. is a straightforward way to write this function that just follows the template.

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1. Challenge:

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1. develop a program to check for duplicated names among

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1. all directories and files in the given tree, not just

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1. subdirectories.
   
   Here's a hint.

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1. Develop the following function:

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   Code Block
   ; flatten-dir-once : symbol file -> (file

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1. or lof) ; Purpose: returns a structure like the original file, except that any (sub)directory with that name is removed and its

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1. contents are promoted up one level in the tree.

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1. Here are two pictorial examples, in both cases removing the directory

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1. named to-remove. These illustrate why this function can

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1. return either a file or a list of files.

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Input

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Output

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Example 1:

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

becomes

      foo
   /  / \  \
bar baz one two

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Example 2:

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 to-remove
   / \ \
foo bar baz

becomes

foo bar baz

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

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foo
/ / \ \
bar baz one two

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Example 2:

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to-remove
/ | \
foo bar baz

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foo bar baz

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

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

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Sample solutions.

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!! Access Permissions

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follow the templates, you are likely to run into difficulty.