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Comp

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211

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Laboratory

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2

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Natural

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Numbers

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&

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List Abbreviations

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.

Exercises

Write each of the following functions on N.

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

   Code Block
   (! 0) = 1 (! (add1 n)) = (* (add1 n) (! n))

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.

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 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)

List Constants

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

Finger Exercises on list constants

1. Evaluate 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.

'(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 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 use formalized arithmetic expressions as trees. For example,

 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.


h3. 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:
{code}
; 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) ... ) ...]))
{code}
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:
{code}
; 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))]))
{code}

*Optional Exercise*
* Use the stepper on {{(add 2 2)}} to see how it works.

Example functions

h4. Exercises

Write each of the functions on *N*. 

# The factorial function {{!}}, which is defined by the equations:
{code}
(! 0) = 1
(! (add1 n)) = (* (add1 n) (! n))
{code}
# 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}}.

h2. 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.

h4. Finger Exercises on List Abbreviations

# Evaluate he following in the DrScheme interactions pane.  You can cut and paste to save time if you want.
{code}
(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)))
{code}

# Rewrite the following using {{list}}.
{code}
(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)
{code}

h3. List Constants

Using {{'}} notation we can abbreviate _constant_ lists even more.

h4. 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}}.

{code}
'(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)))
{code}

This is a
test

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}}.

This is a 
test.

h2. 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.

h3. 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,
{code}
5+(1-8)×(7+1)
{code}

or

...

equivalently,

...

the

...

Scheme

...

code

}
(+ 5 (* (- 1 8) (+ 7 1)))
{code}

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 each addition, subtraction, multiplication, anddivision operation to exactly two subexpressions. We will limit the atomic elements of expressions to numbers.

This is
a
test
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.
{code}
;; Given

(define-struct add (mleft nright)) 
(define-struct sub (mleft nright)) 
(define-struct mul (mleft nright)) 
(define-struct div (mleft nright)) 

;; an Arithmetic-Expression (AExp) is either:
;; - a number ;
;; - (make-add <var>m</var> <var>n</var>l r) where <var>m</var>,<var>n</var>l,r are AExps;
;; - (make-sub <var>m</var> <var>n</var>l r) where <var>m</var>,<var>n</var>l,r are AExps; 
;; - (make-mul <var>m</var> <var>n</var>l r) where <var>m</var>,<var>n</var>l,r are AExps; or
;; - (make-div <var>m</var> <var>n</var>l r) where <var>m</var>,<var>n</var>l,r are AExps,
{code}

Usinbg this data 

Using this data definition,

...

the

...

arithmetic

...

expression

...

above

...

corresponds

...

to

...

the structure ae1 defined by

 structure {{ae1}}
defined by
{code}
(define ae1 (make-add 5 (make-mul (make-sub 1 8) (make-add 7 1))))
{code}

A

...

trival

...

AExp

...

is

...

ae2

...

defined

...

by

}
(define ae2 16)
{code}

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

h4. 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 {eval} using 
for this.  As a further challenge, extend your data definition to accommodate unary operations including
negation and absolute value as unary operators.

h2.  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.
{code}
(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

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is

...

very

...

similar

...

to

...

the

...

descendant

...

trees

...

data

...

structure

...

discussed

...

in

...

class.

...

Tree-based

...

data

...

structures

...

are

...

very

...

common!

...

Directory exercises

1. Create some sample data for the above types.
2. Write the templates for the above types.
3. Develop a function

   Code Block
   ; find? : symbol file

...

1. -> 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.

...

1. Note that this function is a vast simplification of{{find}},

...

1. the

...

1. mother-of-all

...

1. everything-but-the-kitchen-sink

...

1. UNIX

...

1. directory

...

1. traversing

...

1. command.

...

1. If

...

1. open

...

1. a

...

1. terminal

...

1. window

...

1. and

...

1. enter

...

1. Code Block

...

1. man find

...

1. to

...

1. see

...

1. what

...

1. it

...

1. can

...

1. do.

...

1. 
   
   Use

...

1. DrScheme's

...

1. stepper

...

1. to

...

1. step

...

1. through

...

1. an

...

1. example

...

1. use

...

1. of

...

1. find?

...

1. .

...

1. Following

...

1. the

...

1. templates

...

1. leads

...

1. to

...

1. an

...

1. overall

...

1. strategy

...

1. known

...

1. as

...

1. depth-first

...

1. search

...

1. ,

...

1. i.e.

...

1. ,

...

1. it

...

1. explores

...

1. each

...

1. tree

...

1. branch

...

1. to

...

1. the

...

1. end

...

1. before

...

1. moving

...

1. on

...

1. to

...

1. the

...

1. next

...

1. branch.

...

2. Develop

...

1. the

...

1. following

...

1. function:

...

1. Code Block

...

1. ; 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.

...

1. There

...

1. is

...

1. a

...

1. straightforward

...

1. way

...

1. to

...

1. write

...

1. this

...

1. function

...

1. that

...

1. just

...

1. follows

...

1. the

...

1. template.

...

2. Challenge:

...

1. develop

...

1. a

...

1. program

...

1. to

...

1. check

...

1. for

...

1. duplicated

...

1. names

...

1. among

...

1. all

...

1. directories

...

1. and

...

1. files

...

1. in

...

1. the

...

1. given

...

1. tree,

...

1. not

...

1. just

...

1. subdirectories.

...

1. 
   
   Here's

...

1. a

...

1. hint.

...

1. Develop

...

1. the

...

1. following

...

1. function:

...

1. Code Block

...

1. ; 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.

...

1. Here

...

1. are

...

1. two

...

1. pictorial

...

1. examples,

...

1. in

...

1. both

...

1. cases

...

1. removing

...

1. the

...

1. directory

...

1. named to-remove.

...

1. These

...

1. illustrate

...

1. why

...

1. this

...

1. function

...

1. can

...

1. return

...

1. either

...

1. a

...

1. file

...

1. or

...

1. a

...

1. list

...

1. of

...

1. files.

Example 1:

h5. Example 1: 

{quote}
      foo
    / \   \
 bar baz to-remove
          / \
        one two

becomes

      foo
   /  / \  \
bar baz one two
{quote}

h6. Example 2:

...

Example 2:

...

 to-remove
   / \ \
foo bar baz

becomes

foo bar baz
{quote}

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.

...