Functional Programming Language 1 Scheme Language: part 2

FunctionalProgramming Language

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Scheme Language: part 2

Section 4: Conditional Expressions

4.1 Booleans and Relations 4.2 Functions that Test Conditions 4.3 Conditionals and Conditional Functions

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4.1 Booleans and Relations

In mathematics we talk of true and false claims, which are conditions.

For example, if x and y are numbers, we state these three claims about x and y with1. x = y: ``x is equal to y'';2. x < y: ``x is strictly less than y'';3. x > y: ``x is strictly greater than y''.

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In addition to determining whether an atomic claim holds in a given situation.

Consider the three claims above, which we can combine in several ways:1. x = y and x < y and x > y2. x = y or x < y or x > y3. x = y or x < y .

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Like mathematics, Scheme has ''words'' for expressing truth and falsity, for stating atomic claims, for combining claims into compound claims, and for expressing that a claim is true or false.

The ''word'' for true is #t and the ''word'' for false is #f.

If a claim concerns the relationship between two numbers, it can typically be expressed with a RELATIONAL OPERATION, for example, =, <, >, <= and >=. 5


Translating the three mathematical claims from above follows our well-known pattern of writing a left parenthesis, followed by the operator, its arguments, and a right parenthesis:1. (= x y): ``x is equal to y'';2. (< x y): ``x is strictly less than y''; and3. (> x y): ``x is strictly greater than y''. 6


Like the arithmetic operations, the test operations can take more than two arguments.

For example, (< 1 2 3)

is legal Scheme notation for ``1 < 2 and 2 < 3.''

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The and and or operators also take any number of arguments.

The not operation always takes one argument.

The and, or, and not operators work on Boolean values only.

1. (and (= x y) (< y z))2. (or (= x y) (< y z))3. (not (= x y)) 8


Example, Consider the following compound condition:

(and (= 5 5) (< 5 6)) It consists of two atomic claims:

(= 5 5) and (< 5 6). Both evaluate to true.

= (and true true)= true

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Exercise 4.1.1 What are the results of the following Scheme conditions?1. (> 7/2 3)2. (= (* 2 2) 4)3. (and (> 4 3) (<= 10 100))4. (or (> 4 3) (= 10 100))5. (not (= 2 3))

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4.2 Functions that Test Conditions

Here is a simple function that tests some condition about a number:;; is-5? : number -> boolean;; to determine whether n is equal to 5(define (is-5? n) (= n 5))

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Here is a slightly more interesting function with a boolean output:

;; is-between-5-6? : number -> boolean

;; to determine whether n is between 5 and 6 (exclusive)

(define (is-between-5-6? n) (and (< 5 n) (< n 6)))

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The following third function from numbers to boolean values represents the most complicated form of interval:;; is-between-5-6-or-over-10? : number -> boolean;; to determine whether n is between 5 and 6 (exclusive) ;; or larger than or equal to 10(define (is-between-5-6-or-over-10? n)

(or (is-between-5-6? n) (>= n 10)))13


Exercise 4.2.1 Translate the following five intervals on the real line into Scheme functions that accept a number and return true if the number is in the interval and false if it is outside:1. the interval (3,7]:

2. the interval [3,7]:14


3. the interval [3,9):

4. the union of (1,3) and (9,11):

5. the range of numbers outside of [1,3]

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Exercise 4.2.2. Translate the following three Scheme functions into intervals on the line of reals:

1. (define (in-interval-1? x) (and (> -3 x) (< x 0)))

2. (define (in-interval-2? x) (or (< x 1) (> x 2)))

3. (define (in-interval-3? x) (not (and (>= 1 x) (<= x 5)))) 16


Mathematical equations in one variable are claims about an unknown number.

For example, the quadratic equation

(x2 + 2) (x + 1) = 0

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We can use Scheme to formulate equational conditions as a function. ;; equation1 : number -> boolean;; to determine whether x is a solution

;; for (x2 + 2) (x + 1) = 0 (define (equation1 x)

(= (+ (* x x) (+ (* 2 x) 1)) 0))18


When we apply equation1 to some number, we get true or false:

(equation1 -1)= true

and(equation1 +1)= false 19

4.3 Conditionals and Conditional Functions

In many time we interest function must determine which of several conditions holds for the input.

We say that the function is a CONDITIONAL FUNCTION, and we formulate the definition of such functions using CONDITIONAL EXPRESSIONS.20


The general shape of a conditional expression is

Example:

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(cond [(<= n 50) "D"] [(<= n 60) "C"] [(<= n 70) "B"] [(> n 80) "A"])

(cond [(<= n 50) "D"] [(<= n 60) "C"] [(<= n 70) "B"] [else "A"])


Exercise 4.3.1 Decide which of the following two cond-expressions is legal:

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(cond[(< n 10) 20][(> n 20) 0][else 1])

(cond [(< n 10) 20] [(and (> n 20) (<= n 30))] [else 1])


Exercise 4.3.2 Suppose the bank pays 4% for deposits of up to $1,000 (inclusive), 4.5% for deposits of up to $5,000 (inclusive), and 5% for deposits of more than $5,000.

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;; interest-rate : number -> number;; to determine the interest rate for the given amount(define (interest-rate amount)

(cond[(<= amount 1000) 0.040][(<= amount 5000) 0.045][else 0.050])) 24

Section 5: Symbolic Information

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5. Symbolic Information Scheme supports several ways to

express symbolic information: symbols, strings, (keyboard) characters, and images.

A symbol is a sequence of keyboard characters preceded by a single forward quotation mark:

'a 'dog 'chocolate 'cat! 'two^3 'and%so%on?

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5. Symbolic Information

Scheme provides only one basic operation on symbols: symbol=?, a comparison operation.1. (symbol=? 'Hello 'Hello)

= true2. (symbol=? 'Hello 'Howdy)

= false

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Symbols were first introduced to computing by researchers in artificial intelligence who wanted to design functions that could have conversations with people.(define (reply s)

(cond [(symbol=? s 'GoodMorning) 'Hi] [(symbol=? s 'HowAreYou?) 'Fine] [(symbol=? s 'GoodAfternoon)

'INeedANap] [(symbol=? s 'GoodEvening)

'BoyAmITired]))28


A string is a second form of symbolic data. Like a symbol, a string consists of a sequence of keyboard characters, but they are enclosed in string quotes:

"the dog" "isn't" "made of" "chocolate" "two^3" "and so

on?“ The operation needed is string=?,

which compares two strings The way symbol=? compares two

symbols. 29

Section 6: Structures 6.1 Structures 6.2 Extended Exercise: Drawing Simple Pictures 6.3 Structure Definitions

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6.1 Structures Scheme provides many different

methods for compounding data. DrRacket's teachpacks represent

pixels with posn structures. The posn structure type that is also

provided by lang/htdp-beginner. (require lang/posn)

It has an x coordinate, and it has a y coordinate, which tells us where the pixel is located.

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6.1 Structures Here is DrRacket's definition of posn:

(define-struct posn (x y)) When DrRacket evaluates this

structure definition, it creates three operations for us,1. make-posn, the CONSTRUCTOR, which creates posn structures;2. posn-x, a SELECTOR, which extracts the x coordinate;3. posn-y, also a selector, which extracts the y coordinate. 32

6.1 Structures A posn structure combines two

numbers. The posn structure is built into

DrRacket's teaching languages, including make-posn, posn-x, and posn-y.

For example,1. (make-posn 3 4)2. (posn-x (make-posn 7 0)) 3. (posn-y (make-posn 7 0))

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6.1 Structures

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6.1 Structures Example: 1. (distance-to-0 (make-posn 3 4))

= 5 2. (distance-to-0 (make-posn 8 6))

= 10 3. (distance-to-0 (make-posn 5

12))= 13

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6.1 Structures Exercise 6.1.1 Evaluate the

following expressions:1. (distance-to-0 (make-posn 3 4))2. (distance-to-0 (make-posn (* 2 3) (* 2 4)))3. (distance-to-0 (make-posn 12 (- 6 1)))

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6.2 Extended Exercise: Drawing Simple Pictures

DrRacket provides the graphics teachpack draw.rtk, which introduces simple graphics operations:

1. draw-solid-line, – which consumes two posn

structures, – the beginning and the end of the

line on the canvas, – and a color. 37


2. draw-solid-rect, – which consumes four arguments: – a posn structure for the upper-left

corner of the rectangle, – a number for the width of the

rectangle, – another number for its height,

and a color.

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3. draw-solid-disk, – which consumes three arguments: – a posn structure for the center of the

disk, – a number for the radius of the disk,– and a color.

4. draw-circle, which consumes three arguments:

– a posn structure for the center of the circle,

– a number for the radius, – and a color.

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We refer to the action to the canvas as an EFFECT.

Each drawing operation also comes with a matching clear- operation: clear-solid-line, clear-solid-rect, clear-solid-disk, and clear-circle.

Read the documentation for draw.ss in DrRacket's HelpDesk.

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Drawing operations on computers interpret the screen as follows:

The origin of the plane is in the upper-left corner. 41


When you evaluate (start 500 500) with the draw.ss library, DrRacket create a new, empty drawing canvas, like this one:

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Evaluating a drawing expressions,

(draw-solid-line (make-posn 100 100) (make-posn 200 200) 'red)

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Exercise 6.2.1 Evaluate the following expressions in order:

(start 300 300)(draw-solid-line (make-posn 10 10)

(make-posn 110 10) 'red)(draw-solid-rect (make-posn 10 30) 100

50 'blue) (draw-circle (make-posn 110 30) 30 'yellow)

(draw-solid-disk (make-posn 10 80) 50 'green)

(stop) 44

6.3 Structure Definitions

A STRUCTURE DEFINITION is, as the term says, a new form of definition.

In general, the names of these new operations are created by prefixing the name of the structure with ``make-'' and by postfixing the name with all the field names.

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Now consider the following example:(define-struct entry (name zip

phone)) Each entry combines three values. We also say that each entry structure

has three fields: name, zip, and phone.

The constructor make-entry consumes three values.

(make-entry 'PeterLee 15270 '606-7771) 46


The define-struct definition of entry also introduces new selectors:entry-name entry-zip entry-phone

Here is how we can use the first one:(entry-name (make-entry 'PeterLee 15270 '606-7771))= 'PeterLee 47


If we give the structure a name, (define phonebook (make-entry

'PeterLee 15270 '606-7771)) then we can use the selectors in

the Interactions window to extract the data from the three fields:(entry-name phonebook) (entry-zip phonebook)(entry-phone phonebook) 48


Exercise 6.3.1 Consider the following structure definitions:1. (define-struct movie (title producer))2. (define-struct boyfriend (name hair eyes phone))3. (define-struct cheerleader (name number))4. (define-struct CD (artist title price))5. (define-struct sweater (material size producer))

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Functional Programming Language 1 Scheme Language: part 2