Transcript Scheme-part1

Functional Programming Language
1
Functional Programming Language
A program is: Composition of operations on data
 Characteristics (in pure form):
– Name values, not memory locations
– Value binding through parameter passing
– Recursion rather than iteration
 Key operations: Function Assignment, Function
Application, and Function Abstraction– Based on
the Lambda Calculus

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Example: Functional Language
Haskell
 Erlang
 ML
 Scheme
 Ruby
 Pure
 Needle

Clean
J
 Pizza
 Mercury
 Goo
…

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Scheme Language

Book: http://www.schemers.org/
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Section 1: Getting Started
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Download DrRacket
Racket is a Scheme
 Download DrRacket
http://www.informatics.buu.ac.th/~janya/886340_PPL/Progra
mming%20Languages/Scheme/racket-6.1-i386-win32.exe

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The Racket Language

Choose Language: The Racket Language
Definition window
Interaction window
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Section 2: Numbers and Expressions
2.1
2.2
2.3
2.4
Numbers and Arithmetic
Variables and Programs
Errors
Designing Programs
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2.1 Numbers and Arithmetic

Numbers
– Scheme numbers are printed in the standard way, but spaces
are not allowed within a number.
– Numbers come in many different flavors: positive and
negative integers, fractions (also known as rationals), and
reals are the most widely known classes of numbers.
– There is no limit on the size of exact numbers in Scheme
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2.1 Numbers and Arithmetic
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
-5
2/3
17/3
1.4142165237321
The first is an integer, the second one a negative integer,
the next two are fractions, and the last one is an inexact
representation of a real number
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2.1 Numbers and Arithmetic

Numbers
– Thus, -5, with no space between the – and 5, is the negative
integer -5. But – 5, with space in between, is two separate
expressions: the minus operator and the number 5.
– Similarly, 3/2 is the fraction 3/2, while 3 / 2 is three separate
expressions: the number 3, the division operator, and the
number 2.
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2.1 Numbers and Arithmetic
Scheme computes with EXACT integers and rationals as
long as we use primitive operations that produce exact
results.
– Thus, it displays the result of (/ 7 14) as 1/2.
 The square root of 2 is not a rational but a real number,
Scheme uses an INEXACT NUMBER:
– (sqrt 2)
= 1.4142135623730951

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2.1 Numbers and Arithmetic
Arithmetic
 Scheme expression has the shape
(operation A ... B)
 Whenever A ... B are numbers, the expression can be
evaluated; otherwise, A ... B are evaluated first.

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2.1 Numbers and Arithmetic
Like a pocket calculator, the simplest of computers,
Scheme permits programmers to add, subtract, multiply,
and divide numbers:
(+ 5 5) (+ -5 5) (- 5 5) (* 3 4) (/ 8 12)
 All arithmetic expressions are parenthesized and mention
the operation first; the numbers follow the operation and
are separated by spaces.

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2.1 Numbers and Arithmetic

Exercise 2.1.1
1. (+ 5 6 3)
2. (/ 6 -9 -1)
3. (+ 2 (* 3 4))
4. (* (+ 2 2) (-9 4))
5. (+ (- 1 1) (* 2 5) (+ 4 8))
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2.1 Numbers and Arithmetic
As in arithmetic or algebra, we can nest expressions:
(* (+ 2 2) (/ (* (+ 3 5) (/ 30 10)) 2))
 Scheme evaluates these expressions exactly as we do. It
first reduces the innermost parenthesized expressions to
numbers, then the next layer, and so on:

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2.1 Numbers and Arithmetic



Stepper
DrRacket provides a Step tool that shows how a
complex expression is evaluated, step-by-step.
To use the Step tool, put the expression(s) you want
to evaluate into DrRacket's top area, the definitions
window, then click the Step button
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2.1 Numbers and Arithmetic


Stepper
For example, given the expression
(* (+ 2 2) (/ (* (+ 3 5) (/ 30 10)) 2))
= (* 4 (/ (* 8 3) 2))
= (* 4 (/ 24 2))
= (* 4 12)
= 48
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2.1 Numbers and Arithmetic

Exercise 2.1.2
1. (- 2 (+ 10 9))
2. (/ (* (- 2 1) (+ 5 -9)) 6)
3. (* (+ 5 5) (+ (- (* 2 3) (/ 12 6)) (* 10 3)))
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2.1 Numbers and Arithmetic

Scheme not only provides simple arithmetical operations but a
whole range of advanced mathematical operations on numbers.
Here are five examples:
– (sqrt A) computes (A)1/2
– (expt A B) computes AB;
– (remainder A B) computes the remainder of the integer division
A/B;
– (log A) computes the natural logarithm of A;
– (sin A) computes the sine of A radians.
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2.2 Variables and Programs
The Shape of Definitions
 The location of parentheses in a definition must match the
following pattern exactly:

start definition
start program and input names
(define (f arg ...) expression)
end definition
end input names
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2.2 Variables and Programs
In algebra we learn to formulate dependencies between
quantities using VARIABLE EXPRESSIONS.
 A variable is a placeholder that stands for an unknown
quantity.
 Using Scheme, we can formulate such an expression as a
program and use the program as many times as necessary.

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2.2 Variables and Programs

For example,
(n / 3) + 2
 Here is the program that corresponds to the above expression:
(define (f n) (+ (/ n 3) 2))
 First determine the result of the expression at n = 2, n = 5, and
n = 9 by hand
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2.2 Variables and Programs

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2.2 Variables and Programs
A program is such a rule.
 It is a rule that tells us and the computer how to produce data
from some other data.
 Large programs consist of many small programs and
combine them in some manner.
 A good name for our sample expression is area-of-disk.

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2.2 Variables and Programs
Using this name, we would express the rule for computing
the area of a disk as follows:
(define (area-of-disk r) (* 3.14 (* r r)))
 area-of-disk is a rule, that it consumes a single INPUT,
called r, and that the result, or OUTPUT, is going to be (*
3.14 (* r r)) once we know what number r represents.

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2.2 Variables and Programs
We may write expressions whose operation is area-of-disk
followed by a number:
(area-of-disk 5)
 We also say that we APPLY area-of-disk to 5.
(area-of-disk 5)
= (* 3.14 (* 5 5))
= (* 3.14 25)
= 78.5
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2.2 Variables and Programs
Variable Names
 Unlike most programming languages, program and variable
names in Scheme can contain almost any character.
 For example, the following are legal names:
a-b in->ft my-2nd-program positive? !@$%^&/*~program

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2.2 Variables and Programs
Variable Names
 The name a-b does not mean a minus b; it's just a name. Even
*3.14 is a name. Suppose that you forget the space between *
and 3.14 when defining area-of-disk:
(define (area-of-disk r) (*3.14 (* r r)))
 When you test this definition of area-of-disk, DrRacket reports
that the name *3.14 is undefined.

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2.2 Variables and Programs
Variable Names
 Avoid the following characters in variable names:
" ' ` , # ; | \
 Since quotes cannot be used in names, do not attempt to use x'
as ``x prime.'' The pairs of bracket characters [ ] and { } are
also special: they act just like parenthesis pairs ( ).

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2.2 Variables and Programs
Variable Names
 Names are case-sensitive, i.e., an upper-case letter is treated
different from a lower-case letter in a name.
 For example, the name area-of-disk is different from the name
Area-Of-Disk.

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2.2 Variables and Programs
Variable Names
 Many programs consume more than one input. Say we wish to
define a program that computes the area of a ring.


The area of the ring is that of the outer disk minus the area of
the inner disk, which means that the program requires two
unknown quantities: the outer and the inner radii. Let us call
these unknown numbers outer and inner.
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2.2 Variables and Programs
Variable Names
 Then the program that computes the area of a ring is defined as
follows:
(define (area-of-ring outer inner)
(- (area-of-disk outer) (area-of-disk inner)))

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2.2 Variables and Programs
Variable Names
 That the program accepts two inputs, called outer and inner,
and that the result is going to be the difference between (areaof-disk outer) and (area-of-disk inner).
 When we wish to use area-of-ring, we must supply two inputs:
(area-of-ring 5 3)

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2.2 Variables and Programs

Variable Names
(area-of-ring 5 3)
= (- (area-of-disk 5) (area-of-disk 3))
= (- (* 3.14 (* 5 5)) (* 3.14 (* 3 3)))
= ...
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2.2 Variables and Programs
Exercise 2.2.1
 Define the program dollar->euro, which consumes a number of
dollars and produces the euro equivalent.
 1 dollar is 1.17 euros

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2.2 Variables and Programs
Exercise 2.2.2
 Define the program triangle. It consumes the length of a
triangle's side and the perpendicular height. The program
produces the area of the triangle.
 Formula for computing the area of a triangle is
area of a triangle = 1/2 * base * height

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2.2 Variables and Programs
Exercise 2.2.3
 Define the program convert3. It consumes three digits, starting
with the least significant digit, followed by the next most
significant one, and so on. The program produces the
corresponding number. For example, the expected value of
(convert3 1 2 3)
 is 321. Use an algebra book to find out how such a conversion
works.

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2.2 Variables and Programs
Exercise 2.2.4
 Also formulate the following three expressions as programs:
1. n2 + 10
2. (1/2) · n2 + 20
3. 2 - (1/n)
 Determine their results for n = 2 and n = 9 by hand and with
DrRacket.

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2.3 Errors
When an error occurs, DrRacket highlights the source of
the error, whether it is in the definitions window or the
interactions window.
 For example, typing (/ 1 0) in the interactions window
causes that expression to be highlighted (and also produces
the error message ``/: division by zero'').

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2.3 Errors
For example,
 The first one contains an extra pair of parentheses around
the variable x, which is not a compound expression
(define (P x) (+ (x) 10))
 The second contains two atomic expressions, x and 10,
instead of one.
(define (Q x) x 10)

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2.3 Errors

Exercise 2.3.1 Evaluate the following sentences in
DrRacket, one at a time:
1. (+ (10) 20)
5. (+ 10)
2. (10 + 20)
6. (* 10)
3. (10 20)
7. (- 10)
4. (+ +)
8. (/ 10)
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2.3 Errors

Exercise 2.3.2 Enter the following sentences, one by one,
into DrRacket's Definitions window and click Run:
1. (define (f 1) (+ x 10))
2. (define (g x) + x 10)
3. (define h(x) (+ x 10))
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2.3 Errors

Exercise 2.3.3 Evaluate the following grammatically legal
Scheme expressions in DrRacket's Interactions window:
1. (+ 5 (/ 1 0))
2. (sin 10 20)
3. (somef 10)
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2.3 Errors
Exercise 2.3.4 Enter the following grammatically legal
Scheme program into the Definitions window and click the
Run button:
(define (somef x) (sin x x))
 Then, in the Interactions window, evaluate the expressions:
(somef 10 20)
(somef 10)

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2.4 Designing Programs
DrRacket ignores anything between the first semi-colon in a
line and the end of the line.
;; program : input1 input2 ... -> output
 DrRacket ignores block comments starting with #| and ending
with |#
#| This is a
multi-line
block comment. |#

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2.4 Designing Programs

The design recipe: A complete example
;; Contract: area-of-disk : number -> number
;; Purpose: to compute the area of a disk of
radius r.
;; Example: (area-of-disk 5) should produce 78.5
;; Definition: [refines the header]
(define (area-of-disk r) (* 3.14 (* r r)))
;; Tests:
(area-of-disk 5) "=" 78.5
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Section 3: Programs are Function Plus
Variable Definitions
3.1 Composing Functions
3.2 Variable Definitions
3.3 Finger Exercises on Composing Functions
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Programs are Function Plus Variable
Definitions
In general, a program consists not just of one, but of many
definitions.
 The area-of-ring program, for example, consists of two
definitions: the one for area-of-ring and another one for areaof-disk.
(define (area-of-ring outer inner)
(- (area-of-disk outer)
(area-of-disk inner)))

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Programs are Function Plus Variable
Definitions
We refer to both as FUNCTION DEFINITIONs and, using
mathematical terminology in a loose way, say that the
program is COMPOSED of several functions.
 Because the first one, area-of-ring, is the function we really
wish to use, we refer to it as the MAIN FUNCTION; the
second one, area-of-disk, is an AUXILIARY FUNCTION,
occasionally also called HELPER FUNCTION.

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3.1 Composing Functions

The use of auxiliary functions makes the design process
manageable and renders programs readable. Compare the
following two versions of area-of-ring:
(define (area-of-ring outer inner)
(- (area-of-disk outer)
(area-of-disk inner)))
(define (area-of-ring outer inner)
(- (* 3.14 (* outer outer))
(* 3.14 (* inner inner))))
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3.2 Variable Definitions

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3.2 Variable Definitions
The order for variable definitions is significant.
(define RADIUS 5)
(define DIAMETER (* 2 RADIUS))
is fine, but
(define DIAMETER (* 2 RADIUS))
(define RADIUS 5)
DrRacket does not yet know the definition of RADIUS when it
tries to evaluate (* 2 RADIUS).

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3.3 Finger Exercises on Composing Functions
Exercise 3.3.1 Convert English measurements to metric ones
 Here is a table that shows the six major units of length
measurements of the English system:

English
1 inch
1 foot =
1 yard =
1 rod =
1 furlong =
1 mile =
12 in.
3 ft.
5(1/2)yd.
40 rd.
8 fl.
metric
= 2.54 cm
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3.3 Finger Exercises on Composing Functions
Develop the functions inches->cm, feet->inches, yards->feet,
rods->yards, furlongs->rods, and miles->furlongs.
 Then develop the functions feet->cm, yards->cm, rods>inches, and miles->feet.

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3.3 Finger Exercises on Composing Functions

Exercise 3.3.2 Develop the program volume-cylinder. It
consumes the radius of a cylinder's base disk and its height; it
computes the volume of the cylinder.
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3.3 Finger Exercises on Composing Functions

Exercise 3.3.3 Develop area-cylinder. The program
consumes the radius of the cylinder's base disk and its height.
Its result is the surface area of the cylinder.
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การบ้าน




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Exercise 3.3.1 ชื่อไฟล์ ex3_3_1.rkt
Exercise 3.3.2 ชื่อไฟล์ ex3_3_2.rkt
Exercise 3.3.3 ชื่อไฟล์ ex3_3_3.rkt
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