CS220Programming Principles

프로그래밍의 이해2002 가을학기 Class 6

한 태숙

2

Symbolic Data

A Symbol is a new primitive data type Value vs. Symbol(a b c d)

(23 45 17)

((Norah 12)(Molly 9)(Anna 7))

(* (+ 23 45) (+ x 9))

(define (fact n)

(if (= n 1) 1 (* n (fact (- n 1)))))

3

Quote

We create symbol with “quote”(define x 23)

x ; ==> 23

(quote x) ; ==> x

’x

(quote (a b c))

’(a b c)

(car ’(a b c))

(car (quote (a b c)))

4

Symbol Evaluate to? Print as?(define z ’y)

z

(+ x 3)

(list + x 3)

(list ’+ ’x ’3)

’(list + x 3)

’(2 a)

’(2 (b 3))

(car ’(a b c))

5

EQ? EQV? EQUAL?

(eqv? obj1 obj2) returns #t if obj1 and obj2 are – both #t or both #f– both symbol and (string=? (symbol->string obj1)

(symbol->string obj2)) => #t

– both numbers, numerically equal, and either both exact or both inexact

– both characters and the same character according to char=? procedure

6

EQV?

(eqv? obj1 obj2) returns #t if obj1 and obj2 are – the empty list– pairs, vectors or strings that denote the same locatio

ns in the store– procedures whose location tags are equal

7

Eqv? Returns #f

(eqv? obj1 obj2) returns #f if:– obj1 and obj2 are of different types– one of obj1 and obj2 is #t but the other is #f– obj1 and obj2 are symbols but

(string=? (symbol->string obj1)

(symbol->string obj2) =>#f– one of obj1 and obj2 is an exact number but the othe

r is an inexact number

8

Eqv? Returns #f (continue’d)

(eqv? obj1 obj2) returns #f if: – obj1 and obj2 are numbers(characters) for which the

= (char=?) procedure returns #f– one of obj1 and obj2 is the empty list but the other i

s not– obj1 and obj2 are pairs, vectors, or strings that denot

e distinct locations– obj1 and obj2 are procedures that would behave diff

erently for some arguments

9

Examples of Eqv?

(eqv? ’a ’a) ==> #t

(eqv? ’a ’b) ==> #f

(eqv? 2 2) ==> #t

(eqv? ’() ’()) ==> #t

(eqv? 10000 10000) ==> #t

(eqv? (cons 1 2) (cons 1 2)) ==> #f

(eqv? (lambda() 1) (lambda () 2)) ==> #f

(eqv? #f ’nil) ==> #f

(let ((p (lambda (x) x))) (eqv? p p)) ==> #t

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(eqv? ’(a) ’(a)) ==> unspecified

(eqv? ”a” ”a”) ==> unspecified

(eqv? ’(b) (cdr ’(a b))) ==> unspecified

(let ((x ’(a))) eqv? x x )) ==> #t

11

EQ? - similar to but finer than eqv?

guaranteed to have the same behavior on symbols, booleans, the empty list, pairs, procedures, and non-empty strings and vectors

may be implementation-dependent on numbers and characters and will return true only when eqv? would also return true

may behave differently from eqv? on empty vectors and empty strings

12

Examples on Eq?

(eq? ’a ’a) ==> #t

(eq? ’(a) ’(a)) ==> unspecified

(eq? (list ’a) (list ’a)) ==> #f

(eq? ”a” ”a”) ==> unspecified

(eq? ”” ””) ==> unspecified

(eq? ’() ’()) ==> #t

(eq? 2 2) ==> unspecified

(eq? #\A #\A) ==> unspecified

13

(eq? car car) ==> #t

(let ((n (+ 2 3))) (eq? n n)) ==> unspecified

(let ((x ’(a))) (eq? x x)) ==> #t

(let ((x ’#())) (eq? x x)) ==> #t

(let ((p (lambda (x) x))) (eq? p p)) ==> #t

14

EQUAL?

Library procedure recursively compare the contents of pairs, vecto

rs, and strings, applying eqv? on other objects such as symbols and numbers.

Objects are generally equal? if the print the same.

15

Examples on Equal?

(equal? ’a ’a) ==> #t

(equal? ’(a) ’(a)) ==> #t

(equal? ’(a (b) c) ’(a (b) c)) ==> #t

(equal? ”abc” ”abc”) ==> #t

(equal? 2 2) ==> #t

(equal? (make-vector 5 ’a)

(make-vector 5 ’a)) ==> #t

(equal? (lambda (x) x) (lambda (y) y))

==> unspecified

16

Memq - with eq?

(symbol , list ) -> #f if none is the same in list

sublist with symbol otherwise(define (memq item x)(cond ((null? x) #f)

((eq? item (car x)) x)

(else (memq item (cdr x)))))

17

Exercise 2.53

(list ’a ’b ’c)

(list (list ’george))

(cdr ’((x1 x2) (y1 y2)))

(cadr ’((x1 x2) (y1 y2)))

(pair? (cadr ’(a short list)))

(memq ’red ’((red shoes) (blue socks)))

(memq ’red ’(red shoes blue socks))

(car ’ ’abracadabra)

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Exercise 2.54

Implement equal?(equal? ’(this is a list) ’(this is a list))

==> #t

(equal? ’(this is a list) ’(this (is a) list))

==> #f

Plan: both symbols and eq? symbols

both lists and equal? (car a) (car b)

and equal? (cdr a) (cdr b)

19

Differentiation

Numerical Computation(define (numerical-derivative f)

(define epsilon 0.0001)

(lambda (x)

(/ (- (f (+ x epsilon))

(f x))

epsilon)))

20

Symbolic Differentiation

(define (deriv exp var)

(cond ((constant? exp)(make-constant 0))

((variable? exp)

(if (same-variable? exp var)

(make-constant 1)

(make-constant 0)))

((sum? exp)

(make-sum (deriv (addend exp) var)

(deriv (augend exp) var)))

21

((product? exp)

(make-sum

(make-product

(multiplier exp)

(deriv (multiplicand exp) var))

(make-product

(multiplicand exp)

(deriv (multiplier exp) var))))

(else (error ”unknown expression type”

exp))))

22

Math Expression Abstraction(I)

(make-constant <c>) Construct constant <c>

(constant? <e>) Is <e> a constant?

(make-variable <v>) Construct a variable <v>

(variable? <e>) Is <e> a variable?

(same-variable <v1> <v2>)

Are <v1> and <v2> same?

23

Math Expression Abstraction(II)

(make-sum <addend> <augend>)

Construct sum

(sum? <e>) Is <e> a sum?

(addend <e>) Addend of sum <e>

(augend <e>) Augend of sum <e>

(make-product <multiplier> <multiplicand>)

(product? <e>) Is <e> a product?

(multiplier <e>) multiplier of product <e>

(multiplicand <e>) multiplicand of product<e>

24

Math Expression Implementation(I)

Represent (ax+b) with prefix notation such as (+ (* a x) b)

(define (make-constant x) x)

(define (constant? x) (number? x))

25

Math Expression Implementation(II)

(define (make-variable x) x)

(define (variable? x) (symbol? x))

(define (same-variable? v1 v2)

(and (variable? v1)

(variable? v2)

(eq? v1 v2)))

26

Math Expression Implementation(III)

(define (make-sum a1 a2) (list ’+ a1 a2))(define (sum? x) (and (pair? x) (eq? (car x) ’+)))(define (addend s) (cadr s)(define (augend s) (caddr s)(define (make-product m1 m2) (list ’* m1 m2))(define (product? x) (and (pair? x) (eq? (car x) ’*)))(define (multiplier m) (cadr m)(define (multiplicand m) (caddr m)

27

Reducing Math Exp Implementation

(define (make-sum a1 a2)

(cond ((and (constant? a1) (constant? a2))

(make-constant (+ a1 a2)))

((constant? a1)

(if (= a1 0) a2 (list ’+ a1 a2)))

((constant? a2)

(if (= a2 0) a1 (list ’+ a1 a2)))

(else (list ’+ a1 a2))))

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(define (make-product m1 m2)

(cond ((and (constant? m1) (constant? m2))

(make-constant (* m1 m2)))

((constant? m1)

(cond ((= m1 0) (make-constant 0))

((= m1 1) m2)

(else (list ’* m1 m2))))

((constant? m2)

(cond ((= m2 0) (make-constant 0))

((= m2 1) m1)

(else (list ’* m1 m2))))

(else (list ’* m1 m2))))

29

Adding Exponential Expression

(define (deriv exp var)

(cond

……….

((exponential? exp)

(make-product

(make-product (exponent exp)

(make-exponential

(base exp)

(- (exponent exp) 1)))

(deriv (base exp) var)))))

30

(define (make-exponential b e)

(cond ((= e 0) (make-constant 1))

((= e 1) b)

(else (list ’** b e))))

(define exponential? exp)

(and (pair? exp) (eq? (car exp) ’**)))

(define (base exp) (cadr exp))

(define (exponent exp) (caddr exp))

31

Dotted Tail Notation

Exercise 2.20(define (f x . y) <body> )

(f 1 2 3 4)

in <body> x bound to 1

y bound to (2 3 4)

(define (same-parity x . y)

(same-parity 1 2 3 4 5 6 7)

x==> 1 y==>(2 3 4 5 6 7)

32

Implementation(II) of Math Exp

Using Variable # Terms -> (+ a b c)(define (make-sum a1 . a2)

(cons ’+ (cons a1 a2)))

(define (augend s)

(if (null? (cdddr s)) (caddr s)

(cons ’+ (cddr s))))

(define (multiplicand p)

(if (null? (cdddr p)) (caddr p)

(cons ’* (cddr p))))