問題 2.79
equ? を実装せよ。
(define *table* '())
(define (get op type)
(let ((item (filter (lambda (x)
(and (eq? (car x) op)
(equal? (cadr x) type)))
*table*)))
(if (null? item)
#f
(caddr (car item)))))
(define (put op type item)
(set! *table*
(cons (list op type item)
(filter (lambda (x)
(not (and (eq? (car x) op)
(equal? (cadr x) type))))
*table*))))
;;;;;
(define (attach-tag type-tag contents)
(if (eq? type-tag 'scheme-number)
contents
(cons type-tag contents)))
(define (type-tag datum)
(cond ((number? datum) 'scheme-number)
((pair? datum) (car datum))
(else
(error "Bad tagged datum -- TYPE-TAG" datum))))
(define (contents datum)
(cond ((number? datum) datum)
((pair? datum) (cdr datum))
(else
(error "Bad tagged datum -- CONTENTS" datum))))
(define (apply-generic op . args)
(let ((type-tags (map type-tag args)))
(let ((proc (get op type-tags)))
(print ";APPLY-GENERIC op=" op " args=" args)
(if proc
(apply proc (map contents args))
(error
"No method for these types -- APPLY-GENERIC"
(list op type-tags))))))
;;;;;
(define (install-scheme-number-package)
;; システムの他の部分へのインターフェース
(define (tag x)
(attach-tag 'scheme-number x))
(put 'add '(scheme-number scheme-number)
(lambda (x y) (tag (+ x y))))
(put 'sub '(scheme-number scheme-number)
(lambda (x y) (tag (- x y))))
(put 'mul '(scheme-number scheme-number)
(lambda (x y) (tag (* x y))))
(put 'div '(scheme-number scheme-number)
(lambda (x y) (tag (/ x y))))
(put 'equ? '(scheme-number scheme-number)
(lambda (x y) (tag (= x y))))
(put 'make 'scheme-number
(lambda (x) (tag x)))
'done)
(define (install-rational-package)
;; 内部手続き
(define (numer x) (car x))
(define (denom x) (cdr x))
(define (make-rat n d)
(let ((g (gcd n d)))
(cons (/ n g) (/ d g))))
(define (add-rat x y)
(make-rat (+ (* (numer x) (denom y))
(* (numer y) (denom x)))
(* (denom x) (denom y))))
(define (sub-rat x y)
(make-rat (- (* (numer x) (denom y))
(* (numer y) (denom x)))
(* (denom x) (denom y))))
(define (mul-rat x y)
(make-rat (* (numer x) (numer y))
(* (denom x) (denom y))))
(define (div-rat x y)
(make-rat (* (numer x) (denom y))
(* (denom x) (numer y))))
(define (equ? x y)
(and (= (numer x) (numer y))
(= (denom x) (denom y))))
;; システムの他の部分へのインターフェース
(define (tag x) (attach-tag 'rational x))
(put 'add '(rational rational)
(lambda (x y) (tag (add-rat x y))))
(put 'sub '(rational rational)
(lambda (x y) (tag (sub-rat x y))))
(put 'mul '(rational rational)
(lambda (x y) (tag (mul-rat x y))))
(put 'div '(rational rational)
(lambda (x y) (tag (div-rat x y))))
(put 'equ? '(rational rational)
(lambda (x y) (equ? x y)))
(put 'make 'rational
(lambda (n d) (tag (make-rat n d))))
'done)
(define (install-rectangular-package)
;; 内部手続き
(define (real-part z) (car z))
(define (imag-part z) (cdr z))
(define (make-from-real-imag x y) (cons x y))
(define (magnitude z)
(sqrt (+ (square (real-part z))
(square (imag-part z)))))
(define (angle z)
(atan (imag-part z) (real-part z)))
(define (equ? z1 z2)
(and (= (real-part z1) (real-part z2))
(= (imag-part z1) (imag-part z2))))
(define (make-from-mag-ang r a)
(cons (* r (cos a)) (* r (sin a))))
;; システムの他の部分とのインターフェース
(define (tag x) (attach-tag 'rectangular x))
(put 'real-part '(rectangular) real-part)
(put 'imag-part '(rectangular) imag-part)
(put 'magnitude '(rectangular) magnitude)
(put 'angle '(rectangular) angle)
(put 'equ? '(rectangular rectangular)
(lambda (z1 z2) (equ? z1 z2)))
(put 'make-from-real-imag 'rectangular
(lambda (x y) (tag (make-from-real-imag x y))))
(put 'make-from-mag-ang 'rectangular
(lambda (r a) (tag (make-from-mag-ang r a))))
'done)
(define (install-polar-package)
;; 内部手続き
(define (magnitude z) (car z))
(define (angle z) (cdr z))
(define (make-from-mag-ang r a) (cons r a))
(define (real-part z)
(* (magnitude z) (cos (angle z))))
(define (imag-part z)
(* (magnitude z) (sin (angle z))))
(define (equ? z1 z2)
(and (= (magnitude z1) (magnitude z2))
(= (angle z1) (angle z2))))
(define (make-from-real-imag x y)
(cons (sqrt (+ (square x) (square y)))
(atan y x)))
;; システムの他の部分とのインターフェース
(define (tag x) (attach-tag 'polar x))
(put 'real-part '(polar) real-part)
(put 'imag-part '(polar) imag-part)
(put 'magnitude '(polar) magnitude)
(put 'angle '(polar) angle)
(put 'equ? '(polar polar)
(lambda (z1 z2) (equ? z1 z2)))
(put 'make-from-real-imag 'polar
(lambda (x y) (tag (make-from-real-imag x y))))
(put 'make-from-mag-ang 'polar
(lambda (r a) (tag (make-from-mag-ang r a))))
'done)
(define (install-complex-package)
;; 直交座標と極座標パッケージから取り入れた手続き
(define (make-from-real-imag x y)
((get 'make-from-real-imag 'rectangular) x y))
(define (make-from-mag-ang r a)
((get 'make-from-mag-ang 'polar) r a))
;; 内部手続き
(define (add-complex z1 z2)
(make-from-real-imag (+ (real-part z1) (real-part z2))
(+ (imag-part z1) (imag-part z2))))
(define (sub-complex z1 z2)
(make-from-real-imag (- (real-part z1) (real-part z2))
(- (imag-part z1) (imag-part z2))))
(define (mul-complex z1 z2)
(make-from-mag-ang (* (magnitude z1) (magnitude z2))
(+ (angle z1) (angle z2))))
(define (div-complex z1 z2)
(make-from-mag-ang (/ (magnitude z1) (magnitude z2))
(- (angle z1) (angle z2))))
;; システムの他の部分へのインターフェース
(define (tag z) (attach-tag 'complex z))
(put 'add '(complex complex)
(lambda (z1 z2) (tag (add-complex z1 z2))))
(put 'sub '(complex complex)
(lambda (z1 z2) (tag (sub-complex z1 z2))))
(put 'mul '(complex complex)
(lambda (z1 z2) (tag (mul-complex z1 z2))))
(put 'div '(complex complex)
(lambda (z1 z2) (tag (div-complex z1 z2))))
(put 'make-from-real-imag 'complex
(lambda (x y) (tag (make-from-real-imag x y))))
(put 'make-from-mag-ang 'complex
(lambda (r a) (tag (make-from-mag-ang r a))))
(put 'real-part '(complex) real-part)
(put 'imag-part '(complex) imag-part)
(put 'magnitude '(complex) magnitude)
(put 'angle '(complex) angle)
(put 'equ? '(complex complex) equ?)
(put 'equ? '(polar rectangular)
(lambda (pol rec)
(equ? (cons 'polar pol)
(make-from-mag-ang (magnitude (cons 'rectangular rec))
(angle (cons 'rectangular rec))))))
(put 'equ? '(rectangular polar)
(lambda (rec pol)
(equ? (cons 'polar pol) (cons 'rectangular rec))))
'done)
;;;;;
(define (add x y) (apply-generic 'add x y))
(define (sub x y) (apply-generic 'sub x y))
(define (mul x y) (apply-generic 'mul x y))
(define (div x y) (apply-generic 'div x y))
(define (real-part z) (apply-generic 'real-part z))
(define (imag-part z) (apply-generic 'imag-part z))
(define (magnitude z) (apply-generic 'magnitude z))
(define (angle z) (apply-generic 'angle z))
;;;;;
(define (make-scheme-number n)
((get 'make 'scheme-number) n))
(define (make-rational n d)
((get 'make 'rational) n d))
(define (make-complex-from-real-imag x y)
((get 'make-from-real-imag 'complex) x y))
(define (make-complex-from-mag-ang r a)
((get 'make-from-mag-ang 'complex) r a))
(define (equ? x y) (apply-generic 'equ? x y))
;;;;;
(install-scheme-number-package)
(install-rational-package)
(install-rectangular-package)
(install-polar-package)
(install-complex-package)型が異なる場合に対応していませんが、
Complex に関してだけ Rectangular と Polar に対応しています。
Complex パッケージに細工を施していますが、実に微妙な実装です。
また、すべての型の組み合わせを実装するのは大変ですし、
将来型が増えたら目も当てられません。ダメダメです。
そのためのデータ主導プログラミングのはずだったのに、
どうしてこうなった！orz
実は、異なる型同士の演算方法については次節で解決方法を習います。
今はサスマン先生の手のひらの上で転がされておきましょうｗ
polar を rectangular に寄せると誤差が出るため、rectangular を polar に寄せることにした。 (※ polar の値は rectangular を元に作っており、すでに誤差が出ているので、たまたま問題化しないだけである) 真パターン (equ? (make-complex-from-real-imag 1 2) (make-complex-from-real-imag 1 2)) (equ? (make-complex-from-mag-ang 2.23606797749979 1.1071487177940904) (make-complex-from-mag-ang 2.23606797749979 1.1071487177940904)) (equ? (make-complex-from-real-imag 1 2) (make-complex-from-mag-ang 2.23606797749979 1.1071487177940904)) (equ? (make-complex-from-mag-ang 2.23606797749979 1.1071487177940904) (make-complex-from-real-imag 1 2)) 偽パターン (equ? (make-complex-from-real-imag 1 2) (make-complex-from-real-imag 3 4)) (equ? (make-complex-from-mag-ang 1.1 2.2) (make-complex-from-mag-ang 3.3 4.4)) (equ? (make-complex-from-real-imag 1 2) (make-complex-from-mag-ang 3.3 4.4)) (equ? (make-complex-from-mag-ang 3.3 4.4) (make-complex-from-real-imag 1 2))
問題 2.80
=zero? を実装せよ。
(define *table* '())
(define (get op type)
(let ((item (filter (lambda (x)
(and (eq? (car x) op)
(equal? (cadr x) type)))
*table*)))
(if (null? item)
#f
(caddr (car item)))))
(define (put op type item)
(set! *table*
(cons (list op type item)
(filter (lambda (x)
(not (and (eq? (car x) op)
(equal? (cadr x) type))))
*table*))))
;;;;;
(define (attach-tag type-tag contents)
(if (eq? type-tag 'scheme-number)
contents
(cons type-tag contents)))
(define (type-tag datum)
(cond ((number? datum) 'scheme-number)
((pair? datum) (car datum))
(else
(error "Bad tagged datum -- TYPE-TAG" datum))))
(define (contents datum)
(cond ((number? datum) datum)
((pair? datum) (cdr datum))
(else
(error "Bad tagged datum -- CONTENTS" datum))))
(define (apply-generic op . args)
(let ((type-tags (map type-tag args)))
(let ((proc (get op type-tags)))
(print ";APPLY-GENERIC op=" op " args=" args)
(if proc
(apply proc (map contents args))
(error
"No method for these types -- APPLY-GENERIC"
(list op type-tags))))))
;;;;;
(define (install-scheme-number-package)
;; システムの他の部分へのインターフェース
(define (tag x)
(attach-tag 'scheme-number x))
(put 'add '(scheme-number scheme-number)
(lambda (x y) (tag (+ x y))))
(put 'sub '(scheme-number scheme-number)
(lambda (x y) (tag (- x y))))
(put 'mul '(scheme-number scheme-number)
(lambda (x y) (tag (* x y))))
(put 'div '(scheme-number scheme-number)
(lambda (x y) (tag (/ x y))))
(put 'equ? '(scheme-number scheme-number)
(lambda (x y) (tag (= x y))))
(put '=zero? '(scheme-number)
(lambda (x) (zero? x)))
(put 'make 'scheme-number
(lambda (x) (tag x)))
'done)
(define (install-rational-package)
;; 内部手続き
(define (numer x) (car x))
(define (denom x) (cdr x))
(define (make-rat n d)
(if (= d 0)
(error "zero denominator -- MAKE-RAT")
(let ((g (gcd n d)))
(cons (/ n g) (/ d g)))))
(define (add-rat x y)
(make-rat (+ (* (numer x) (denom y))
(* (numer y) (denom x)))
(* (denom x) (denom y))))
(define (sub-rat x y)
(make-rat (- (* (numer x) (denom y))
(* (numer y) (denom x)))
(* (denom x) (denom y))))
(define (mul-rat x y)
(make-rat (* (numer x) (numer y))
(* (denom x) (denom y))))
(define (div-rat x y)
(make-rat (* (numer x) (denom y))
(* (denom x) (numer y))))
(define (equ? x y)
(and (= (numer x) (numer y))
(= (denom x) (denom y))))
(define (=zero? x) (zero? (numer x)))
;; システムの他の部分へのインターフェース
(define (tag x) (attach-tag 'rational x))
(put 'add '(rational rational)
(lambda (x y) (tag (add-rat x y))))
(put 'sub '(rational rational)
(lambda (x y) (tag (sub-rat x y))))
(put 'mul '(rational rational)
(lambda (x y) (tag (mul-rat x y))))
(put 'div '(rational rational)
(lambda (x y) (tag (div-rat x y))))
(put 'equ? '(rational rational)
(lambda (x y) (equ? x y)))
(put '=zero? '(rational)
(lambda (x) (=zero? x)))
(put 'make 'rational
(lambda (n d) (tag (make-rat n d))))
'done)
(define (install-rectangular-package)
;; 内部手続き
(define (real-part z) (car z))
(define (imag-part z) (cdr z))
(define (make-from-real-imag x y) (cons x y))
(define (magnitude z)
(sqrt (+ (square (real-part z))
(square (imag-part z)))))
(define (angle z)
(atan (imag-part z) (real-part z)))
(define (equ? z1 z2)
(and (= (real-part z1) (real-part z2))
(= (imag-part z1) (imag-part z2))))
(define (=zero? z)
(and (zero? (real-part z)) (zero? (imag-part z))))
(define (make-from-mag-ang r a)
(cons (* r (cos a)) (* r (sin a))))
;; システムの他の部分とのインターフェース
(define (tag x) (attach-tag 'rectangular x))
(put 'real-part '(rectangular) real-part)
(put 'imag-part '(rectangular) imag-part)
(put 'magnitude '(rectangular) magnitude)
(put 'angle '(rectangular) angle)
(put 'equ? '(rectangular rectangular)
(lambda (z1 z2) (equ? z1 z2)))
(put '=zero? '(rectangular)
(lambda (z) (=zero? z)))
(put 'make-from-real-imag 'rectangular
(lambda (x y) (tag (make-from-real-imag x y))))
(put 'make-from-mag-ang 'rectangular
(lambda (r a) (tag (make-from-mag-ang r a))))
'done)
(define (install-polar-package)
;; 内部手続き
(define (magnitude z) (car z))
(define (angle z) (cdr z))
(define (make-from-mag-ang r a) (cons r a))
(define (real-part z)
(* (magnitude z) (cos (angle z))))
(define (imag-part z)
(* (magnitude z) (sin (angle z))))
(define (equ? z1 z2)
(and (= (magnitude z1) (magnitude z2))
(= (angle z1) (angle z2))))
(define (=zero? z) (zero? (magnitude z)))
(define (make-from-real-imag x y)
(cons (sqrt (+ (square x) (square y)))
(atan y x)))
;; システムの他の部分とのインターフェース
(define (tag x) (attach-tag 'polar x))
(put 'real-part '(polar) real-part)
(put 'imag-part '(polar) imag-part)
(put 'magnitude '(polar) magnitude)
(put 'angle '(polar) angle)
(put 'equ? '(polar polar)
(lambda (z1 z2) (equ? z1 z2)))
(put '=zero? '(polar)
(lambda (z) (=zero? z)))
(put 'make-from-real-imag 'polar
(lambda (x y) (tag (make-from-real-imag x y))))
(put 'make-from-mag-ang 'polar
(lambda (r a) (tag (make-from-mag-ang r a))))
'done)
(define (install-complex-package)
;; 直交座標と極座標パッケージから取り入れた手続き
(define (make-from-real-imag x y)
((get 'make-from-real-imag 'rectangular) x y))
(define (make-from-mag-ang r a)
((get 'make-from-mag-ang 'polar) r a))
;; 内部手続き
(define (add-complex z1 z2)
(make-from-real-imag (+ (real-part z1) (real-part z2))
(+ (imag-part z1) (imag-part z2))))
(define (sub-complex z1 z2)
(make-from-real-imag (- (real-part z1) (real-part z2))
(- (imag-part z1) (imag-part z2))))
(define (mul-complex z1 z2)
(make-from-mag-ang (* (magnitude z1) (magnitude z2))
(+ (angle z1) (angle z2))))
(define (div-complex z1 z2)
(make-from-mag-ang (/ (magnitude z1) (magnitude z2))
(- (angle z1) (angle z2))))
;; システムの他の部分へのインターフェース
(define (tag z) (attach-tag 'complex z))
(put 'add '(complex complex)
(lambda (z1 z2) (tag (add-complex z1 z2))))
(put 'sub '(complex complex)
(lambda (z1 z2) (tag (sub-complex z1 z2))))
(put 'mul '(complex complex)
(lambda (z1 z2) (tag (mul-complex z1 z2))))
(put 'div '(complex complex)
(lambda (z1 z2) (tag (div-complex z1 z2))))
(put 'make-from-real-imag 'complex
(lambda (x y) (tag (make-from-real-imag x y))))
(put 'make-from-mag-ang 'complex
(lambda (r a) (tag (make-from-mag-ang r a))))
(put 'real-part '(complex) real-part)
(put 'imag-part '(complex) imag-part)
(put 'magnitude '(complex) magnitude)
(put 'angle '(complex) angle)
(put 'equ? '(complex complex) equ?)
(put 'equ? '(polar rectangular)
(lambda (pol rec)
(equ? (cons 'polar pol)
(make-from-mag-ang (magnitude (cons 'rectangular rec))
(angle (cons 'rectangular rec))))))
(put 'equ? '(rectangular polar)
(lambda (rec pol)
(equ? (cons 'polar pol) (cons 'rectangular rec))))
(put '=zero? '(complex) =zero?)
'done)
;;;;;
(define (add x y) (apply-generic 'add x y))
(define (sub x y) (apply-generic 'sub x y))
(define (mul x y) (apply-generic 'mul x y))
(define (div x y) (apply-generic 'div x y))
(define (real-part z) (apply-generic 'real-part z))
(define (imag-part z) (apply-generic 'imag-part z))
(define (magnitude z) (apply-generic 'magnitude z))
(define (angle z) (apply-generic 'angle z))
;;;;;
(define (make-scheme-number n)
((get 'make 'scheme-number) n))
(define (make-rational n d)
((get 'make 'rational) n d))
(define (make-complex-from-real-imag x y)
((get 'make-from-real-imag 'complex) x y))
(define (make-complex-from-mag-ang r a)
((get 'make-from-mag-ang 'complex) r a))
(define (equ? x y) (apply-generic 'equ? x y))
(define (=zero? x) (apply-generic '=zero? x))
;;;;;
(install-scheme-number-package)
(install-rational-package)
(install-rectangular-package)
(install-polar-package)
(install-complex-package)こちらは equ? と違い引数が 1 つなので難しい事はありません。
ただし、各型のゼロ判定については多少の数学の知識が必要です。
各型は以下のように判定します。
数値 : zero? で判定。 rational : numer が 0 なら 0。denom が 0 はありえない(make-rat 時にエラー)。 rectangular : real-part が 0 かつ imag-part が 0 なら 0。 polar : magnitude が 0 なら 0。magnitude が 0 の場合 angle は意味を為さない。 complex : rectangular もしくは polar に判定を任せる。
