SICP 問題 2.77（complex型の為の選択子が必要っす。）

問題

louis Reasonerは、zを図2.24に示すオブジェクトとして、式（magnitude z）を評価しようとした。
驚いたことに答えの5の代わりに、型（complex）には演算magunitudeは対応する手続きは無いという。apply-genericのエラーメッセージを得た。この対話をAlyssa P.Hackerに見せると彼女は「complex数に対して複素数の選択子は定義されておらず、polerとrectangularだけ定義されているのが問題である。これが働くようにするには、complexパッケージに次のものを追加しなければならない。」という。

(put 'real-part '(complex) real-part)

(put 'imag-part '(complex) imag-part)

(put 'magnitude '(complex) magnitude)

(put 'angle '(complex) angle)

どうしてこれが働くか詳しく述べよ。例えばzが図2.24に示すオブジェクトとして、式(magnitude z)を評価する時、呼び出される全ての手続きをトレースせよ。得にapply-genericは何回呼び出されるか。それぞれの場合どの手続きが振り分けられるか。

解答

追加する前の状態での動作を確認しよう。
おっとその前に準備をちゃんとせねば。
このエントリでまとめたハッシュテーブルを使ったgetとputを使えるようにgoshに読み込ませる。あ、ここで読み込ませるのは追記した、equal?を使ったハッシュテーブルの方なので注意されたし。

更に、前回のエントリでまとめた、「有理数」「複素数（直交座標形式／極座標形式）」の各種演算をまとめたパッケージを全て使えるようにgoshに読み込ませる。こっちは不足している手続きがいっぱいあったので、ここで改めて掲載しよう。

;;;==================================================
;;;汎用手続き
;;;==================================================
(define (apply-generic op . args)
  (let* ((type-tags (map type-tag args))
	 (proc (get op type-tags)))
    (if proc
	(apply proc (map contents args))
	(error
	 "No method for these types -- APPLY-GENERIC"
	 (list op type-tags)))))

(define (attach-tag type-tag contents)
  (cons type-tag contents))

(define (type-tag datum)
  (if (pair? datum)
      (car datum)
      (error "Bad tagged datum -- TYPE-TAG" datum)))

(define (contents datum)
  (if (pair? datum)
      (cdr datum)))

(define (square x)
  (* x x))

(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))

;;;==================================================
;;;scheme型の数字に関連するパッケージ
;;;==================================================
(define (install-scheme-number-package)
  (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 'make 'scheme-number
       (lambda (x) (tag x)))
  'done)
(install-scheme-number-package) ;インストールする

;;;構成子
(define (make-scheme-number n)
  ((get 'make 'scheme-number) n))


;;;==================================================
;;;有理数に関連する演算パッケージ
;;;==================================================
(define (install-rational-package)
  (define (gcd a b)
    (if (= b 0)
	a
	(gcd b (remainder a b))))

  ;private
  (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))))

  ;public
  (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 'make 'rational
       (lambda (n d) (tag (make-rat n d))))
  'done)
(install-rational-package) ;インストールする

;;;構成子
(define (make-rational n d)
  ((get 'make 'rational) n d))


;;;==================================================
;;;直交座標形式のパッケージ
;;;==================================================
(define (install-rectangular-package)
  ;内部手続き
  ;※ install-rectangular-package内で定義されているので、
  ;   次に定義してある install-polar-package 内の同一名称の手続きとは競合が発生しない！
  (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 (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 '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)
(install-rectangular-package) ;インストールする


;;;==================================================
;;;極座標形式のパッケージ
;;;==================================================
(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 (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 '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)
(install-polar-package) ;インストールする

;;;==================================================
;;;統合型の複素数の演算パッケージ
;;;==================================================
(define (install-complex-package)
  (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))

  ;private
  (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 (* (magunitude z1) (magunitude z2))
		       (+ (angle z1) (angle z2))))
  (define (div-complex z1 z2)
    (make-from-mag-ang (/ (magunitude z1) (magunitude z2))
		       (- (angle z1) (angle z2))))

  ;public
  (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))))
  'done)
(install-complex-package) ;インストールする

;;;構成子
(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))

;;;選択子
(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))

では実験。

gosh> (define z (make-complex-from-real-imag 3 4))

z

gosh> z

(complex rectangular 3 . 4)

gosh> (magnitude z)


    
ERROR: No method for these types -- APPLY-GENERIC (magnitude (complex))
    
Stack Trace:

_______________________________________

gosh> 

よ〜しよしよし。じゃ、Alyssa P.Hackerの言う通りにcomplex型に対する選択子がちゃんとインストールされるように、install-complex-packageに、指示された４つの手続きを登録して改めて評価してみよう。
コメントで目立つようにしてあります。

(define (apply-generic op . args)
  (let* ((type-tags (map type-tag args))
	 (proc (get op type-tags)))
    (if proc
	(apply proc (map contents args))
	(error
	 "No method for these types -- APPLY-GENERIC"
	 (list op type-tags)))))

(define (attach-tag type-tag contents)
  (cons type-tag contents))

(define (type-tag datum)
  (if (pair? datum)
      (car datum)
      (error "Bad tagged datum -- TYPE-TAG" datum)))

(define (contents datum)
  (if (pair? datum)
      (cdr datum)))

(define (square x)
  (* x x))

(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 (install-scheme-number-package)
  (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 'make 'scheme-number
       (lambda (x) (tag x)))
  'done)
(install-scheme-number-package)


(define (make-scheme-number n)
  ((get 'make 'scheme-number) n))


(define (install-rational-package)
  (define (gcd a b)
    (if (= b 0)
	a
	(gcd b (remainder a b))))

  (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 (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 'make 'rational
       (lambda (n d) (tag (make-rat n d))))
  'done)
(install-rational-package)

(define (make-rational n d)
  ((get 'make 'rational) n d))


(define (install-rectangular-package)
  (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 (make-from-mag-ang r a)
    (cons (* r (cos a)) (* r (sin a))))

  (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 '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)
(install-rectangular-package)


(define (install-polar-package)
  (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 (make-from-real-imag x y)
    (cons (sqrt (+ (square x) (square y)))
	  (atan y x)))

  (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 '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)
(install-polar-package)

(define (install-complex-package)
  (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))

  (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 (* (magunitude z1) (magunitude z2))
		       (+ (angle z1) (angle z2))))
  (define (div-complex z1 z2)
    (make-from-mag-ang (/ (magunitude z1) (magunitude z2))
		       (- (angle z1) (angle z2))))
  ;;;==============================
  ;;;ここに選択子の定義を追加
  ;;;==============================
  (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 (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))))

  ;;;==============================
  ;;;ここにcomplex用の選択子を追加
  ;;;==============================
  (put 'real-part '(complex) real-part)
  (put 'imag-part '(complex) imag-part)
  (put 'magnitude '(complex) magnitude)
  (put 'angle '(complex) angle)
  'done)
(install-complex-package)


(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 (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))

では改めて実験。

gosh> (magnitude z)

5.0

gosh> 

いいね〜。じゃ、これでapply-genericが何回呼び出されているかチェック。

gosh> (use slib)

#

gosh> (require 'trace)

#t

gosh> (trace magnitude)

#

gosh> (trace apply-generic)

#

gosh> (magnitude z)

CALL magnitude (complex rectangular 3 . 4)

 CALL apply-generic magnitude (complex rectangular 3 . 4)

  CALL apply-generic magnitude (rectangular 3 . 4)

  RETN apply-generic 5.0

 RETN apply-generic 5.0

RETN magnitude 5.0

5.0

gosh> 

２回ですね。