Scheme - 記号データ(例: 記号微分(前置演算子、中置き、加算と乗算(括弧の省略)))

計算機プログラムの構造と解釈
Gerald Jay Sussman, Julie Sussman
Harold Abelson (原著) 和田 英一 (翻訳)

開発環境

• OS X Lion - Apple(OS)
• Emacs、BBEdit - Bare Bones Software, Inc. (Text Editor)
• プログラミング言語: MIT/GNU Scheme

計算機プログラムの構造と解釈(Gerald Jay Sussman(原著)、Julie Sussman(原著)、Harold Abelson(原著)、和田 英一(翻訳)、ピアソンエデュケーション)の2(データによる抽象の構築)、2.3(記号データ)、2.3.2(例: 記号微分)の問題 2.58、a、bを解いてみる。

その他参考書籍

• Instructor's Manual to Accompany Structure & Interpretation of Computer Programs

問題 2.58

a.

コード

sample.scm

(define (deriv exp var)
  (cond ((number? exp) 0)
        ((variable? exp)
         (if (same-variable? exp var) 1 0))
        ((sum? exp)
         (make-sum (deriv (addend exp) var)
                   (deriv (augend exp) var)))
        ((product? exp)
         (make-sum  (make-product (multiplier exp)
                                  (deriv (multiplicand exp) var))
                    (make-product (deriv (multiplier exp) var)
                                  (multiplicand exp))))
        ((exponentiation? exp)
           (let ((n (exponent exp))
                 (u (base exp)))
             (make-product n
                           (make-product (make-exponentiation u
                                                              (make-sum n -1))
                                         (deriv u var)))))
        (else (error "unkown expression type -- DERIV" exp))))

(define (make-sum a1 a2)
  (cond ((=number? a1 0) a2)
        ((=number? a2 0) a1)
        ((and (number? a1) (number? a2)) (+ a1 a2))
        (else (list a1 '+ a2))))

(define (addend s) (car s))

(define (augend s) (caddr s))

(define (make-product m1 m2) 
  (cond ((or (=number? m1 0) (=number? m2 0)) 0)
        ((=number? m1 1) m2)
        ((=number? m2 1) m1)
        ((and (number? m1) (number? m2)) (* m1 m2))
        (else (list m1 '* m2))))

(define (multiplier p) (car p))

(define (multiplicand p) (caddr p))

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

(define (same-variable? v1 v2)
  (and (variable? v1) (variable? v2) (eq? v1 v2)))

(define (sum? x)
  (and (pair? x) (eq? (cadr x) '+)))

(define (product? x)
  (and (pair? x) (eq? (cadr x) '*)))

(define (exponentiation? x)
  (and (pair? x) (eq? (cadr x) '**)))

(define (base x) (car x))

(define (exponent x) (caddr x))

(define (make-exponentiation a b)
  (cond ((=number? b 0) 1)
        ((=number? b 1) a)
        (else (list a '** b))))

(define (=number? exp num)
  (and (number? exp) (= exp num)))

(define a '(x + (3 * (x + (y + 2)))))

入出力結果(Terminal, REPL(Read, Eval, Print, Loop))

1 ]=> (deriv a 'x)

;Value: 4

b.

コード

sample.scm

(define (deriv exp var)
  (cond ((number? exp) 0)
        ((variable? exp)
         (if (same-variable? exp var) 1 0))
        ((sum? exp)
         (make-sum (deriv (addend exp) var)
                   (deriv (augend exp) var)))
        ((product? exp)
         (make-sum  (make-product (multiplier exp)
                                  (deriv (multiplicand exp) var))
                    (make-product (deriv (multiplier exp) var)
                                  (multiplicand exp))))
        ((exponentiation? exp)
           (let ((n (exponent exp))
                 (u (base exp)))
             (make-product n
                           (make-product (make-exponentiation u
                                                              (make-sum n -1))
                                         (deriv u var)))))
        (else (error "unkown expression type -- DERIV" exp))))

(define (make-sum a1 a2)
  (cond ((=number? a1 0) a2)
        ((=number? a2 0) a1)
        ((and (number? a1) (number? a2)) (+ a1 a2))
        ; 追加
        ((eq? a1 a2) (list 2 '* a1))
        (else (list a1 '+ a2))))

; 修正箇所
(define (addend s)
  (define (iter result s)
    (if (eq? (cadr s) '+)
        (if (null? result)
            (car s)
            (append result (list (car s))))
          (iter (append result (list (car s))) (cdr s))))
  (iter '() s))

; 修正箇所
(define (augend s)
  (if (eq? (cadr s) '+)
      (let ((a (cddr s)))
        (if (and (pair? a) (null? (cdr a)))
            (car a)
            a))
      (augend (cddr s))))

(define (make-product m1 m2) 
  (cond ((or (=number? m1 0) (=number? m2 0)) 0)
        ((=number? m1 1) m2)
        ((=number? m2 1) m1)
        ((and (number? m1) (number? m2)) (* m1 m2))
        (else (list m1 '* m2))))

(define (multiplier p) (car p))

; 修正箇所
(define (multiplicand p)
  (let ((a (cddr p)))
    (if (and (pair? a) (null? (cdr a)))
        (car a)
        a)))

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

(define (same-variable? v1 v2)
  (and (variable? v1) (variable? v2) (eq? v1 v2)))

;(define (sum? x)
;  (and (pair? x) (eq? (cadr x) '+)))
(define (sum? x) (memq '+ x))

(define (product? x) (not (memq '+ x)))

(define (exponentiation? x)
  (and (pair? x) (eq? (cadr x) '**)))

(define (base x) (car x))

(define (exponent x) (caddr x))

(define (make-exponentiation a b)
  (cond ((=number? b 0) 1)
        ((=number? b 1) a)
        (else (list a '** b))))

(define (=number? exp num)
  (and (number? exp) (= exp num)))

(define a '(x + (3 * (x + (y + 2)))))

入出力結果(Terminal, REPL(Read, Eval, Print, Loop))

 ]=> (deriv '(x + x * x) 'x)

;Value 2: (1 + (2 * x))

1 ]=> (deriv '(x + (x * x)) 'x)

;Value 3: (1 + (2 * x))

1 ]=> (deriv '((x + x) * x) 'x)

;Value 4: ((x + x) + (2 * x))

1 ]=> (deriv '(2 * x) 'x)

;Value: 2

1 ]=> (deriv '(2 * x * x) 'x)

;Value 5: (2 * (2 * x))

1 ]=> (deriv '(2 * x * x * x) 'x)

;Value 6: (2 * ((x * (2 * x)) + (x * x)))

1 ]=> (deriv '(2 * x * x * x * x) 'x)

;Value 7: (2 * ((x * ((x * (2 * x)) + (x * x))) + (x * x * x)))

ということで、不要な括弧は省き、乗算は加算より前に行う、微分プログラムが動作するような、適切な述語、選択子、構成子は設計できる。