今までは手続きの抽象化、ここからはデータの抽象化。
リンク先のサイトの Figure 2.1: Data-abstraction barriers in the rational-number package が分かりやすい。
Programs that use rational numbers- (Rational numbers in problem domain)
add-rat sub-rat ...- (Rational numbers as numerators and denominators)
make-rat numer denom- (Rational numbers as pairs)
cons car cdr
前準備
(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 (equal-rat? x y)
(= (* (numer x) (denom y))
(* (numer y) (denom x))))
(define (make-rat n d)
(let ((g (gcd n d)))
(cons (/ n g) (/ d g))))
(define (numer x) (car x))
(define (denom x) (cdr x))
(define (print-rat x)
(newline)
(display (numer x))
(display "/")
(display (denom x)))
2.1
(define (make-rat n d)
(define (sign x)
(cond ((positive? x) 1)
((negative? x) -1)
(else 0)))
(let ((g (* (gcd n d) (sign d))))
(cons (/ n g) (/ d g))))
2.2
(define (make-point x y)
(cons x y))
(define (x-point p)
(car p))
(define (y-point p)
(cdr p))
(define (print-point p)
(newline)
(display "(")
(display (x-point p))
(display ",")
(display (y-point p))
(display ")"))
(define (make-segment s e)
(cons s e))
(define (start-segment s)
(car s))
(define (end-segment s)
(cdr s))
(define (midpoint-segment s)
(let ((start (start-segment s))
(end (end-segment s)))
(make-point (/ (+ (x-point start) (x-point end)) 2)
(/ (+ (y-point start) (y-point end)) 2))))
2.3
(define (make-rect-tb top bottom)
(cons top bottom))
(define (top-rect r)
(car r))
(define (bottom-rect r)
(cdr r))
(define (make-rect-lr left right)
(cons left right))
(define (left-rect r)
(car r))
(define (right-rect r)
(cdr r))
(define (length-segment s)
(define (square x) (expt x 2))
(let ((start (start-segment s))
(end (end-segment s)))
(sqrt (+ (square (- (x-point start) (x-point end)))
(square (- (y-point start) (y-point end)))))))
(define (perimeter-rect-lr r)
(* (+ (length-segment (left-rect r))
(length-segment (right-rect r)))
2))
(define (perimeter-rect-tb r)
(* (+ (length-segment (top-rect r))
(length-segment (bottom-rect r)))
2))
(perimeter-rect-lr
(make-rect-lr
(make-segment
(make-point 1 2)
(make-point 4 5))
(make-segment
(make-point 10 15)
(make-point 3 7))))
2.4 (cons)
(define (%cons x y) (lambda (m) (m x y))) (define (%car z) (z (lambda (x y) x))) (define (%cdr z) (z (lambda (x y) y)))
2.5 (cons)
(define (%cons a b)
(* (expt 2 a) (expt 3 b)))
(define (pow-of x y)
(if (zero? (remainder x y))
(+ 1 (pow-of (/ x y) y))
0))
(define (%car c)
(pow-of c 2))
(define (%cdr c)
(pow-of c 3))
2.6 (チャーチ数)
前もやったな。
(define zero (lambda (f) (lambda (x) x))) (define (add-1 n) (lambda (f) (lambda (x) (f ((n f) x))))) ; (add-1 zero) ; => (lambda (f) (lambda (x) (f (((lambda (f) (lambda (x) x)) f) x)))) ; => (lambda (f) (lambda (x) (f ( (lambda (x) x) x)))) ; => (lambda (f) (lambda (x) (f x ))) (define one (lambda (f) (lambda (x) (f x)))) ; (add-1 one) ; => (lambda (f) (lambda (x) (f (((lambda (f) (lambda (x) (f x))) f) x)))) ; => (lambda (f) (lambda (x) (f ( (lambda (x) (f x)) x)))) ; => (lambda (f) (lambda (x) (f (f x) ))) (define two (lambda (f) (lambda (x) (f (f x))))) (define (to-number n) ((n (lambda (x) (+ x 1))) 0)) (define (add n m) (lambda (f) (lambda (x) ((m f) ((n f) x))))) (define (mul n m) (lambda (f) (n (m f))))
2.1.4 Extended Exercise: Interval Arithmetic 前準備
区間演算。
(define (add-interval x y)
(make-interval (+ (lower-bound x) (lower-bound y))
(+ (upper-bound x) (upper-bound y))))
(define (mul-interval x y)
(let ((p1 (* (lower-bound x) (lower-bound y)))
(p2 (* (lower-bound x) (upper-bound y)))
(p3 (* (upper-bound x) (lower-bound y)))
(p4 (* (upper-bound x) (upper-bound y))))
(make-interval (min p1 p2 p3 p4)
(max p1 p2 p3 p4))))
(define (div-interval x y)
(mul-interval x
(make-interval (/ 1.0 (upper-bound y))
(/ 1.0 (lower-bound y)))))
2.7
(define (make-interval l u) (cons l u)) (define (lower-bound i) (car i)) (define (upper-bound i) (cdr i))
2.8
(define (sub-interval x y)
(make-interval (- (lower-bound x) (lower-bound y))
(- (upper-bound x) (upper-bound y))))
2.9
(define (width-interval i) (/ (- (upper-bound i) (lower-bound i)) 2))
2.10
(define (div-interval x y)
(if (and (not (positive? (lower-bound y)))
(not (negative? (upper-bound y))))
(error "div-interval: y spans zero")
(mul-interval x
(make-interval (/ 1.0 (upper-bound y))
(/ 1.0 (lower-bound y))))))
2.11
(define (mul-interval x y)
(if (and (not (positive? (lower-bound x)))
(not (negative? (upper-bound x)))
(not (positive? (lower-bound y)))
(not (negative? (upper-bound y))))
(let ((p1 (* (lower-bound x) (lower-bound y)))
(p2 (* (lower-bound x) (upper-bound y)))
(p3 (* (upper-bound x) (lower-bound y)))
(p4 (* (upper-bound x) (upper-bound y))))
(make-interval (min p1 p2 p3 p4)
(max p1 p2 p3 p4)))
(let ((p1 (* (lower-bound x) (lower-bound y)))
(p2 (* (upper-bound x) (upper-bound y))))
(make-interval (min p1 p2)
(max p1 p2)))
))自信なし。
2.12
(define (make-center-width c w) (make-interval (- c w) (+ c w))) (define (center i) (/ (+ (lower-bound i) (upper-bound i)) 2)) (define (width i) (/ (- (upper-bound i) (lower-bound i)) 2))
(define (make-center-percent c p) (make-center-width c (* c p 0.01))) (define (percent i) (* (/ (width i) (center i)) 100))
2.13 - 2.16
桁落ちとかそういうことかな。数学的な話。省略
