#+BEGIN_HTML
---
title: 2.1 - Introduction to Data Abstraction
layout: org
---
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* Example: Arithmetic Operations for Rational Numbers

  #+begin_src scheme :tangle yes
    ;; ===================================================================
    ;; 2.1.1: Example: Arithmetic Operators for Rational Numbers
    ;; ===================================================================

    (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) (cons n d))

    (define (numer x) (car x))

    (define (denom x) (cdr x))

    (define (print-rat x)
      (newline)
      (display (numer x))
      (display "/")
      (display (denom x)))

    (define (gcd a b)
      (if (= b 0)
          a
          (gcd b (remainder a b))))

    (define (make-rat n d)
      (let ((g (gcd n d)))
        (cons (/ n g) (/ d g))))
  #+end_src
** Exercise 2.1:
   Define a better version of `make-rat' that handles
   both positive and negative arguments.  `Make-rat' should normalize
   the sign so that if the rational number is positive, both the
   numerator and denominator are positive, and if the rational number
   is negative, only the numerator is negative.

   ----------------------------------------------------------------------

   #+begin_src scheme :tangle yes
     ;; -------------------------------------------------------------------
     ;; Exercise 2.1
     ;; -------------------------------------------------------------------

     (define (make-rat n d)
       (cond ((and (negative? n) (negative? d)) (make-rat (abs n) (abs d)))
             ((negative? d) (make-rat (- n) (- d)))
             (else (let ((g (gcd n d)))
                     (cons (/ n g) (/ d g))))))
   #+end_src

* Abstraction Barriers
** Exercise 2.2
   Consider the problem of representing line segments in a plane.
   Each segment is represented as a pair of points: a starting point
   and an ending point.  Define a constructor `make-segment' and
   selectors `start-segment' and `end-segment' that define the
   representation of segments in terms of points.  Furthermore, a
   point can be represented as a pair of numbers: the x coordinate and
   the y coordinate.  Accordingly, specify a constructor `make-point'
   and selectors `x-point' and `y-point' that define this
   representation.  Finally, using your selectors and constructors,
   define a procedure `midpoint-segment' that takes a line segment as
   argument and returns its midpoint (the point whose coordinates are
   the average of the coordinates of the endpoints).  To try your
   procedures, you'll need a way to print points:

   #+begin_src scheme :tangle yes
     ;; -------------------------------------------------------------------
     ;; Excercise 2.2
     ;; -------------------------------------------------------------------

     (define (print-point p)
       (newline)
       (display "(")
       (display (x-point p))
       (display ",")
       (display (y-point p))
       (display ")"))
   #+end_src

   ----------------------------------------------------------------------

   #+begin_src scheme :tangle yes
     (define make-point cons)
     (define x-point car)
     (define y-point cdr)

     (define (midpoint-segment p1 p2)
       (let ((average (lambda (x y) (/ (+ x y) 2))))
         (make-point
          (average (x-point p1) (x-point p2))
          (average (y-point p1) (y-point p2)))))
   #+end_src
** Exercise 2.3:
   Implement a representation for rectangles in a plane.  (Hint: You
   may want to make use of *Note Exercise 2-2::.)  In terms of your
   constructors and selectors, create procedures that compute the
   perimeter and the area of a given rectangle.  Now implement a
   different representation for rectangles.  Can you design your
   system with suitable abstraction barriers, so that the same
   perimeter and area procedures will work using either
   representation?

   ----------------------------------------------------------------------

   #+begin_src scheme :tangle yes
     ;; -------------------------------------------------------------------
     ;; Exercise 2.3
     ;; -------------------------------------------------------------------

     (define (perimeter-rectangle r)
       (+ (* 2 (width-rectangle r))
          (* 2 (height-rectangle r))))

     (define (area-rectangle r)
       (* (width-rectangle r)
          (height-rectangle r)))

     ;; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
     ;; Hard mode - Expose the 4 points of the rectangle
     ;;             Width and Height have their own abstraction layer
     ;;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

     (define (width-rectangle r)
       (abs (- (x2-rectangle r)
               (x1-rectangle r))))

     (define (height-rectangle r)
       (abs (- (y2-rectangle r)
               (y1-rectangle r))))

     (define (x1-rectangle r) (x-point (top-left-point-rectangle r)))
     (define (x2-rectangle r) (x-point (bottom-right-point-rectangle r)))
     (define (y1-rectangle r) (y-point (top-left-point-rectangle r)))
     (define (y2-rectangle r) (y-point (bottom-right-point-rectangle r)))

     ;; -------------------------------------------------------------------
     ;; Rectangle implementation using two points on a plane

     (define make-rectangle cons)
     (define top-left-point-rectangle car)
     (define bottom-right-point-rectangle cdr)
     (define (top-right-point-rectangle r)
       (make-point (x-point (top-left-point-rectangle r))
                   (y-point (bottom-right-point-rectangle r))))
     (define (bottom-left-point-rectangle r)
       (make-point (x-point (top-left-point-rectangle r))
                   (y-point (bottom-right-point-rectangle r))))

     ;; -------------------------------------------------------------------
     ;; Rectangle implementation using an origin point, width and height

     (define (make-rectangle origin width height)
       (cons origin (cons width height)))
     (define (top-left-point-rectangle r) (car r))
     (define (top-right-point-rectangle r)
       (let ((x (x-point (car r)))
             (y (y-point (car r)))
             (width (car (cdr r))))
         (make-point (+ x width) y)))
     (define (bottom-left-point-rectangle r)
       (let ((x (x-point (car r)))
             (y (y-point (car r)))
             (height (cdr (cdr r))))
         (make-point x (+ y height))))
     (define (bottom-right-point-rectangle r)
       (let ((x (x-point (car r)))
             (y (y-point (car r)))
             (width (car (cdr r)))
             (height (cdr (cdr r))))
         (make-point (+ x width) (+ y height))))

     ;; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
     ;; Simpler solution - Expose only width + height
     ;; ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

     ;; -------------------------------------------------------------------
     ;; Rectangle implementation using two points on a plane

     (define make-rectangle cons)
     (define (width-rectangle r)
       (let ((p1 (car r))
             (p2 (cdr r)))
         (abs (- (x-point p1)
                 (x-point p2)))))
     (define (height-rectangle r)
       (let ((p1 (car r))
             (p2 (cdr r)))
         (abs (- (y-point p1)
                 (y-point p2)))))

     ;; -------------------------------------------------------------------
     ;; Rectangle implementation using an origin point, width and height

     (define (make-rectangle origin width height)
       (cons origin (cons width height)))
     (define (width-rectangle r) (car (cdr r)))
     (define (height-rectangle r) (cdr (cdr r)))
   #+end_src
* What is Meant by Data
** Exercise 2.4
   Here is an alternative procedural representation of pairs.  For
   this representation, verify that `(car (cons x y))' yields `x' for
   any objects `x' and `y'.

   #+begin_src scheme :tangle yes
     ;; -------------------------------------------------------------------
     ;; Exercise 2.4
     ;; -------------------------------------------------------------------

     (define (cons x y)
       (lambda (m) (m x y)))

     (define (car z)
       (z (lambda (p q) p)))
   #+end_src

   What is the corresponding definition of `cdr'? (Hint: To verify
   that this works, make use of the substitution model of section
   *Note 1-1-5::.)

   ----------------------------------------------------------------------

   #+begin_src scheme :tangle yes
     (define (cdr z)
       (z (lambda (p q) q)))
   #+end_src
** Exercise 2.5
   Show that we can represent pairs of nonnegative integers using only
   numbers and arithmetic operations if we represent the pair a and b
   as the integer that is the product 2^a 3^b.  Give the corresponding
   definitions of the procedures `cons', `car', and `cdr'.

   ----------------------------------------------------------------------

   #+begin_src scheme :tangle yes
     ;; -------------------------------------------------------------------
     ;; Exercise 2.5
     ;; -------------------------------------------------------------------

     (define (cons a b)
       (* (expt 2 a) (expt 3 b)))

     (define (factor-count n x count)
       (if (= 0 (remainder x n))
           (factor-count n (/ x n) (+ 1 count))
           count))

     (define (car p)
       (factor-count 2 p 0))

     (define (cdr p)
       (factor-count 3 p 0))
   #+end_src
** Exercise 2.6
   In case representing pairs as procedures wasn't mind-boggling
   enough, consider that, in a language that can manipulate
   procedures, we can get by without numbers (at least insofar as
   nonnegative integers are concerned) by implementing 0 and the
   operation of adding 1 as

   #+begin_src scheme
     (define zero (lambda (f) (lambda (x) x)))

     (define (add-1 n)
       (lambda (f) (lambda (x) (f ((n f) x)))))
   #+end_src

   This representation is known as "Church numerals", after its
   inventor, Alonzo Church, the logician who invented the [lambda]
   calculus.

   Define `one' and `two' directly (not in terms of `zero' and
   `add-1').  (Hint: Use substitution to evaluate `(add-1 zero)').
   Give a direct definition of the addition procedure `+' (not in
   terms of repeated application of `add-1').

   ----------------------------------------------------------------------

   #+begin_src scheme :tangle yes
     (define one (lambda (f) (lambda (x) (f x))))
     (define two (lambda (f) (lambda (x) (f (f x)))))

     (define (add a b)
       (lambda (f)
         (lambda (x)
           ((a f) ((b f) x)))))
   #+end_src

* Extended Exercise: Interval Arithmetic
  #+begin_src scheme :tangle yes
    ;; ===================================================================
    ;; 2.1.4: Extended Exercise: Interval Arithmetic
    ;; ===================================================================

    (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)))))

  #+end_src

** Exercise 2.7
   Alyssa's program is incomplete because she has not specified the
   implementation of the interval abstraction.  Here is a definition
   of the interval constructor:

   #+begin_src scheme :tangle yes
     ;; -------------------------------------------------------------------
     ;; Exercise 2.7
     ;; -------------------------------------------------------------------

     (define (make-interval a b) (cons a b))
   #+end_src

   Define selectors `upper-bound' and `lower-bound' to complete the
   implementation.

   ----------------------------------------------------------------------

   #+begin_src scheme :tangle yes
     (define (upper-bound p)
       (max (car p) (cdr p)))

     (define (lower-bound p)
       (min (car p) (cdr p)))
   #+end_src

** Exercise 2.8:
   Using reasoning analogous to Alyssa's, describe how the difference
   of two intervals may be computed.  Define a corresponding
   subtraction procedure, called `sub-interval'.

   ----------------------------------------------------------------------

   #+begin_src scheme :tangle yes
     ;; -------------------------------------------------------------------
     ;; Exercise 2.8
     ;; -------------------------------------------------------------------
   #+end_src