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---
title: 2.4 - Multiple Representations for Abstract Data
layout: org
---
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* Representations for Complex Numbers
* Tagged Data
  #+begin_src scheme :tangle yes
    ;; ===================================================================
    ;; 2.4.2: Tagged Data
    ;; ===================================================================

    (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)
          (error "Bad tagged datum -- CONTENTS" datum)))

    (define (rectangular? z)
      (eq? (type-tag z) 'rectangular))

    (define (polar? z)
      (eq? (type-tag z) 'polar))

    (define (real-part-rectangular z) (car z))

    (define (imag-part-rectangular z) (cdr z))

    (define (magnitude-rectangular z)
      (sqrt (+ (square (real-part-rectangular z))
               (square (imag-part-rectangular z)))))

    (define (angle-rectangular z)
      (atan (imag-part-rectangular z)
            (real-part-rectangular z)))

    (define (make-from-real-imag-rectangular x y)
      (attach-tag 'rectangular (cons x y)))

    (define (make-from-mag-ang-rectangular r a)
      (attach-tag 'rectangular
                  (cons (* r (cos a)) (* r (sin a)))))

    (define (real-part-polar z)
      (* (magnitude-polar z) (cos (angle-polar z))))

    (define (imag-part-polar z)
      (* (magnitude-polar z) (sin (angle-polar z))))

    (define (magnitude-polar z) (car z))

    (define (angle-polar z) (cdr z))

    (define (make-from-real-imag-polar x y)
      (attach-tag 'polar
                  (cons (sqrt (+ (square x) (square y)))
                        (atan y x))))

    (define (make-from-mag-ang-polar r a)
      (attach-tag 'polar (cons r a)))

    (define (real-part z)
      (cond ((rectangular? z)
             (real-part-rectangular (contents z)))
            ((polar? z)
             (real-part-polar (contents z)))
            (else (error "Unknown type -- REAL-PART" z))))

    (define (imag-part z)
      (cond ((rectangular? z)
             (imag-part-rectangular (contents z)))
            ((polar? z)
             (imag-part-polar (contents z)))
            (else (error "Unknown type -- IMAG-PART" z))))

    (define (magnitude z)
      (cond ((rectangular? z)
             (magnitude-rectangular (contents z)))
            ((polar? z)
             (magnitude-polar (contents z)))
            (else (error "Unknown type -- MAGNITUDE" z))))

    (define (angle z)
      (cond ((rectangular? z)
             (angle-rectangular (contents z)))
            ((polar? z)
             (angle-polar (contents z)))
            (else (error "Unknown type -- ANGLE" z))))

    (define (add-complex z1 z2)
      (make-from-real-imag (+ (real-part z1) (real-part z2))
                           (+ (imag-part z1) (imag-part z2))))

    (define (make-from-real-imag x y)
      (make-from-real-imag-rectangular x y))

    (define (make-from-mag-ang r a)
      (make-from-mag-ang-polar r a))
  #+end_src
* Data-Directed Programming and Additivity
  #+begin_src scheme :tangle yes
    ;; ===================================================================
    ;; 2.4.3: Data-Directed Programming and Additivity
    ;; ===================================================================

    (define (install-rectangular-package)
      ;; internal procedures
      (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))))

      ;; interface to the rest of the system
      (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)

    (define (install-polar-package)
      ;; internal procedures
      (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)))

      ;; interface to the rest of the system
      (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)

    (define (apply-generic op . args)
      (let ((type-tags (map type-tag args)))
        (let ((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 (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-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))
  #+end_src
** Exercise 2.73
   Section *Note 2-3-2:: described a program that performs symbolic
   differentiation:

   #+begin_src scheme
     (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))))
             <MORE RULES CAN BE ADDED HERE>
             (else (error "unknown expression type -- DERIV" exp))))
   #+end_src

   We can regard this program as performing a dispatch on the type of
   the expression to be differentiated.  In this situation the "type
   tag" of the datum is the algebraic operator symbol (such as `+')
   and the operation being performed is `deriv'.  We can transform
   this program into data-directed style by rewriting the basic
   derivative procedure as

   #+begin_src scheme
     (define (deriv exp var)
        (cond ((number? exp) 0)
              ((variable? exp) (if (same-variable? exp var) 1 0))
              (else ((get 'deriv (operator exp)) (operands exp)
                                                 var))))

     (define (operator exp) (car exp))

     (define (operands exp) (cdr exp))
   #+end_src
   
   a. Explain what was done above.  Why can't we assimilate the
      predicates `number?' and `same-variable?' into the
      data-directed dispatch?

      ----------------------------------------------------------------------
      
      Rather than embed the logic for each operator we want to support
      in the ~deriv~ function, we'll dispatch them based on the
      operator in the expression.

      ~number?~ and ~same-variable~ cannot be dispatched this way
      because they're scalar values, not compound expressions tagged
      with an operator to dispatch on.

   b. Write the procedures for derivatives of sums and products,
      and the auxiliary code required to install them in the table
      used by the program above.

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

      #+begin_src scheme
        (define (install-deriv-code)
          (define (deriv-sum exp var)
            (make-sum (deriv (addend exp) var)
                      (deriv (augend exp) var)))
          (define (deriv-product expr var)
            (make-sum
             (make-product (multiplier exp)
                           (deriv (multiplicand exp) var))
             (make-product (deriv (multiplier exp) var)
                           (multiplicand exp))))
          (put 'deriv '+ deriv-sum)
          (put 'deriv '* deriv-product))
      #+end_src

   c. Choose any additional differentiation rule that you like,
      such as the one for exponents (*Note Exercise 2-56::), and
      install it in this data-directed system.

   d. In this simple algebraic manipulator the type of an
      expression is the algebraic operator that binds it together.
      Suppose, however, we indexed the procedures in the opposite
      way, so that the dispatch line in `deriv' looked like

      #+begin_src scheme
        ((get (operator exp) 'deriv) (operands exp) var)
      #+end_src

      What corresponding changes to the derivative system are
      required?

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

      Nothing, only the implementations of the dispatch table storage
      / lookup methods ( ~put~ / ~get~ ) would change.

** Exercise 2.74:
   Insatiable Enterprises, Inc., is a highly decentralized
   conglomerate company consisting of a large number of independent
   divisions located all over the world.  The company's computer
   facilities have just been interconnected by means of a clever
   network-interfacing scheme that makes the entire network appear to
   any user to be a single computer.  Insatiable's president, in her
   first attempt to exploit the ability of the network to extract
   administrative information from division files, is dismayed to
   discover that, although all the division files have been
   implemented as data structures in Scheme, the particular data
   structure used varies from division to division.  A meeting of
   division managers is hastily called to search for a strategy to
   integrate the files that will satisfy headquarters' needs while
   preserving the existing autonomy of the divisions.

   Show how such a strategy can be implemented with data-directed
   programming.  As an example, suppose that each division's personnel
   records consist of a single file, which contains a set of records
   keyed on employees' names.  The structure of the set varies from
   division to division.  Furthermore, each employee's record is
   itself a set (structured differently from division to division)
   that contains information keyed under identifiers such as `address'
   and `salary'.  In particular:

   a. Implement for headquarters a `get-record' procedure that
      retrieves a specified employee's record from a specified
      personnel file.  The procedure should be applicable to any
      division's file.  Explain how the individual divisions' files
      should be structured.  In particular, what type information
      must be supplied?

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

      #+begin_src scheme
        (define division-identifier car)
        (define division-data cdr)
        (define tag-division cons)

        (define (get-record name tagged-file)
          (let ((division (division-identifier tagged-file))
                (file (division-data tagged-file)))
            (tag-division division (apply-generic 'get-record
                                                  (division-identifier file)
                                                  name
                                                  (division-data file)))))
      #+end_src

      Division files must be tagged with a unique identifier for the
      division.

   b. Implement for headquarters a `get-salary' procedure that
      returns the salary information from a given employee's record
      from any division's personnel file.  How should the record be
      structured in order to make this operation work?

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

      #+begin_src scheme
        (define (get-record-field tagged-record field)
          (let ((division (division-identifier tagged-record))
                (record (division-data tagged-record)))
            (apply-generic 'get-record-field
                           division
                           record
                           field)))
      #+end_src
      
   c. Implement for headquarters a `find-employee-record'
      procedure.  This should search all the divisions' files for
      the record of a given employee and return the record.  Assume
      that this procedure takes as arguments an employee's name and
      a list of all the divisions' files.

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

      #+begin_src scheme
        (define (find-employee-record name division-files)
          (let* ((division-file (car division-files))
                 (rest (cdr division-files))
                 (found-file (get-record name division-file)))
            (if (nil? found-file)
                (find-employee-record name rest)
                found-file)))
      #+end_src
   d. When Insatiable takes over a new company, what changes must
      be made in order to incorporate the new personnel information
      into the central system?

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

      The new company's personnel information must be representable in
      scheme, and will have to be tagged with a new unique
      identifier. New implementations for ~get-record~ and
      ~get-record-field~ will have to be implemented for the new data
      format.
* Message Passing
  #+begin_src scheme
    ;; ===================================================================
    ;; 2.4.4: Message Passing
    ;; ===================================================================

    (define (make-from-real-imag x y)
      (define (dispatch op)
        (cond ((eq? op 'real-part) x)
              ((eq? op 'imag-part) y)
              ((eq? op 'magnitude)
               (sqrt (+ (square x) (square y))))
              ((eq? op 'angle) (atan y x))
              (else
               (error "Unknown op -- MAKE-FROM-REAL-IMAG" op))))
      dispatch)

    (define (apply-generic op arg) (arg op))
  #+end_src
** Exercise 2.75
   Implement the constructor `make-from-mag-ang' in message-passing
   style.  This procedure should be analogous to the
   `make-from-real-imag' procedure given above.

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

   #+begin_src scheme
     (define (make-from-mag-ang r a)
       (define (dispatch op)
         (cond ((eq? op 'real-part)
                (* r (cos a)))
               ((eq? op 'imag-part)
                (* r (sin a)))
               ((eq? op 'magnitude) r)
               ((eq? op 'angle) a)
               (else
                (error "Unknown op -- MAKE-FROM-MAG-ANG" op))))
       dispatch)
   #+end_src
** Exercise 2.76
   As a large system with generic operations evolves, new types of data
   objects or new operations may be needed.  For each of the three
   strategies--generic operations with explicit dispatch, data-directed
   style, and message-passing-style--describe the changes that must be
   made to a system in order to add new types or new operations.  Which
   organization would be most appropriate for a system in which new
   types must often be added?  Which would be most appropriate for a
   system in which new operations must often be added?

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

   * Generic operations with explicit dispatch
     * A new constructor must be built to represent a new data format
       and uniquely tag it
     * Each generic accessor method must be updated to support a new
       tagged data format
     * New generics must be written to support all possible data formats
     (Not additive)
   * Data-directed style
     * To add a new format, operations must be registered with a
       global lookup table using a unique tag
     * To add a new operation, each type implementation must be
       updated to support the new operation, and a new generic
       function must be made to dispatch it
   * Message-passing style
     * To add a new type, a new constructor must be built that handles
       the supported operations
     * To add a new operation, all constructors must be updated to
       support it

   When new types must often be added, data-directed is more
   appropriate, as people creating new types don't have to worry about
   the operations contract changing frequently.

   When new operations must often be added, message-passing is more
   appropriate, as operations can be added independently from the type
   implementations (which can be caught up later).