Abstraction

1
Abstraction

• A way of managing complexity for large programs
• A means of separating details of computation from
  use of computation
• Types of Abstraction
• Data Abstraction (Today)
• Procedural Abstraction (Later)

2
Data Abstraction Motivation

• Languages come with primitives, which are built
  in data types and operations.
• Ex numbers, strings, booleans
• Ex , -, , /, substring, eq?
• Sometimes, we want to group pieces of
  information.

3
Abstract Data Types ADTs

• We want to create our own data types, which we
  can treat as a unit.
• We want to be able to do the following
• glue data together to make them
• Access parts of them
• Apply procedures to them
• Generate new versions of them
• Modify them (later. For now, ADTs are all
  immutable)
• Create expressions that include them as subunits

4
ADTs in Scheme cons

• Scheme provides 2 ways of gluing things together
  cons and list
• Cons stands for constructor. Joings two pieces
  of data together.
• (cons x y) creates a pair
• (cons 1 2) -gt (1 . 2)
• (car p) returns the first element
• (cdr p) returns second element
• (pair? p) returns whether p is a pair

A cons cell
cdr
car
5
ADTs in Scheme list

• A list is a bunch of cons cells linked together
  used to join multiple things
• (list a b c ) creates a list
• (list 1 2 3) -gt
• (1 2 3)
• (car l) returns the first element
• (cdr l) returns a list of the rest of the
  elements
• (list-ref l n) returns the nth element of

A list
cdr l
l
car l
At the end of a list is a null. Null is written
as ()
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Practice with car, cdr, cons, list

• What do the following print out?
• (list 1 (list 2 (list 3 4)))
• (car (cdr (list 1 2 3)))
• (cons (list 1 2 3) 4)
• What prints the following?
• (1 (2 3) 4)
• (1 (2 . 3) 4)
• (((1) 2) 3)

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Example Rational Numbers

• A rational number is a number that can be
  represented as n/d where n, d are integers.
• We can represent a rational number with a cons
  cell
• Constructor
• (define (make-rat n d) (cons n d))
• Accessors
• (define (numer r) (car r))
• (define (denom r) (cdr r))
• Contract This must hold regardless of
  implementation
• (numer (make-rat n d)) -gt n
• (denom (name-rat n d)) -gt d

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Example Rational Number

• The ADT must fulfill the following contract
  regardless of implementation
• (numer (make-rat n d)) -gt n
• (denom (name-rat n d)) -gt d
• Rational Numbers can also be implemented with a
  list. Still fulfills contract.
• (define (make-rat n d) (list n d))
• (define (numer r) (car r))
• (define (denom r) (cadr r))

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Example Rational Numbers

• Some operations
• (define (rat x y)
• (make-rat
• ( (numer x) (numer y))
  ( (denom x) (denom y))))
• (define (eq-rat x y)
• (and ( (numer x) (numer y))
  ( (denom x) (denom y))))
• Do you see a problem with this???

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Example Rational Numbers

• Suppose we have the rational number 1/2 and 2/4.
  They are equal, but our eq-rat method says
  otherwise.
• How to fix it? Make sure our rational numbers are
  always in lowest terms.
• The following function finds the greatest common
  denominator
• (define (gcd a b)
• (if ( b 0)
• a
• (gcd b (remainder a b))))

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Example Rational Numbers

• We can add a call to gcd in our constructor so
  that we always make rational numbers in lowest
  terms
• (define (make-rat n d)
• (cons (/ n (gcd n d)) (/ d (gcd n d))))
• Because the methods that produce new rational
  numbers (rat, rat) are defined in terms of
  make-rat, this is the only place we have to make
  changes -gt Advantage of abstraction!

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Summary Rational Numbers

• Constructor
• (define (make-rat n d) (let ((g (gcd n d))
• (cons (/ n g) (/ d g))))
• Accessors
• (define (numer r) (car r))
• (define (denom r) (cdr r))
• Contract
• (numer (make-rat n d)) -gt n
• (denom (make-rat n d)) -gt d
• Sample operation
• (define (rat x y)
• (make-rat ( (numer x) (numer y))
• ( (denom x) (denom y))))