1 data abstraction. pairs and lists. (sicp sections 2.1.1 – 2.2.1)
DESCRIPTION
3 Data-object abstraction Export only what is needed. Publish: constructors, selectors Guarantee: the behavior Hide: local variables and procedures, way of implementation, internal details, etc. Interface ImplementationTRANSCRIPT
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Data Abstraction. Pairs and Lists.(SICP Sections 2.1.1 – 2.2.1)
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Procedural abstraction
Export only what is needed.
• Publish: name, number and type of arguments (and conditions they must satisfy) type of procedure’s return value• Guarantee: the behavior of the procedure
• Hide: local variables and procedures, way of implementation, internal details, etc.
Interface
Implementation
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Data-object abstraction
Export only what is needed.
• Publish: constructors, selectors
• Guarantee: the behavior
• Hide: local variables and procedures, way of implementation, internal details, etc.
Interface
Implementation
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An example: Rational numbers
We would like to represent rational numbers.A rational number is a quotient a/b of two integers.
Constructor: (make-rat a b)
Selectors: (numer r)
(denom r)
Guarantee: (numer (make-rat a b)) = a
(denom (make-rat a b)) = b
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An example: Rational numbers
We would like to represent rational numbers.A rational number is a quotient a/b of two integers.
Constructor: (make-rat a b)
Selectors: (numer r)
(denom r)
A betterGuarantee:
(numer (make-rat a b))(denom (make-rat a b))
= ab
A weaker condition, but still sufficient!
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(add-rat x y) (sub-rat x y) (mul-rat x y) (div-rat x y) (equal-rat? x y) (print-rat x)
We can now use the constructors and selectors to implement operations on rational numbers:
A form of wishful thinking: we don’t know how make-rat numer and denom are implemented, but we use them.
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(define (equal-rat? x y) (= (* (numer x) (denom y)) (* (numer y) (denom x))))
Implementing the operations
(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 (sub-rat x y) …
(define (add-rat x y) ;n1/d1 + n2/d2 = (n1.d2 + n2.d1) / (d1.d2) (make-rat (+ (* (numer x) (denom y)) (* (numer y) (denom x))) (* (denom x) (denom y))))
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Using the rational package(define (print-rat x) (newline) (display (numer x)) (display ”/”) (display (denom x)))
(define one-half (make-rat 1 2))(print-rat one-half) 1/2 (define one-third (make-rat 1 3))(print-rat (add-rat one-half one-third)) 5/6(print-rat (add-rat one-third one-third)) 6/9
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Abstraction barriers
Programs that use rational numbers
add-rat sub-rat mul-rat…
make-rat numer denom
rational numbers in problem domain
rational numbers as numerators and denumerators
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Gluing things together
We still have to implement numer, denom, and make-rat
A pair:
We need a way to glue things together…
(define x (cons 1 2))
(car x) 1
(cdr x) 2
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Pair: A primitive data type.
Constructor: (cons a b)
Selectors: (car p)
(cdr p)
Guarantee: (car (cons a b)) = a
(cdr (cons a b)) = b
Abstraction barrier: We say nothing about the representation or implementation of pairs.
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Pairs
(define x (cons 1 2))
(define y (cons 3 4))
(define z (cons x y))
(car (car z)) 1 ;(caar z)
(car (cdr z)) 3 ;(cadr z)
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Implementing make-rat, numer, denom
(define (make-rat n d) (cons n d))
(define (numer x) (car x))
(define (denom x) (cdr x))
7מבוא מורחב שיעור 14
Abstraction barriers
Programs that use rational numbers
add-rat sub-rat mul-rat...
make-rat numer denom
cons car cdr
rational numbers in problem domain
rational numbers as numerators and denumerators
rational numbers as pairs
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Alternative implementation for add-rat
(define (add-rat x y) (cons (+ (* (car x) (cdr y)) (* (car y) (cdr x))) (* (cdr x) (cdr y))))
AbstractionViolation
If we bypass an abstraction barrier,changes to one level may affect many levels above it.Maintenance becomes more difficult.
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A solution: change the constructor
(define (make-rat a b) (let ((g (gcd a b))) (cons (/ a g) (/ b g))))
In our current implementation we keep 10000/20000as such and not as 1/2.
This: • Makes the computation more expensive.• Prints out clumsy results.
No other changes are required!
Rationals - Alternative Implementation
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Reducing to lowest terms, another way
(define (make-rat n d) (cons n d))
(define (numer x) (let ((g (gcd (car x) (cdr x)))) (/ (car x) g)))
(define (denom x) (let ((g (gcd (car x) (cdr x)))) (/ (cdr x) g)))
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How can we implement pairs? (first solution – “lazy” implementation)
(define (cons x y) (lambda (f) (f x y)))
(define (car z) (z (lambda (x y) x)))
(define (cdr z) (z (lambda (x y) y)))
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(define (cons x y) (lambda (f) (f x y)))
(define (car z) (z (lambda (x y) x)))
(define (cdr z) (z (lambda (x y) y)))
How can we implement pairs? (first solution, cont’)
> (define p (cons 1 2))Name Value
(lambda(f) (f 1 2)) p
> (car p)
( (lambda(f) (f 1 2)) (lambda (x y) x))
( (lambda(x y) x) 1 2 )
> 1
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How can we implement pairs?(Second solution: “eager” implementation)
(define (cons x y) (lambda (m) (cond ((= m 0) x) ((= m 1) y) (else (error "Argument not 0 or
1 -- CONS" m))))))
(define (cdr z) (z 1))
(define (car z) (z 0))
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(define (cons x y) (lambda (m) (cond ((= m 0) x) ((= m 1) y) (else ...))) (define (car z) (z 0))
(define (cdr z) (z 1))
Implementing pairs (second solution, cont’)
> (define p (cons 3 4)) (lambda(m) (cond ((= m 0) 3) ((= m 1) 4) (else ..)))
p
> (car p)
((lambda(m) (cond ..)) 0)
(cond ((= 0 0) 3) ((= 0 1) 4) (else ...)))
> 3
Name Value
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Box and Pointer Diagram
A pair can be implemented directly using two “pointers”.
Originally on IBM 704: (car a) Contents of Address part of Register (cdr a) Contents of Decrement part of Register
(define a (cons 1 2)) 2
1
a
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Box and pointer diagrams (cont.)
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1
3
2
(cons (cons 1 (cons 2 3)) 4)
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Compound Data
A closure property: The result obtained by creating a compound data structure can itself be treated as a primitive object and thus be input to the creation of another compound object.
Pairs have the closure property:We can pair pairs, pairs of pairs etc.
(cons (cons 1 2) 3)
3
2
1
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Lists
(cons 1 (cons 3 (cons 2 ’() )))
1 3 2
Syntactic sugar: (list 1 3 2)
The empty list (a.k.a. null or nill)
7מבוא מורחב שיעור 27
Formal Definition of a ListA list is either
• ’() -- The empty list• A pair whose cdr is a list.
Lists are closed under the operations cons and cdr:
• If lst is a non-empty list, then (cdr lst) is a list.
• If lst is a list and x is arbitrary, then (cons x lst) is a list.
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Lists
(list <x1> <x2> ... <xn>)
is syntactic sugar for
(cons <x1> (cons <x2> ( … (cons <xn> ’() ))))
<x1> <x2> <xn>
…
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Lists (examples)
(cdr (list 1 2 3)) (cdr (cons 1 (cons 2 (cons 3 ’() ))))(cons 2 (cons 3 ’() )) (list 2 3)
(cons 3 (list 1 2))(cons 3 (cons 1 (cons 2 ’() ))) (list 3 1 2)
3 1 2
2 3
The following expressions all result in the same structure:
and similarly the following
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More Elaborate Lists(list 1 2 3 4)
(cons (list 1 2) (list 3 4))
(list (list 1 2) (list 3 4))
1 2 3 4
1
3 4
2
1 3 42
Prints as (1 2 3 4)
Prints as ((1 2) 3 4)
Prints as ((1 2) (3 4))
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Yet More Examples
p2( (1 . 2) (1 . 2) )
1 2
p
3
p1
p2
(define p (cons 1 2))p(1 . 2)
(define p1 (cons 3 p)
p1(3 1 . 2)
(define p2 (list p p))
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The Predicate Null?
null? : anytype -> boolean
(null? <z>) #t if <z> evaluates to empty list
#f otherwise
(null? 2) #f (null? (list 1)) #f (null? (cdr (list 1))) #t (null? ’()) #t (null? null) #t
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The Predicate Pair?
pair? : anytype -> boolean
(pair? <z>) #t if <z> evaluates to a pair
#f otherwise. (pair? (cons 1 2)) #t (pair? (cons 1 (cons 1 2))) #t (pair? (list 1)) #t (pair? ’()) #f (pair? 3) #f (pair? pair?) #f
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The Predicate Atom?atom? : anytype -> boolean
(define (atom? z) (and (not (pair? z)) (not (null? z))))
(define (square x) (* x x))(atom? square) #t (atom? 3) #t(atom? (cons 1 2)) #f
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More examples(define digits (list 1 2 3 4 5 6 7 8 9))
?(0 1 2 3 4 5 6 7 8 9)
(define digits1 (cons 0 digits))digits1
(define l (list 0 digits))l?(0 (1 2 3 4 5 6 7 8 9))
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The procedure length(define digits (list 1 2 3 4 5 6 7 8 9))(length digits)
9(define l null)(length l)0(define l (cons 1 l))(length l)1
(define (length l) (if (null? l) 0
(+ 1 (length (cdr l)))))
The procedure append
(define (append list1 list2) (cond ((null? list1) list2) ; base (else (cons (car list1) ; recursion (append (cdr list1) list2)))))
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Constructor• Type: Number * T -> LIST(T)> (make-list 7 ’foo)(foo foo foo foo foo foo foo)> (make-list 5 1) (1 1 1 1 1)
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List type• Pairs:• For every type assignment TA and type expressions
S,S1,S2:• TA |- cons:[S1*S2 -> PAIR(S1,S2)]• TA |- car:[PAIR(S1,S2) -> S1]• TA |- cdr:[PAIR(S1,S2) -> S2]• TA |- pair?:[S -> Boolean]• TA |- equal?:[PAIR(S1,S2)*PAIR(S1,S2) -> Boolean]
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For every type environment TA and type expression S:TA |- list:[Unit -> LIST(S)]TA |- cons:[T*LIST(S) -> LIST(S)]TA |- car:[LIST(S) -> S]TA |- cdr:[LIST(S) -> LIST(S)]TA |- null?:[LIST(S) -> Boolean]TA |- list?:[S -> Boolean]TA |- equal?:[LIST(S)*LIST(S) -> Boolean]
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Cont.• For every type environment TA and type expression S:• TA |- list:[Unit -> LIST]• TA |- cons:[S*LIST -> LIST]• TA |- car:[LIST -> S]• TA |- cdr:[LIST -> LIST]• TA |- null?:[LIST -> Boolean]• TA |- list?:[S -> Boolean]• TA |- equal?:[LIST*LIST -> Boolean]
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