מבוא מורחב למדעי המחשב Scheme בשפת

10תרגול

Streams

3.5, pages 316-352

definitions file on web

2

cons, car, cdr

(define s (cons 9 (begin (display 7) 5)))-> prints 7

The display command is evaluated whileevaluating the cons.

(car s) -> 9(cdr s) -> 5

3

cons-stream, stream-car, stream-cdr

(define s (cons-stream 9 (begin (display 7) 5)))

Due to the delay of the second argument, cons-stream does not activate the display command

(stream-car s) -> 9(stream-cdr s) -> prints 7 and returns 5 stream-cdr activates the display whichprints 7, and then returns 5.

4

List enumerate(define (enumerate-interval low high)

(if (> low high) nil

(cons low

(enumerate-interval

(+ low 1) high))))

(enumerate-interval 2 8)

-> (2 3 4 5 6 7 8)

(car (enumerate-interval 2 8))

-> 2

(cdr (enumerate-interval 2 8))

-> (3 4 5 6 7 8)5

Stream enumerate(define (stream-enumerate-interval low high)

(if (> low high) the-empty-stream

(cons-stream low

(stream-enumerate-interval

(+ low 1) high))))

(stream-enumerate-interval 2 8)

-> (2 . #<promise>)

(stream-car (stream-enumerate-interval 2 8))

-> 2

(stream-cdr (stream-enumerate-interval 2 8))

-> (3 . #<promise>)6

List map

(map <proc> <list>)

(define (map proc s) (if (null? s) nil (cons (proc (car s)) (map proc (cdr s)))))

(map square (enumerate-interval 2 8))

-> (4 9 16 25 36 49 64)

7

Stream map

(map <proc> <stream>)

(define (stream-map proc s) (if (stream-null? s) the-empty-stream (cons-stream (proc (stream-car s)) (stream-map proc (stream-cdr s)) )))

(stream-map square (stream-enumerate-interval 2 8))-> (4 . #<promise>)

8

List of squares

(define squares

(map square

(enumerate-interval 2 8)))

squares

-> (4 9 16 25 36 49 64)

(car squares)

-> 4

(cdr squares)

-> (9 16 25 36 49 64)9

Stream of squares

(define stream-squares

(stream-map square

(stream-enumerate-interval 2 8)))

stream-squares

-> (4 . #<promise>)

(stream-car stream-squares)

-> 4

(stream-cdr stream-squares)

-> (9 . #<promise>)10

List reference

(define (list-ref s n)

(if (= n 0) (car s)

(list-ref (cdr s) (- n 1))))

(define squares

(map square

(enumerate-interval 2 8)))

(list-ref squares 3)

-> 25

11

Stream reference

(define (stream-ref s n)

(if (= n 0) (stream-car s)

(stream-ref (stream-cdr s) (- n 1))))

(define stream-squares

(stream-map square

(stream-enumerate-interval 2 8)))

(stream-ref stream-squares 3)

-> 25

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List filter

(filter <predicate> <list>)

(define (filter pred s) (cond ((null? s) nil) ((pred (car s)) (cons (car s) (filter pred (cdr s)))) (else (filter pred (cdr s)))))

(filter even? (enumerate-interval 1 20))-> (2 4 6 8 10 12 14 16 18 20)

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Stream filter(stream-filter <predicate> <stream>)

(define (stream-filter pred s) (cond ((stream-null? s) the-empty-stream) ((pred (stream-car s)) (cons-stream (stream-car s) (stream-filter pred (stream-cdr s)))) (else (stream-filter pred (stream-cdr s))))))(stream-filter even?

(stream-enumerate-interval 1 20)) -> (2 . #<promise>)

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Generalized list map

(generalized-map <proc> <list1> … <listn>)

(define (generalized-map proc . arglists) (if (null? (car arglists)) nil (cons (apply proc (map car arglists)) (apply generalized-map (cons proc (map cdr arglists))))))

(generalized-map + squares squares squares)

-> (12 27 48 75 108 147 192)

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Generalized stream map

(generalized-stream-map <proc> <stream1> … <streamn>)

(define (generalized-stream-map proc . argstreams) (if (stream-null? (car argstreams)) the-empty-stream (cons-stream (apply proc (map stream-car argstreams)) (apply generalized-stream-map (cons proc (map stream-cdr argstreams))))))

(generalized-stream-map + stream-squares stream-squares stream-squares)-> (12 . #<promise>)

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List for each

(define (for-each proc s)

(if (null? s) 'done

(begin

(proc (car s))

(for-each proc (cdr s)))))

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Stream for each

(define (stream-for-each proc s) (if (stream-null? s) 'done (begin (proc (stream-car s)) (stream-for-each proc (stream-cdr s)))))

useful for viewing (finite!) streams(define (display-stream s) (stream-for-each display s))

(display-stream (stream-enumerate-interval 1 20)) -> prints 1 … 20 done

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Lists(define sum 0)(define (acc x) (set! sum (+ x sum)) sum)

(define s (map acc (enumerate-interval 1 20)))s -> (1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 210) sum -> 210(define y (filter even? s))y -> (6 10 28 36 66 78 120 136 190 210)

sum -> 210

(define z (filter (lambda (x) (= (remainder x 5) 0)) s))z -> (10 15 45 55 105 120 190 210) sum -> 210

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(list-ref y 7)

-> 136 sum -> 210

(display z)

-> prints (10 15 45 55 105 120 190 210)

sum -> 210

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Streams

(define sum 0)(define (acc x) (set! sum (+ x sum)) sum)

(define s (stream-map acc (stream-enumerate-interval 1 20)))s -> (1 . #<promise>) sum -> 1

(define y (stream-filter even? s))y -> (6 . #<promise>) sum -> 6

(define z (stream-filter (lambda (x) (= (remainder x 5) 0)) s))z -> (10 . #<promise>) sum -> 10

21

(stream-ref y 7)

-> 136 sum -> 136

(display-stream z)

-> prints 10 15 45 55 105 120 190 210 done

sum -> 210

22

Defining streams implicitlyby delayed evaluation

Suppose we needed an infinite list of Dollars.We can(define bill-gates (cons-stream ‘dollar bill-gates))

If we need a Dollar we can take the car

(stream-car bill-gates) -> dollar

The cdr would still be an infinite list of

Dollars.

(stream-cdr bill-gates)->(dollar . #<promise>)23

24

Infinite Streams

Formulate rules defining

infinite series

wishful thinking is key

24

1,ones

(define ones

(cons-stream 1 ones))

1,1,1,… = ones =

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2,twos(define twos (cons-stream 2 twos))

ones + onesadding two infinite series of ones(define twos (stream-map + ones ones))

2 * oneselement-wise operations on an infinite series of ones(define twos (stream-map (lambda (x) (* 2 x)) ones)) or (+ x x)

2,2,2,… = twos =

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1,2,3,… = integers =

1,ones + integers

1,1,1…

1,2,3,…

2,3,4,…

(define integers

(cons-stream 1

(stream-map + ones integers)))

+

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0,1,1,2,3,… = fibs =

0,1,fibs + (fibs from 2nd position) 0,1,1,2,… 1,1,2,3,… 1,2,3,5,…

(define fibs (cons-stream 0 (cons-stream 1 (stream-map + fibs (stream-cdr fibs)))))

+

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1,doubles + doubles

1,2,4,8,…

1,2,4,8,…

2,4,8,16,…

(define doubles (cons-stream 1

(stream-map + doubles doubles)))

1,2,4,8,… = doubles =

+

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1,2 * doubles

(define doubles (cons-stream 1

(stream-map (lambda (x) (* 2 x)) doubles)))

or (+ x x)

1,2,4,8,… = doubles =

30

1,factorials * integers from 2nd

position 1, 1*2, 1*2*3,… 2, 3, 4,… 1*2,1*2*3,1*2*3*4,…

(define factorials (cons-stream 1 (stream-map * factorials (stream-cdr integers))))

1,1x2,1x2x3,... = factorials =

x

31

(1),(1 2),(1 2 3),… = runs =

(1), append runs with a list of integers from 2nd position

(1), (1 2), (1 2 3),… (2), (3), (4),… (1 2),(1 2 3),(1 2 3 4),…

(define runs (cons-stream (list 1) (stream-map append runs (stream-map list (stream-cdr integers)))))

append

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a0,a0+a1,a0+a1+a2,… = partial sums =

a0,partial sums + (stream from 2nd pos) a0, a0+a1, a0+a1+a2,… a1, a2, a3,… a0+a1,a0+a1+a2,a0+a1+a2+a3,…

(define (partial-sums a) (cons-stream (stream-car a) (stream-map + (partial-sums a) (stream-cdr a))))

+

33

34

Partial Sums (cont.)

(define (partial-sums a)(define sums

(cons-stream (stream-car a) (stream-map + sums (stream-cdr a)))) sums)

This implementation is more efficient since it uses the stream itself rather than recreating it recursively

34

Approximating the natural logarithm of 2

1,-1,1,-1,…(define alternate

(cons-stream 1 (stream-map - alternate)))

1/1,-1/2,1/3,…(define ln2-series (stream-map / alternate integers))

4

1

3

1

2

112ln

35

Approximating the natural logarithm of 2

(define ln2 (partial-sums ln2-series))1,1/2,5/6,7/12,…

using Euler’s sequence acceleration(define ln2-euler (euler-transform ln2))7/10,29/42,25/36,457/660,…

using super acceleration(define ln2-accelerated (accelerated-sequence euler-transform ln2))1,7/10,165/238,380522285/548976276,…

4

1

3

1

2

112ln

36

Power series

2345234232

15432

xxxxxex

2342

1cos42 xx

x

234523

sin53 xx

xx

37

Power series

The series

is represented as the stream whoseelements are the coefficient

a0,a1,a2,a3…

33

2210 xaxaxaa

38

Power series integral

The integral of the series

is the series

where c is any constant

33

2210 xaxaxaa

43

32

210 4

1

3

1

2

1xaxaxaxac

39

Power series integral

Input: representing a powerseries

Output: coefficients of the

non-constant term of the integral of theseries

(define (integrate-series a) (stream-map / a integers))

,4

1,

3

1,

2

1, 3210 aaaa

,,,, 3210 aaaa

40

Exponent seriesThe function is its own derivative and the integral of are the sameexcept for the constant term

According to this rule, a definition ofthe exponent series is:(define exp-series (cons-stream 1 (integrate-series exp-series)))

which results in 1,1,1/2,1/6,1/24,1/120…as expected

xexxe xe

10 e

2345234232

15432

xxxxxex

41

Sine and cosine seriesThe derivate of sine is cosineThe derivate of cosine is (- sine)

(define cosine-series (cons-stream 1 (stream-map – (integrate-series sine-series))))

(define sine-series (cons-stream 0

(integrate-series cosine-series)))

Which results incosine-series: 1,0,-1/2,0,1/24,…sine-series: 0,1,0,-1/6,0,1/120,…As expected

2342

1cos42 xx

x

234523

sin53 xx

xx42

RepeatInput: procedure f of one argument, number of repetitionsOutput: f*…*f, n times

(define (repeated f n) (if (= n 1) f (compose f (repeated f (- n 1)))))

(define (compose f g) (lambda (x) (f (g x))))

43

Repeat streamf,f*f,f*f*f,… = repeat =f,compose f,f,f,… with repeat

(define f-series (cons-stream f f-series))

(define stream-repeat (cons-stream f (stream-map compose f-series stream-repeat)))

We would like f to be a parameter44

Repeat streamf,f*f,f*f*f,… = repeat =f,compose f,f,f,… with repeat

(define (repeated f) (define f-series (cons-stream f f-series)) (define stream-repeat (cons-stream f (stream-map compose f-series stream-repeat))) stream-repeat)

45

Interleave

1,1,1,2,1,3,1,4,1,5,1,6,…(interleave ones integers)

s0,t0,s1,t1,s2,t2,… interleave =s0,interleave (t, s from 2nd position)(define (interleave s t) (if (stream-null? s) t (cons-stream (stream-car s) (interleave t (stream-cdr s)))))

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