1. מבוא מורחב למדעי המחשב בשפת Scheme תרגול 13

2. Streams 3.5, pages 316-352 definitions file on web 2

3. cons, car, cdr (define s (cons 9 (begin (display 7) 5))) -> prints 7 The display command is evaluated while evaluating the cons. (car s) -> 9 (cdr s) -> 5 3

4. 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 which prints 7, and then returns 5. 4

5. 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

6. 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. # ) (stream-car (stream-enumerate-interval 2 8)) -> 2 (stream-cdr (stream-enumerate-interval 2 8)) -> (3. # ) 6

7. List map (map ) (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

8. Stream map (map ) (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. # ) 8

9. 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

10. Stream of squares (define stream-squares (stream-map square (stream-enumerate-interval 2 8))) stream-squares -> (4. # ) (stream-car stream-squares) -> 4 (stream-cdr stream-squares) -> (9. # ) 10

11. 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

12. 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 12

13. List filter (filter ) (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) 13

14. Stream filter (stream-filter ) (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. # ) 14

15. Generalized list map (generalized-map … ) (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) 15

16. Generalized stream map (generalized-stream-map … ) (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. # ) 16

17. List for each (define (for-each proc s) (if (null? s) 'done (begin (proc (car s)) (for-each proc (cdr s))))) 17

18. 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 18

19. 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 19

20. (list-ref y 7) -> 136 sum -> 210 (display z) -> prints (10 15 45 55 105 120 190 210) sum -> 210 20

21. Streams (define sum 0) (define (acc x) (set! sum (+ x sum)) sum) (define s (stream-map acc (stream-enumerate-interval 1 20))) s -> (1. # )sum -> 1 (define y (stream-filter even? s)) y -> (6. # )sum -> 6 (define z (stream-filter (lambda (x) (= (remainder x 5) 0)) s)) z -> (10. # )sum -> 10 21

22. (stream-ref y 7) -> 136 sum -> 136 (display-stream z) -> prints 10 15 45 55 105 120 190 210 done sum -> 210 22

23. Defining streams implicitly by 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. # ) 23

24. 24 Infinite Streams Formulate rules defining infinite series wishful thinking is key 24

25. 1,ones (define ones (cons-stream 1 ones)) 1,1,1,… = ones = 25

26. 2,twos (define twos (cons-stream 2 twos)) ones + ones adding two infinite series of ones (define twos (stream-map + ones ones)) 2 * ones element-wise operations on an infinite series of ones (define twos (stream-map (lambda (x) (* 2 x)) ones)) or (+ x x) 2,2,2,… = twos = 26

27. 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))) + 27

28. 0,1,1,2,3,… = fibs = 0,1,fibs + (fibs from 2 nd 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))))) + 28

29. 1,doubles + doubles 1,2,4,8,… 2,4,8,16,… (define doubles (cons-stream 1 (stream-map + doubles doubles))) 1,2,4,8,… = doubles = + 29

30. 1,2 * doubles (define doubles (cons-stream 1 (stream-map (lambda (x) (* 2 x)) doubles))) or (+ x x) 1,2,4,8,… = doubles = 30

31. 1,factorials * integers from 2 nd 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

32. (1),(1 2),(1 2 3),… = runs = (1), append runs with a list of integers from 2 nd 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 32

33. a0,a0+a1,a0+a1+a2,… = partial sums = a0,partial sums + (stream from 2 nd 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. 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

35. 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)) 35

36. 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,… 36

37. Power series 37

38. Power series The series is represented as the stream whose elements are the coefficient a0,a1,a2,a3… 38

39. Power series integral The integral of the series is the series where c is any constant 39

40. Power series integral Input: representing a power series Output: coefficients of the non-constant term of the integral of the series (define (integrate-series a) (stream-map / a integers)) 40

41. Exponent series The function is its own derivative and the integral of are the same except for the constant term According to this rule, a definition of the 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 41

42. Sine and cosine series The derivate of sine is cosine The 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 in cosine-series: 1,0,-1/2,0,1/24,… sine-series: 0,1,0,-1/6,0,1/120,… As expected 42

43. Repeat Input: procedure f of one argument, number of repetitions Output: 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

44. Repeat stream f,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 parameter 44

45. Repeat stream f,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

46. 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 2 nd position) (define (interleave s t) (if (stream-null? s) t (cons-stream (stream-car s) (interleave t (stream-cdr s))))) 46