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missing slides from lecture #8

missing slides from lecture #8. How does the interpreter prints lists and pairs ??. First version, using dot notation. ( define (print-list-structure x) (define (print-contents x) (print-list-structure (car x)) (display " . ") (print-list-structure (cdr x)))

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missing slides from lecture #8

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  1. missing slides fromlecture #8 מבוא מורחב

  2. How does the interpreter prints lists and pairs ?? מבוא מורחב

  3. First version, using dot notation (define (print-list-structure x) (define (print-contents x) (print-list-structure (car x)) (display " . ") (print-list-structure (cdr x))) (cond ((null? x) (display "()")) ((atom? x) (display x)) (else (display "(") (print-contents x) (display ")")))) (p-l-s (list 1 2 3)) ==> (1 . (2 . (3 . ()))) מבוא מורחב

  4. Second version, try to identify lists (define (print-list-structure x) (define (print-contents x) (print-list-structure (car x)) (cond ((null? (cdr x)) nil) ((atom? (cdr x)) (display " . ") (print-list-structure (cdr x))) (else (display " ") (print-contents (cdr x))))) (cond ((null? x) (display "()")) ((atom? x) (display x)) (else (display "(") (print-contents x) (display ")")))) (p-l-s (list 1 2 3)) ==> (1 2 3)

  5. More examples How to create the following output ? ( 1 2 . 3) (cons 1 (cons 2 3)) (1. 2 3) cannot מבוא מורחב

  6. Lecture #10 מבוא מורחב

  7. A bigger example: symbolic diffrentiation מבוא מורחב

  8. 3. A better implementation 1. Use cond instead of nested if expressions 2. Use data abstraction • To use cond: • write a predicate that collects all tests to get to a branch:(define sum-expr? (lambda (e) (and (pair? e) (eq? (car e) '+)))); type: Expr -> boolean • do this for every branch:(define variable? (lambda (e) (and (not (pair? e)) (symbol? e)))) מבוא מורחב

  9. Use data abstractions • To eliminate dependence on the representation: (define make-sum (lambda (e1 e2) (list '+ e1 e2))(define addend (lambda (sum) (cadr sum))) מבוא מורחב

  10. A better implementation (define deriv (lambda (expr var) (cond ((number? expr) 0) ((variable? expr) (if (eq? expr var) 1 0)) ((sum-expr? expr) (make-sum (deriv (addend expr) var) (deriv (augend expr) var))) ((product-expr? expr) <handle product expression>) (else (error "unknown expression type" expr)) )) מבוא מורחב

  11. Deriv: Reduction problem (deriv '(+ x 3) 'x) ==> (+ 1 0) (deriv '(* x y) 'x) ==> (+ (* x 0) (* 1 y)) (deriv '(* (* x y) (+ x 3)) 'x) ==> (+ (* (* x y) (+ 1 0)) (* (+ (* x 0) (* 1 y)) (+ x 3))) מבוא מורחב

  12. Changes are isolated (deriv '(+ x y) 'x) ==> (+ 1 0) (a list!) (define (make-sum a1 a2) (cond ((=number? a1 0) a2) ((=number? a2 0) a1) ((and (number? a1) (number? a2)) (+ a1 a2)) (else (list '+ a1 a2)))) (define (=number? exp num) (and (number? exp) (= exp num))) (deriv '(+ x y) 'x) ==> 1 (deriv '(* x y) 'x) ==> y מבוא מורחב

  13. Variable number of arguments in sums and products (deriv’(+ (* x 3) 10 (+ 2 x)) ’x) (define (augend s) (if (null? (cdddr s)) (caddr s) (cons '+ (cddr s)))) (augend ’(+ (* x 3) 10 (+ 2 x))) ==> (+ 10 (+ 2 x)) 4 (deriv’(+ (* x 3) 10 (+ 2 x)) ’x) ==> מבוא מורחב

  14. Variable number of summands and multipliers (deriv '(+ (* x a) (* x b) (* x c)) 'x) => (+ a (+ b c)) (define (make-sum a1 a2) (cond ((=number? a1 0) a2) ((=number? a2 0) a1) ((and (number? a1) (number? a2)) (+ a1 a2)) ((sum? a2) (cons '+ (cons a1 (cdr a2)))) (else (list '+ a1 a2)))) (deriv '(+ (* x a) (* x b) (* x c)) 'x) => (+ a b c) מבוא מורחב

  15. Representing sets מבוא מורחב

  16. Definitions A set is a collection of distinct items (element-of-set? x set) (adjoin-set x set) (union-set s1 s2) (intersection-set s1 s2) מבוא מורחב

  17. Version 1: Represent a set as an unordered list (define (element-of-set? x set) (cond ((null? set) false) ((equal? x (car set)) true) (else (element-of-set? x (cdr set))))) equal? : Like eq? for symbols. Works for numbers Works recursively for compounds: Apply equal? to the components. (eq? (list ‘a ‘b) (list ‘a ‘b)) (equal? (list ‘a ‘b) (list ‘a ‘b)) מבוא מורחב

  18. Version 1: Represent a set as a list (define (adjoin-set x set) (if (element-of-set? x set) set (cons x set))) מבוא מורחב

  19. Version 1: Represent a set as a list (define (intersection-set set1 set2) (cond ((or (null? set1) (null? set2)) '()) ((element-of-set? (car set1) set2) (cons (car set1) (intersection-set (cdr set1) set2))) (else (intersection-set (cdr set1) set2)))) מבוא מורחב

  20. Version 1: Represent a set as a list (define (union-set set1 set2) (cond ((null? set1) set2)) ((not (element-of-set? (car set1) set2)) (cons (car set1) (union-set (cdr set1) set2))) (else (union-set (cdr set1) set2)))) (define (union-set set1 set2) (cond ((null? set1) set2)) (else (adjoin-set (car set1) (union-set (cdr set1) set2))))) מבוא מורחב

  21. Complexity Element-of-set Adjoin-set Intersection-set Union-set (n) (n) (n2) (n2) מבוא מורחב

  22. Version 2: Representing a set as an ordered list (define (element-of-set? x set) (cond ((null? set) false) ((= x (car set)) true) ((< x (car set)) false) (else (element-of-set? x (cdr set))))) n/2 steps on average  (n) Adjoin-set is similar, please try by yourself מבוא מורחב

  23. Ordered lists (cont.) (define (intersection-set set1 set2) (cond ((or (null? set1) (null? set2)) '()) ((element-of-set? (car set1) set2) (cons (car set1) (intersection-set (cdr set1) set2))) (else (intersection-set (cdr set1) set2)))) Can we do it better ? מבוא מורחב

  24. Ordered lists (cont.) (define (intersection-set set1 set2) (if (or (null? set1) (null? set2)) '() (let ((x1 (car set1)) (x2 (car set2))) (cond ((= x1 x2) (cons x1 (intersection-set (cdr set1) (cdr set2)))) ((< x1 x2) (intersection-set (cdr set1) set2)) ((< x2 x1) (intersection-set set1 (cdr set2))))))) מבוא מורחב

  25. Ordered lists (Cont.) set1 set2 intersection (1 3 7 9) (1 4 6 7) (1 (3 7 9) (4 6 7) (1 (7 9) (4 6 7) (1 (7 9) (6 7) (1 (7 9) (7) (1 (9) () (1 7) Time and space  (n) Union -- similar מבוא מורחב

  26. unordered Element-of-set Adjoin-set Intersection-set Union-set (n) (n) (n2) (n2) Complexity ordered (n) (n) (n) (n) מבוא מורחב

  27. 3 7 7 1 9 3 5 9 12 5 1 12 Representing sets as binary trees מבוא מורחב

  28. 7 9 3 12 5 1 Representing sets as binary trees (Cont.) (define (entry tree) (car tree)) (define (left-branch tree) (cadr tree)) (define (right-branch tree) (caddr tree)) (define (make-tree entry left right) (list entry left right)) מבוא מורחב

  29. Representing sets as binary trees (Cont.) (define (element-of-set? x set) (cond ((null? set) false) ((= x (entry set)) true) ((< x (entry set)) (element-of-set? x (left-branch set))) ((> x (entry set)) (element-of-set? x (right-branch set))))) Complexity: (d) If tree is balanced d  log(n) מבוא מורחב

  30. 1 7 3 9 3 5 12 5 1 7 9 12 Representing sets as binary trees (Cont.) מבוא מורחב

  31. Representing sets as binary trees (Cont.) (define (adjoin-set x set) (cond ((null? set) (make-tree x '() '())) ((= x (entry set)) set) ((< x (entry set)) (make-tree (entry set) (adjoin-set x (left-branch set)) (right-branch set))) ((> x (entry set)) (make-tree (entry set) (left-branch set) (adjoin-set x (right-branch set)))))) מבוא מורחב

  32. unordered ordered Element-of-set Adjoin-set Intersection-set Union-set (n) (n) (n) (n) (n) (n2) (n) (n2) Complexity trees (log(n)) (log(n)) (nlog(n)) (nlog(n)) מבוא מורחב

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