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Recursion

Recursion. To understand recursion , one must first understand recursion. Specifying Incremental Work. 0 -> f(0). D 1. 1 -> f(1). n -> f(n). D n. n+1 -> f(n+1). Divide and Conquer. Problem. decomposition. Subproblem1. Subproblem2. recursion. primitive.

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Recursion

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  1. Recursion To understand recursion, one must first understand recursion. L14Recur+Clojure

  2. Specifying Incremental Work 0 -> f(0) D1 1 -> f(1) ... n -> f(n) Dn n+1 -> f(n+1) L14Recur

  3. Divide and Conquer Problem decomposition Subproblem1 Subproblem2 recursion primitive Subsolution1 Subsolution2 composition Solution L14Recur

  4. Cartesian Product > (cart2 '(1 2) '(a b)) ( (1 a) (1 b) (2 a) (2 b) ) > (cart2 '(0 1 2) '(a b)) ( (0 a) (0 b) (1 a) (1 b) (2 a) (2 b) ) > (couple ' 1 '(b c)) ( (1 b) (1 c) ) > (couple ' 1 '(a b c)) ( (1 a) (1 b) (1 c) ) L14Recur

  5. Divide and Conquer (cart2 '(a b) '(1)) car/cdr (couple ' a '(1)) (cart2 '(b) '(1)) recursion “primitive” ((a 1)) ((b 1)) append ((a 1) (b 1)) L14Recur

  6. Different Problems; Same pattern L14Recur

  7. Scheme Definition (define (cart2 x y) (if (null? x) '() (append(couple (car x) y) (cart2 (cdr x) y) ) ) ) (define (couple a y) (if (null? y) '() (cons(list a (car y)) (couple a (cdr y)) ) ) ) L14Recur

  8. Alternate Syntax : lambda (define cart2 (lambda (x y) (if (null? x) '() (append(couple (car x) y) (cart2 (cdr x) y) ) ) ) ) (define couple (lambda (a y) (if (null? y) '() (cons(list a (car y)) (couple a (cdr y)) ) ) ) ) L14Recur

  9. Clojure Syntax (defn couple [a y] (if (empty? y) '() (cons(list a (first y)) (couple a (rest y)) ) ) ) (defn cart2 [x y] (if (empty? x) '() (concat (couple (first x) y) (cart2 (rest x) y) ) ) ) L14Recur

  10. Powerset > (powerset '(2)) ( ( ) (2) ) > (powerset '(1 2) ) ( (1) (1 2) () (2) ) > (powerset '(0 1 2)) ( (0 1) (0 1 2) (0) (0 2) (1) (1 2) () (2) ) Subsets of {0,1,2} with 0 are equinumerous with subsets of {0,1,2} without 0. L14Recur

  11. (define (powerset s) (if (null? s) '(()) (append (ins (car s) (powerset (cdr s)) ) (powerset (cdr s)) ) )) (define (ins a ss) (if (null? ss) '() (cons (cons a (car ss)) (ins a (cdr ss)) ) )) L14Recur

  12. Alternate Syntax : let (define (powerset s) (if (null? s) '(()) (let ( (aux (powerset (cdr s)))) (append (ins (car s) aux) aux ) ) )) (define (ins a ss) (if (null? ss) '() (cons (cons a (car ss)) (ins a (cdr ss)) ) )) L14Recur

  13. Clojure Syntax (defn ins [a ss] (if (empty? ss) '() (cons (cons a (first ss)) (ins a (rest ss)) ) )) (defn powerset [s] (if (empty? s) '(()) (let [aux (powerset (rest s))] (concat (ins (first s) aux) aux ) ) )) L14Recur

  14. Related problems; Different Patterns L14Recur

  15. Remove-First-TopLevel (define (rm-fst-top sym lis) (if (null? lis) '() (if (eq? sym (car lis))(cdrlis) (cons (car lis) (rm-fst-top sym (cdrlis)) ) ) ) ) > (rm-fst-top 'a '(b (b a) a b a)) = (b (b a) b a) • Linear recursion L14Recur

  16. Alternate Syntax : if => cond (define (rm-fst-top sym lis) (cond ( (null? lis)'()) ( (eq? sym (car lis))(cdrlis) ) ( else (cons (car lis) (rm-fst-top sym (cdrlis)) ) ) ) ) > (rm-fst-top 'a '(b (b a) a b a)) = (b (b a) b a) • Linear recursion L14Recur

  17. Remove-All-TopLevel (define (rm-all-top sym lis) (cond ( (null? lis)'()) ( (eq? sym (car lis)) (rm-all-top sym(cdrlis) ) ) ( else(cons (car lis) (rm-all-top sym (cdrlis))) ) ) ) > (rm-all-top 'a '(b (b a) a b a)) = (b (b a) b) > (rm-all-top ' (b a) '(b (b a) a b a)) = (b (b a) a b a) • Linear recursion L14Recur

  18. Remove-All-Expression-TopLevel (define (rm-all-top exp lis) (cond ( (null? lis)'()) ( (equal? exp (car lis)) (rm-all-top exp(cdrlis) ) ) ( else(cons (car lis) (rm-all-top exp (cdrlis)))) ) ) > (rm-all-top ' (b a) '(b (b a) a b a)) = (b a b a) • Linear recursion L14Recur

  19. Remove-All (define (rm-all sym ll) (cond ( (null? ll)'()) ( (symbol? (car ll)) (if (eq? sym (car ll)) ( rm-all sym (cdrll) ) ( cons (car ll) (rm-all sym (cdrll)) ) ) ) (else (cons (rm-all sym (car ll)) (rm-all sym (cdrll)) ) ) ) ) > (rm-all ' a '(b (b a) a b a)) = (b (b) b) • Double recursion L14Recur

  20. rm-all : Structural recursion • Empty list (Basis case) (rm-all 'a '()) • First – Atom (Recursive case) (rm-all 'a '(a b c a)) (rm-all 'a '(bb c a)) • First - Nested list (Recursive case) (rm-all 'a '((a b c) d (a))) (rm-all 'a '(b (a b c) d (a))) L14Recur

  21. Insertion sort (Note: creates a sorted copy) (define (insert n lon) (cond ((null? lon) (list n)) ((> n (car lon)) (cons (car lon) (insert n (cdr lon)))) (else (cons n lon)) ) ) (define (ins-sort lon) (if(null? lon)'() (insert (car lon) (ins-sort (cdr lon))) ) ) • Precondition: second arg to insert ordered. • Postcondition: insert returns an ordered list. L14Recur

  22. Clojure Syntax (defn insert [n lon] (cond(empty? lon) (list n) (> n (first lon)) (cons (first lon) (insert n (rest lon))) :true (cons n lon)) ) ) (defn ins-sort [lon] (if(empty? lon)'() (insert (first lon) (ins-sort (rest lon))) ) ) • Precondition: second arg to insert ordered. • Postcondition: insert returns an ordered list. L14Recur

  23. Subset (uses OR and AND instead of IF) (define (subset? ss s) (or (null? ss) (and (member (car ss) s) (subset? (cdr ss) s) ) ) ) 1=> (subset? '(a b c) '(A p 1 C 2 B q r)) #t 2=> (subset? '(a b c) '(p)) #f L14Recur

  24. Anonymous functions and list membership test in Clojure > ( (fn [xy] (+ xy)) 25 30) 55 > ( #(+ %1%2) 25 30) 55 > (some #(= 5 %)'(5 30)) true > (some #(= 5 %)'(15 30)) nil L14Recur

  25. Subset in Clojure (cf. case sensitive Scheme) (defn subset? [ss s] (or (empty? ss) (and (some #(= (first ss) %) s) (subset? (rest ss) s) ) ) ) > (subset? '() '(A p 1 C 2 B q r)) true > (subset? '(a b c) '(A p 1 C 2 B q r)) nil > (subset? '(a b c) '(p a c b)) true L14Recur

  26. Expression evaluation :A simple syntax directed translation expr -> x | y| z expr -> (+exprexpr) expr -> (ifexprexprexpr) Write a recursive definition for a function ops that counts the number of “+”s. L14Recur

  27. (define (ops e) ; e is assumed to be a symbol or a list (cond ((symbol? e) 0) ((eq? (car e) '+) (+ 1 (ops (cadr e)) (ops (caddr e))) ) ((eq? (car e) 'if) (+ (ops (cadr e)) (ops (caddr e)) (ops (cadddr e)))) (else (display 'ILLEGAL)) ) ) L14Recur

  28. (defn third [x] (second (rest x))) (defn ops [e] ; Clojure code "e is assumed to be a symbol or a list" (cond (symbol? e) 0 (= (first e) '+) (+ 1 (ops (second e)) (ops (third e)) ) (= (first e) 'if) (+ (ops (second e)) (ops (third e)) (ops (last e))) true 'ILLEGAL )) L14Recur

  29. (Alternative Scheme syntax with member and case-construct (not equivalent)) (define (ops e) ; e is assumed to be a symbol or a list (if (membere '(x y z)) 0 (if (symbol? e) (display 'ILLEGAL) (case(car e) ((+) (+ 1 (ops (cadr e)) (ops (caddr e))) ) ((if) (+ (ops (cadr e)) (ops (caddr e)) (ops (cadddr e))) ) (else (display 'ILLEGAL))) ) )) L14Recur

  30. Examples (ops 'x) 0 (ops '(+ y z)) 1 (ops '(if x (+ y z) (+ x z))) 2 L14Recur

  31. (Alternative syntax in Clojure (not equivalent or robust)) (defn ops [e] ;e is assumed to legal (if (some #(= e %) '(x y z)) 0 (if (symbol? e) (println 'ILLEGAL) (case (first e) (+) (+ 1 (ops (second e)) (ops (last e))) (if) (+ (ops (second e)) (ops (third e)) (ops (last e))) (println 'ILLEGAL))) ) ) L14Recur

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