title: CS450 layout: true author: Tiago Cogumbreiro lecture: 07 abstract: foldr, looping first-to-last sep: &nbsp;☽&nbsp; <img id="logo" src="data:image/svg+xml;base64,<?xml version="1.0" encoding="UTF-8"?>
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height="64" /> .bottom-bar[ {{title}} {{sep}} {{abstract}} {{sep}} Lecture {{lecture}} {{sep}} {{author}} ] --- class: middle, full # {{title}} ## Structure of Higher Level Languages ### Lecture {{lecture}}: {{abstract}} ### {{author}} --- class: impact # Exercises on lists --- class: full, center, middle # Searching --- # Element in list? .col-12[ ### Spec ```scheme (require rackunit) (check-true (member? "d" (list "a" "b" "c" "d"))) (check-false (member? "f" (list "a" "b" "c" "d"))) ``` ] -- .col-6[ ### Solution ```scheme (define (member? x l) (match l [(list) #f] [(cons h l) (define result (member? x l)) `(or (equal? x h) result)`])) ``` ] .col-6[ ### Solution in Python ```py def member(x, l): for h in l: if h == x: return True return False ``` ] --- # Element in list? .col-12[ ### Spec ```scheme (require rackunit) (check-true (member? "d" (list "a" "b" "c" "d"))) (check-false (member? "f" (list "a" "b" "c" "d"))) ``` ] .col-6[ ### Alternative solution ```scheme (define (member? x l) (match l [(list) #f] [(cons h _) #:when (equal? x h) #t] [(cons _ l) (member? x l)])) ``` ] .col-6[ ### Solution in Python ```py def member(x, l): for h in l: if h == x: return True return False ``` ] --- # Prefix in list? .col-12[ ### Spec ```scheme (require rackunit) (check-true (string-prefix? "Racket" "R")) ; available in standard library (check-true (match-prefix? "R" (list "foo" "Racket"))) (check-false (match-prefix? "R" (list "foo" "bar"))) ``` ] -- .col-7[ ### Solution ```scheme (define (match-prefix? p l) (match l [(list) #f] [(cons h _) #:when (string-prefix? h p) #t] [(cons _ l) (match-prefix? p l)])) ``` ] --- # Can we generalize the search algorithm? .col-6.small[ ```scheme ; Example 1 (define (member? x l) (match l [(list) #f] [(cons h _) #:when `(equal? h x)` #t] [(cons _ l) (member? x l)])) ``` ] .col-6.small[ ```scheme ; Example 2 (define (match-prefix? p l) (match l [(list) #f] [(cons h _) #:when `(string-prefix? h p)` #t] [(cons _ l) (match-prefix? p l)])) ``` ] -- .row[] ### Solution .col-6[ ```scheme (define (exists? found? l) (match l [(list) #f] [(cons h _) #:when `(found? h)` #t] [(cons _ l) (exists? found? l)])) ``` ] .col-6[ ```scheme ; Example 1 (define (member? `x` l) (exists? (lambda (y) (equal? `x` y)) l)) ; Example 2 (define (match-prefix? `x` l) (exists? (lambda (y) (string-prefix? y `x`))) l) ``` ] --- class: full, center, middle # Removing elements from list --- # Remove zeros from a list ### Spec ```scheme (require rackunit) (check-equal? (list 1 3 4) (remove-0s (list 0 1 3 0 4))) (check-equal? (list 1 2 3) (remove-0s (list 1 2 3))) ``` -- .col-7[ ### Solution ```scheme (define (remove-0s l) (match l [(list) (list)] [(cons h l) #:when (not (equal? h 0)) (cons h (remove-0s l))] [(cons _ l) (remove-0s l)])) ``` ] .col-5[ ### Solution in Python ```py def remove_0s(l): result = [] for h in l: if h != 0: result.append(h) return result ``` ] --- # Can we generalize this functional pattern? .col-6.small[ ### Original ```scheme (define (remove-0s l) (match l [(list) (list)] [(cons h l) #:when (not (equal? h 0)) (cons h (remove-0s l))] [(cons _ l) (remove-0s l)])) ``` ] .col-6[ ### Generalized ```scheme (define (filter keep? l) (match l [(list) (list)] [(cons h l) #:when `(keep? h)` (cons h (filter keep? l))] [(cons _ l) (filter keep? l)])) ;; Usage example (define (remove-0s l) (filter (lambda (x) (not (equal? x 0))) l)) ``` ] --- class: full, center, middle # Concatenate two lists --- # Concatenate two lists Implement function `(append l1 l2)` that appends two lists together. ### Spec ```scheme (check-equal? (append (list 1 2) (list 3 4)) (list 1 2 3 4)) ``` -- ### Solution ```scheme (define (append l1 l2) ; l2 = (list 4 5 6) (match l1 ; l1 = (list 1 2 3) [(list) `l2`] [(cons h l1) ; h = 1 l1 = (list 2 3) (define result (append l1 l2)) ; result = (append '(2 3) '(4 5 6)) ; result = (list 2 3 4 5 6) `(cons h result)`])) ; output = (list 1 2 3 4 5 6) ``` --- class: full, no-counter <iframe src="append-trace-standalone.html" style="position: absolute; inset: 0; width: 100%; height: 100%; transform: none;"></iframe> --- class: impact # Generalizing # order-preserving loops --- # An order-preserving recursion pattern .col-5[ 1. .alt[Case `(list)`] (handle-base) 2. .alt[Case `(cons h l)`] (handle-step) 3. Recursive call handles "smaller" ] <div class="col-7" markdown> ```scheme (define (rec l) (match l [(list) `handle-base`] [(cons h l) (define result (rec l)) `(handle-step h result)`])) ``` </div> .row[] .col-6.small[ ### Example 1 ```scheme (define (map f l) (match l [(list) `(list)`] [(cons h l) (define result (map f l)) `(cons (f h) result)`])) ; = (handle-step h result) ``` ] .col-6.small[ ### Example 2 ```scheme (define (append l1 l2) (match l1 [(list) `l2`] [(cons h l1) (define result (append l1 l2)) `(cons h result)`])) ``` ] --- # A note about side-effects - We need to be mindful when implementing `map` as the order of side-effects may matter. - The **standard** implementation of `map` invokes `f` from left-to-right. - Our implementation implementation of `map` invokes `f` from right-to-left. - The reason we implement r-t-l is to allow for generalization with `foldr` - In terms of code, to obtain an l-t-r ordering, call `f` before recursing - .alt[Run `(map displayln (list 1 2 3)` on either version to observe the difference] <div class="col-6" markdown> ### Side-effects from right-to-left ```scheme (define (map f l) (match l [(list) (list)] [(cons h l) (define result (map f l)) (cons `(f h)` result)])) ``` </div> <div class="col-6" markdown> ### Side effects from left-to-right ```scheme (define (map f l) (match l [(list) (list)] [(cons h l) `(define x (f h))` (define result (map f l)) (cons `x` result)])) ``` </div> --- # An order-preserving recursion pattern .col-6[ ### Searching ```scheme (define (exists? found? l) (match l [(list) #f] [(cons h _) #:when (found? h) #t] [(cons _ l) (exists? found? l)])) ``` ] .col-6[ ### Following the recursion pattern ```scheme (define (exists? found? l) (match l [(list) `#f`] [(cons h l) (define result (exists? found? l)) `(or (found? h) result)`])) ``` ] --- # An order-preserving recursion pattern .col-6[ ### Removing ```scheme (define (filter keep? l) (match l [(list) (list)] [(cons h l) #:when (keep? h) (cons h (filter keep? l))] [(cons _ l) (filter keep? l)])) ``` ] .col-6[ ### Following the recursion pattern ```scheme (define (filter keep? l) (match l [(list) `(list)`] [(cons h l) (define result (filter keep? l)) `(if (keep? h)` `(cons h result) result)`])) ``` ] --- class: full, center, middle # Implementing this recursion pattern --- # Implementing this recursion pattern ### Recursive pattern for lists ```scheme (define (rec l) (match l [(list) `base-case`] [(cons h l) (define result (rec l)) `(handle-step h result)`])) ``` ### Fold right reduction .col-8[ ```scheme (define (foldr `handle-step` `base-case` l) (match l [(list) `base-case`] [(cons h l) (define result (foldr handle-step base-case l)) `(handle-step h result)`])) ``` ] .col-4.small[ ```py # In Python def foldr(handle_step, base_case, l): result = base_case for h in reversed(l): result = handle_step(h, result) return result ``` ] --- # Implementing map with foldr ```scheme (define (map f l) (match `l` [(list) `(list)`] [(cons h l) (define result (map f l)) `(cons (f h) result)`])) ``` -- ### Solution .col-8[ ```scheme (define (map f l) (foldr ; step: how do you build the next result (lambda (h result) `(cons (f h) result)`) ; what to return when the list is empty `(list)` ; iterate/match over l `l`)) ``` ] .col-4.small[ ```python # Python pseudo-code result = [] for h in reversed(l): # result = cons(f(h), result) result.append(f(h)) ``` ] --- # Implementing append with foldr ```scheme (define (append l1 l2) (match `l1` [(list) `l2`] [(cons h l1) (define result (append l1 l2)) `(cons h result)`])) ``` -- ### Solution ```scheme (define (append l1 l2) (foldr ; step: add the element to the list being built (lambda (h result) `(cons h result)`) ; base-case: start with list l2 `l2` ; iterate/match over l1 `l1`)) ``` --- # Implementing filter with foldr ```scheme (define (filter keep? l) (match `l` [(list) `(list)`] [(cons h l) (define result (filter keep? l)) `(if (keep? h) (cons h result) result)`])) ``` -- ### Solution ```scheme (define (filter keep? l) (foldr ; handle-step (lambda (h result) `(if (keep? h) (cons h result) result)`) ; base-case `(list)` ; iterate/match over l `l`)) ``` --- # Implementing exists? with foldr ```scheme (define (exists? found? l) (match `l` [(list) `#f`] [(cons h l) (define result (exists? found? l)) `(or (found? h) result)`])) ``` -- ### Solution ```scheme (define (exists? found? l) (foldr ; handle-step (lambda (h result) `(or (found? h) result)`) ; base-case `#f` ; iterate over l `l`)) ```