PPL

1. PPL Sequence Interface

3. Midterm 2011

5. (define make-tree (λ (value children) (λ (sel) (sel value children)))) (define root-value (λ (tree) (tree (λ (value children) (value))))) (define tree-children (λ (tree) (tree (λ (value children) (children)))))

6. The Sequence Interface • OOP languages support collections • FP is better: sequence operations • Java 8 new feature is sequence operations… Scheme had it for years!

7. What is Sequence Interface? • ADT for lists • In other words: a barrier between clients running sequence applications and their implementations • Abstracts-away element by element manipulation

8. Map • Applies a procedure to all elements of a list. Returns a list as a result ;Signature: map(proc,sequence) ;Purpose: Apply ’proc’ to all ’sequence’. ;Type: [[T1 -> T2]*LIST(T1) -> LIST(T2)] ;Examples: ;(map abs (list -10 2.5 -11.6 17)) ; ==> (10 2.5 11.6 17) ;(map (lambda (x) (* x x)) (list 1 2 3 4)) ; ==> (1 4 9 16) ;Post-condition: For all i=1..length(sequence): resulti = proc(sequencei)

9. Map Example: scale-list Scaling list of numbers by a factor ;Signature: scale-list(items,factor) ;Purpose: Scaling elements of a number list by a factor. ;Type: [LIST(Number)*Number -> LIST(Number)] > (scale-list (list 1 2 3 4 5) 10) (10 20 30 40 50)

10. Implementation of scale-list No Map Map (define scale-list (lambda (items factor) (map (lambda (x) (* x factor)) items)) (define scale-list (lambda (items factor) (if (null? items) (list) (cons (* (car items) factor) (scale-list (cdr items) factor)))))

11. Map Example: scale-tree Mapping over hierarchical lists >(scale-tree (list 1 (list 2 (list 3 4) 5) (list 6 7)) 10) (10 (20 (30 40) 50) (60 70))

12. Implementation of scale-tree No Map Map (define scale-tree (lambda (tree factor) (map (lambda (sub-tree) (if (list? sub-tree) (scale-tree sub-tree factor) (* sub-tree factor))) tree))) (define scale-tree (lambda (tree factor) (cond ((null? tree) (list)) ((not (list? tree)) (* tree factor)) (else (cons (scale-tree (car tree) factor) (scale-tree (cdr tree) factor))))))

13. Map in Java List<Integer> list = Arrays.asList(1, 2, 3, 4, 5, 6, 7); //Old way: for(Integer n: list) { System.out.println(n); } //New way: list.forEach(n -> System.out.println(n)); //Scheme (map list (lambda(n) (display n))

14. Implementation of Map ;Signature: map(proc,items) ;Purpose: Apply ’proc’ to all ’items’. ;Type: [[T1 -> T2]*LIST(T1) -> LIST(T2)] (define map (lambda (proc items) (if (null? items) (list) (cons (proc (car items)) (map proc (cdr items))))))

15. A More General Map • So far, the procedure can get only a single parameter: an item in the list • Map in Scheme is more general: n-ary procedure and n lists (with same length) Example: > (map + (list 1 2 3) (list 40 50 60) (list 700 800 900)) (741 852 963)

16. Haskell Curry

17. Function Currying: Reminder • Technique for turning a function with n parameters to n functions with a single parameter • Good for partial evaluation

18. Currying ;Type: [Number*Number -> Number] (define add (λ (x y) (+ x y))) ;Type: [Number -> [Number -> Number]] (define c-add (λ (x) (λ (y) (add x y)))) (define add3 (c-add 3)) (add3 4) 7

19. Why Currying? (define add-fib (lambda (x y) (+ (fib x) y))) (define c-add-fib (lambda (x) (lambda (y) (+ (fib x) y)))) (define c-add-fib (lambda (x) (let ((fib-x (fib x))) (lambda (y) (+ fib-x y)))))

20. Curried Map Delayed List Naïve Version: ;Signature: c-map-proc(proc) ;Purpose: Create a delayed map for ’proc’. ;Type: [[T1 -> T2] -> [LIST(T1) -> LIST(T2)]] (define c-map-proc (lambda (proc) (lambda (lst) (if (empty? lst) lst (cons (proc (car lst)) ((c-map-proc proc) (cdrlst)))))))

21. Curried Map – delay the list (define c-map-proc (lambda (proc) (letrec ((iter (lambda (lst) (if (empty? lst) lst (cons (proc (car lst)) (iter (cdrlst))))))) iter)))

22. Curried Map – delay the proc ;Signature: c-map-list(lst) ;Purpose: Create a delayed map for ’lst’. ;Type: [LIST(T1) -> [[T1 -> T2] -> LIST(T2)]] (define c-map-list (lambda (lst) (if (empty? lst) (lambda (proc) lst) ;c-map-list returns a procedure (let ((mapped-cdr (c-map-list (cdrlst)))) ;Inductive Currying (lambda (proc) (cons (proc (car lst)) (mapped-cdr proc)))))))

23. Filter Homogenous List Signature: filter(predicate, sequence) Purpose: return a list of all sequence elements that satisfy the predicate Type: [[T-> Boolean]*LIST(T) -> LIST(T)] Example: (filter odd? (list 1 2 3 4 5)) ==> (1 3 5) Post-condition: result = sequence - {el|el∈sequence and not(predicate(el))}``

24. Accumulate Procedure Application Signature: accumulate(op,initial,sequence) Purpose: Accumulate by ’op’ all sequence elements, starting (ending) with ’initial’ Type: [[T1*T2 -> T2]*T2*LIST(T1) -> T2] Examples: (accumulate + 0 (list 1 2 3 4 5)) ==> 15 (accumulate * 1 (list 1 2 3 4 5)) ==> 120

25. Interval Enumeration Signature: enumerate-interval(low, high) Purpose: List all integers within an interval: Type: [Number*Number -> LIST(Number)] Example: (enumerate-interval 2 7) ==> (2 3 4 5 6 7) Pre-condition: high > low Post-condition: result = (low low+1 ... high)

26. Enumerate Tree Signature: enumerate-tree(tree) Purpose: List all leaves of a number tree Type: [LIST union T -> LIST(Number)] Example: (enumerate-tree (list 1 (list 2 (list 3 4)) 5)) ==> (1 2 3 4 5) Post-condition: result = flatten(tree)

27. Sum-odd-squares ;Signature: sum-odd-squares(tree) ;Purpose: return the sum of all odd square leaves ;Type: [LIST -> Number] (define sum-odd-squares (lambda (tree) (accumulate + 0 (map square (filter odd? (enumerate-tree tree))))))

28. Even Fibonacci Numbers Without sequence operations: With sequence operations: (define even-fibs (lambda (n) (accumulate cons (list) (filter even? (map fib (enumerate-interval 0 n)))))) (define even-fibs (lambda (n) (letrec ((next (lambda(k) (if (> k n) (list) (let ((f (fib k))) (if (even? f) (cons f (next (+ k 1))) (next (+ k 1)))))))) (next 0))))

29. Filter implementation ;; Signature: filter(predicate, sequence) ;; Purpose: return a list of all sequence elements that satisfy the predicate ;; Type: [[T-> Boolean]*LIST(T) -> LIST(T)] (define filter (lambda (predicate sequence) (cond ((null? sequence) sequence) ((predicate (car sequence)) (cons (car sequence) (filter predicate (cdr sequence)))) (else (filter predicate (cdr sequence))))))

30. Accumulate Implementation ;;Signature: accumulate(op,initial,sequence) ;;Purpose: Accumulate by ’op’ all sequence elements, starting (ending) ;;with ’initial’ ;;Type: [[T1*T2 -> T2]*T2*LIST(T1) -> T2] (define accumulate (lambda (op initial sequence) (if (null? sequence) initial (op (car sequence) (accumulate op initial (cdr sequence))))))

31. Enumerate-interval Implementation ;;Signature: enumerate-interval(low, high) ;;Purpose: List all integers within an interval: ;;Type: [Number*Number -> LIST(Number)] (define enumerate-interval (lambda (low high) (if (> low high) (list) (cons low (enumerate-interval (+ low 1) high)))))

32. Enumerate-Tree Implementation ;;Signature: enumerate-tree(tree) ;;Purpose: List all leaves of a number tree ;;Type: [LIST union T -> LIST(Number)] (define enumerate-tree (lambda (tree) (cond ((null? tree) (tree)) ((not (list? tree)) (list tree)) (else (append (enumerate-tree (car tree)) (enumerate-tree (cdr tree)))))))