PPL

Sequence Interface

Midterm 2011

(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)))))

The Sequence Interface

• OOP languages support collections• FP is better: sequence operations• Java 8 new feature is sequence operations…

Scheme had it for years!

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

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)

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)

Implementation of scale-list

No Map(define scale-list (lambda (items factor) (if (null? items) (list) (cons (* (car items) factor) (scale-list (cdr items) factor)))))

Map(define scale-list (lambda (items factor) (map (lambda (x) (* x factor)) items))

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))

Implementation of scale-tree

No Map(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))))))

Map(define scale-tree (lambda (tree factor) (map (lambda (sub-tree) (if (list? sub-tree) (scale-tree sub-tree factor) (* sub-tree factor))) tree)))

Map in JavaList<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))

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))))))

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)

Haskell Curry

Function Currying: Reminder

• Technique for turning a function with n parameters to n functions with a single parameter

• Good for partial evaluation

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

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)))))

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) (cdr lst)))))))

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

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 (cdr lst)))) ;Inductive Currying (lambda (proc) (cons (proc (car lst)) (mapped-cdr proc)))))))

Filter Homogenous List

Signature: filter(predicate, sequence)Purpose: return a list of all sequence elements

that satisfy the predicateType: [[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))}``∈

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

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 > lowPost-condition: result = (low low+1 ... high)

Enumerate Tree

Signature: enumerate-tree(tree)Purpose: List all leaves of a number treeType: [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)

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))))))

Even Fibonacci Numbers

Without sequence operations:(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))))

With sequence operations:(define even-fibs (lambda (n) (accumulate cons (list) (filter even? (map fib (enumerate-interval 0 n))))))

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))))))

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))))))

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)))))

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)))))))