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### Closures and Streams

### Modular Designs with Lists

### Streams: Motivation

L11Clos

Contemporary Interest in Closures

- The concept of closures was developed in the 1960s and was first fully implemented in 1975 as a language feature in the Scheme programming language to support lexically scoped first-class functions.
- Project Lambda makes it easier to write code for multi-core processors by adding closures to the Java language and extending the Java API to support parallelizable operations upon streamed data.
- Rick Hickey’s Clojure (a dialect of LISP for Java platform) is a pure functional language with support for rich set of data structures, and constructs for concurrent programming.

L11Clos

Models of Evaluation

Substitution-based

(define (square x) (* x x))

((lambda (xy)

(+ (square x) (square y)))

(- 5 3)5)

- (+ (square 2) (square 5))
- (+ (* 2 2) (* 5 5))
- (+ 4 25)
- 29

L11Clos

Expression Evaluation Options

To evaluate: (operator operand1 operand2 operand3 ...)

- Applicative-Order Evaluation (call by value)
- evaluate each of the sub-expressions.
- apply the leftmost result to the rest.

- Normal-Order Evaluation (call by name)
- apply the leftmost (lambda) sub-expression to the rest and expand. (Argument sub-expressions get evaluated when necessary.)

L11Clos

Models of Evaluation

Environment-based

((lambda (xy)

(+ (square x) (square y)))

(- 5 3)5)

- (+ (square x)(square y)) x=2,y=5
- (+ (* x x)x=2,y=5
(* x x) ) x=5,y=5

- (+ 4 25)
- 29

L11Clos

An extended example

(define square (lambda (x) (* x x)))

(define sum-of-squares

(lambda (x y)

(+ (square x) (square y))))

(define f (lambda (a)

(sum-of-squares (+ a 1) (* a 2))))

L11Clos

Initial Global Environment

L11Clos

Executing (f 5) and (sum-of-squares 6 10)

L11Clos

(define x

(* 5 5))

x

25

(define y

(lambda ()

(* 5 5))

(y)

25

Partial Evaluation : CURRYING

(define add

(lambda (x)

(lambda (y)

(+ x y)))

(define ad4

(add 4))

(ad4 8)

12

L11Clos

(lambda (y)

(+ 4 y)

)

Substitution model is inadequate for mutable data structures.

Environment

< (lambda (y)

(+ x y)) ,

[x <- 4] >

Need to distinguish location and contents of the location.

Closure and ModelsL11Clos

L11Clos

Higher-order functions and lists

- Use of lists and generic higher-order functions enable abstraction and reuse
- Can replace customized recursive definitions with more readable definitions built using “library” functions
- The HOF approach may be less efficient.
- Promotes MODULAR DESIGNS – improves programmer productivity

L11Clos

(define (next k)

(if (> k n) ’()

(let ((f (fib k)))

(if (even? f)

(cons f (next (+ k 1)))

(next (+ k 1)) )) ))

(next 0))

- Take a number n and construct a list of first n even Fibonacci numbers.

L11Clos

enumerate integers from 0 to n

compute the Fibonacci number for each integer

filter them, selecting even ones

accumulate the results using cons, starting with ()

Abstract DescriptionL11Clos

(cond ((null? seq) ’())

((pred (car seq))

(cons (car seq) (filter pred (cdr seq))))

(else (filter pred (cdr seq)))

))

(define (accumulate op init seq)

(if (null? seq) init

(op (car seq) (accumulate op init (cdr seq)))

))

L11Clos

(define (enum-interval low high)

(if (> low high) ’()

(cons low (enum-interval (+ low 1) high)) ))

(define (even-fibs n)

(accumulate cons ’()

(filter even?

(map fib

(enum-interval 0 n)))))

L11Clos

L11Clos

- Modeling real-world objects (with state) and real-world phenomena
- Use computational objects with local variables and implement time variation of states using assignments
- Alternatively, use sequences to model time histories of the states of the objects.

- Possible Implementations of Sequences
- Using Lists
- Using Streams
- Delayed evaluation (demand-based evaluation) useful (necessary) when large (infinite) sequences are considered.

L11Clos

Streams : phenomenaEquational Reasoning

(define s (cons 0 s))

- Illegal. (Solution: infinite sequence of 0’s.)
(0 . (0. (0. (0. … ))))

- (cf. Ada, Pascal,…)
type s = record

car : integer;

cdr : s

end;

- (cf. Ada, Pascal,…)
- How do we represent potentially infinite structures?

L11Clos

(0.(0.(0. … ))) phenomena

(0. Function which when

executed generates

an infinite structure)

Recursive winding and unwinding

(0. )

(0. )

(0. . . . )

L11Clos

>(define stream-car car) phenomena

>(define (stream-cdr s)

( (cadr s) ) )

- Unwrap by executing the second.
>(define stream-zeros

(cons 0 (lambda()

stream-zeros) ) )

- Wrap by forming closure (thunk).

L11Clos

>(stream-car phenomena

(stream-cdr stream-zeros) )

>(define (numbers-from n)

(cons n

(lambda ()

(numbers-from (+ 1 n))

)))

>(define stream-numbers

(numbers-from 0)

)

L11Clos

Recapitulating Stream Primitives phenomena

(define stream-car car)

(define (stream-cdr s)

( (cdr s) ) )

(define (stream-cons x s)

(cons x ( lambda ( ) s) ) )

(define the-empty-stream () )

(define stream-null? null?)

L11Clos

(define (stream-filter p s) phenomena

(cond ((stream-null? s) the-empty-stream)

((p (stream-car s))

(stream-cons (stream-car s)

(stream-filter p (stream-cdr s))))

(else (stream-filter p (stream-cdr s)))

))

(define (stream-enum-interval low high)

(if (> low high) the-empty-stream

(stream-cons low

(stream-enum-interval (+ 1 low) high))))

L11Clos

(stream-car phenomena

(stream-cdr

(stream-filter prime?

(stream-enum-interval 100 1000))))

(define (fibgen f1 f2)

(cons f1 (lambda () (fibgen f2 (+ f1 f2)))

))

(define fibs (fibgen 0 1))

L11Clos

Factorial Revisited phenomena

(define (trfac n)

(letrec

( (iter (lambda (i a)

(if (zero? i) a

(iter (- i 1) (* a i)))))

)

(iter n 1)

)

)

L11Clos

(define (ifac n) phenomena

(let (( i n ) ( a 1 ))

(letrec

( (iter (lambda ()

(if (zero? i) a

(begin

(set! a (* a i))

(set! i (- i 1))

(iter)

))

)

)

)

(iter)

)

))

L11Clos

Factorial Stream phenomena

(define (str n r)

(cons r (lambda ()

(str (+ n 1) (* n r))

)

)

)

(define sfac (str 1 1))

(car ((cdr ((cdr ((cdr sfac)) )) )) )

… (stream-cdr … )

- Demand driven generation of list elements.
- Caching/Memoing necessary for efficiency.
- Avoids assignment.

L11Clos

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