Closures and Streams

1. Closures and Streams L11Clos

2. 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

3. 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

4. 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

5. 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

6. 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

7. Initial Global Environment L11Clos

8. Executing (f 5) and (sum-of-squares 6 10) L11Clos

9. Delayed Evaluation : THUNKS (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

10. Substitution (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 Models L11Clos

11. Modular Designs with Lists L11Clos

12. 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

13. (define (even-fibs n) (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

14. 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 Description L11Clos

15. (define (filter pred seq) (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

16. (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

17. Streams: Motivation L11Clos

18. 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

19. Streams : Equational 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; • How do we represent potentially infinite structures? L11Clos

20. (0.(0.(0. … ))) (0. Function which when executed generates an infinite structure) Recursive winding and unwinding (0. ) (0. ) (0. . . . ) L11Clos

21. >(define stream-car car) >(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

22. >(stream-car (stream-cdr stream-zeros) ) >(define (numbers-from n) (cons n (lambda () (numbers-from (+ 1 n)) ))) >(define stream-numbers (numbers-from 0) ) L11Clos

23. Recapitulating Stream Primitives (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

24. (define (stream-filter p s) (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

25. (stream-car (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

26. Factorial Revisited (define (trfac n) (letrec ( (iter (lambda (i a) (if (zero? i) a (iter (- i 1) (* a i))))) ) (iter n 1) ) ) L11Clos

27. (define (ifac n) (let (( i n ) ( a 1 )) (letrec ( (iter (lambda () (if (zero? i) a (begin (set! a (* a i)) (set! i (- i 1)) (iter) )) ) ) ) (iter) ) )) L11Clos

28. Factorial Stream (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