# Streams and Evalutation Strategies

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1 Data and Program Structure Streams and Evalutation Strategies Lecture V Ahmed Rezine Linköpings Universitet TDDA69, VT 2014

2 Lecture 2: Class descriptions - message passing ( define ( make-account balance ) ;; public methods ( define ( withdraw amount ) ( if (>= balance amount ) ( begin ( set! balance (- balance amount )) balance ) " Insufficient funds ")) ( define ( deposit amount ) ( set! balance (+ balance amount )) balance ) ;; message passing procedure ( define ( dispatch m) ( cond (( eq? m withdraw ) withdraw ) (( eq? m deposit ) deposit ) ( else ( error " Unknown request " m)))) dispatch ) >(define acc1 (make-account 100)) >(define acc2 (make-account 100)) >((acc1 withdraw) 40) 60 >((acc1 deposit) 60) 120 >((acc2 withdraw) 110) Insufficient funds >((acc1 deposit) 110) 10

3 Lecture 2: A half-adder ( define a ( make-wire )) ( define b ( make-wire )) ( define s ( make-wire )) ( define c ( make-wire )) ( define ( half-adder a b s c) ( let ((d ( make-wire )) (e ( make-wire ))) ( or-gate a b d) ( and-gate a b c) ( inverter c e) ( and-gate d e s) ok)) Figure : 3.25 in SICP

4 Lecture 2: A sample simulation > (define the-agenda (make-agenda)) > (define inverter-delay 2) > (define and-gate-delay 3) > (define or-gate-delay 5) > (define input-1 (make-wire)) > (define input-2 (make-wire)) > (define sum (make-wire)) > (define carry (make-wire)) > (probe sum sum) sum 0 New-value = 0 > (probe carry carry) > (set-signal! input-1 1) done > (propagate) sum 8 New-value = 1done > (set-signal! input-2 1) done > (propagate) carry 11 New-value = 1 sum 16 New-value = 0done carry 0 New-value = 0 > (half-adder input-1 input-2 sum carry) ok

5 Lecture 2: Fahrenheit-Celsius converter See Section in SICP for the implementation Recall 9C = 5(F-32) Captured by combining two multipliers, an adder and three constants constraints Figure : 3.28 in SICP

6 Lecture 2 : Fahrenheit-Celsius converter (cont.) ( define ( celsius-fahrenheit-converter c f) ( let ((u ( make-connector ))(v ( make-connector )) (w ( make-connector ))(x ( make-connector )) (y ( make-connector ))) ( multiplier c w u) ( multiplier v x u) ( adder v y f) ( constant 9 w) ( constant 5 x) ( constant 32 y) ok)) > (define C (make-connector)) > (define F (make-connector)) > (celsius-fahrenheit-converter C F) ok > (probe "Celsius" C) > (probe "Farenheit" F) > (set-value! C 25 user) Probe: Celsius = 25 Probe: Farenheit = 77 done > (forget-value! C user) Probe: Celsius =? Probe: Farenheit =? done > (set-value! F 212 user) Probe: Farenheit = 212 Probe: Celsius = 100 done

7 Outline Streams (SICP sec 3.5) Modeling the Modules Efficient Implementation Delayed evaluation and Parameter passing

8 Outline Streams (SICP sec 3.5) Modeling the Modules Efficient Implementation Delayed evaluation and Parameter passing

9 Streams We can use streams to achieve modularity with a new type of modeling: Similar to the way signal processing systems are described. Signals flow through the modules that perform different transformations We build sequences representing the history of the modeled system. The resulting system has a state without assignments or mutable data.

10 Example 1: sum odd squares of a tree s leaves Given a binary tree, compute the sum of the squares of its odd leaves: ( define ( sum- odd- squares tree ) (if ( leaf-node? tree ) (if ( odd? tree ) ( square tree ) 0) (+ ( sum- odd- squares ( left-branch tree )) ( sum- odd- squares ( right-branch tree )))))

11 Example 2: enumerate the odd Fibonacci numbers Given a natural n, build a list of all the odd Fibonacci numbers fib(k) where k is smaller or equal than n. ( define ( sum-odd-fibs n) ( define ( next k) (if (> k n) () ( let ((f ( fib k))) (if ( odd? f) ( cons f ( next (+ k 1))) ( next (+ k 1)))))) ( next 1))

12 Similarities when abstracting The two procedures are structurally very different. Several similarities when abstracting. When summing the squares of the odd leaves of a tree, the procedure: ( define ( sum-odd-squares tree ) (if ( leaf-node? tree ) (if ( odd? tree ) ( square tree ) 0) (+ ( sum-odd-squares ( left-branch tree )) ( sum-odd-squares ( right-branch tree ))))) Enumerates all leaves Keeps only the odd ones Squares them Accumulates + from 0

13 Similarities when abstracting (cont.) When enumerating the odd Fibonacci numbers, the procedure: ( define ( sum-odd-fibs n) ( define ( next k) (if (> k n) () ( let ((f ( fib k))) (if ( odd? f) ( cons f ( next (+ k 1))) ( next (+ k 1)))))) ( next 1)) Enumerates from 0 to n Compute Fibonacci numbers Filter them, keeping the odd ones Accumulates cons from ()

14 Similarities when abstracting (cont.) The two procedures fail to exhibit these flow structures. We build a modular solution based on higher order functions and let data flow between the modules.

15 Streams to increase conceptual clarity: (sum-odd-squares revisited) ( define ( sum-odd-squares tree ) ( sum ( map-square ( filter-odd ( enumerate-tree tree ))))) ( define ( sum-odd-squares tree ) (if ( leaf-node? tree ) (if ( odd? tree ) ( square tree ) 0) (+ ( sum-odd-squares ( left-branch tree )) ( sum-odd-squares ( right-branch tree )))))

16 Streams to increase conceptual clarity: (sum-odd-fibs revisited) ( define ( sum-odd-fibs n) ( accumulate-cons ( filter-odd ( map-fib ( enumerate-interval 1 n))))) ( define ( sum-odd-fibs n) ( define ( next k) (if (> k n) () ( let ((f ( fib k))) (if ( odd? f) ( cons f ( next (+ k 1))) ( next (+ k 1)))))) ( next 1))

17 Streams as flow of data Increase the conceptual clarity More elegant and succinct What data structure to use? What about efficiency? We will use the following building tools: Constructor: cons-stream Selectors: stream-car, stream-cdr Recognizer : stream-null? The empty object : the-empty-stream

18 Streams as flow of data (cont.) The description exactly matches the one for lists and sequences: cons-stream! cons stream-car! car stream-cdr! cdr stream-null?! null? the-empty-stream! () In the beginning, we can think of, and implement, streams as usual sequences. Later, we will adopt more efficient implementations that will even allow us to manipulate infinite sequences!

19 Outline Streams (SICP sec 3.5) Modeling the Modules Efficient Implementation Delayed evaluation and Parameter passing

20 enumerate-tree in sum-odd-squares We can now describe the procedure for summing the squares of the odd leaves of a tree with the new approach: ( define ( sum-odd-squares tree ) ( sum ( map-square ( filter-odd ( enumerate-tree tree ))))) ( define ( enumerate-tree tree ) (if ( leaf-node? tree ) ( cons-stream tree the-empty-stream ) ( append-streams ( enumerate-tree ( left-branch tree )) ( enumerate-tree ( right-branch tree )))))

21 enumerate-tree in sum-odd-squares (cont.) We can now describe the procedure for summing the squares of the odd leaves of a tree with the new approach: ( define ( sum-odd-squares tree ) ( sum ( map-square ( filter-odd ( enumerate-tree tree ))))) ( define ( append-streams s1 s2) ( if ( stream-null? s1) s2 ( cons-stream ( stream-car s1) ( append-streams ( stream-cdr s1) s2))))

22 filter-odd in sum-odd-squares ( define ( sum-odd-squares tree ) ( sum ( map-square ( filter-odd ( enumerate-tree tree ))))) ( define ( filter-odd s) ( cond (( stream-null? s) the-empty-stream ) (( odd? ( stream-car s)) ( cons-stream ( stream-car s) ( filter-odd ( stream-cdr s)))) ( else ( filter-odd ( stream-cdr s)))))

23 map-square in sum-odd-squares ( define ( sum-odd-squares tree ) ( sum ( map-square ( filter-odd ( enumerate-tree tree ))))) ( define ( map-square s) ( if ( stream-null? s) the-empty-stream ( cons-stream ( square ( stream-car s)) ( map-square ( stream-cdr s)))))

24 sum-streams in sum-odd-squares ( define ( sum-odd-squares tree ) ( sum ( map-square ( filter-odd ( enumerate-tree tree ))))) ( define ( sum-stream s) ( if ( stream-null? s) 0 (+ ( stream-car s) ( sum-stream ( stream-cdr s)))))

25 enumerate-interval in sum-odd-fibs We can also describe the procedure for building the sequence of all odd Fibonacci numbers Fib(k) where k n: ( define ( sum-odd-fibs n) ( accumulate-cons ( filter-odd ( map-fib ( enumerate-interval 1 n))))) ( define ( enumerate-interval low high ) ( if (> low high ) the-empty-stream ( cons-stream low ( enumerate-interval (+ low 1) high ))))

26 map-fib in sum-odd-fibs We can also describe the procedure for building the sequence of all odd Fibonacci numbers Fib(k) where k n: ( define ( sum-odd-fibs n) ( accumulate-cons ( filter-odd ( map-fib ( enumerate-interval 1 n))))) ( define ( map-fib s) ( if ( stream-null? s) the-empty-stream ( cons-stream ( fib ( stream-car s)) ( map-fib ( stream-cdr s)))))

27 accumulate-cons in sum-odd-fibs We can also describe the procedure for building the sequence of all odd Fibonacci numbers Fib(k) where k n: ( define ( sum-odd-fibs n) ( accumulate-cons ( filter-odd ( map-fib ( enumerate-interval 1 n))))) ( define ( accumulate-cons s) ( if ( stream-null? s) () ( cons ( stream-car s) ( accumulate-cons ( stream-cdr s)))))

28 Higher order functions for streams: accumulate We can abstract stream operations such as sum, product and accumulate (see labs for corresponding abstractions on lists) ( define ( accumulate combiner initial-value stream ) ( if ( stream-null? stream ) initial-value ( combiner ( stream-car stream ) ( accumulate combiner initial-value ( stream-cdr stream ))))) ( define ( sum-stream stream ) ( accumulate + 0 stream )) ( define ( accumulate-cons stream ) ( accumulate cons () stream ))

29 Higher order functions for streams: stream-map We can abstract stream operations such as sum, product and accumulate (see labs for corresponding abstractions on lists) ( define ( stream-map proc stream ) ( if ( stream-null? stream ) the-empty-stream ( cons-stream ( proc ( stream-car stream )) ( stream-map proc ( stream-cdr stream ))))) ( define ( map-square s) ( stream-map square s)) ( define ( map-fib s) ( stream-map fib s))

30 Higher order functions for streams: stream-filter We can abstract stream operations such as sum, product and accumulate (see labs for corresponding abstractions on lists) ( define ( stream-filter predicate stream ) ( cond (( stream-null? stream ) the-empty-stream ) (( predicate ( stream-car stream )) ( cons-stream ( stream-car stream ) ( stream-filter predicate ( stream-cdr stream )))) ( else ( stream-filter predicate ( stream-cdr stream ))))) ( define ( filter-odd s) ( stream-filter odd? s))

31 Some examples ( define ( prod-square-odd-elements-stream stream ) ( accumulate * 1 ( stream-map square ( stream-filter odd? stream )))) ( define ( salary-of-highest-paid-programmer records ) ( accumulate max 0 ( stream-map salary ( stream-filter programmer? records ))))

32 Outline Streams (SICP sec 3.5) Modeling the Modules Efficient Implementation Delayed evaluation and Parameter passing

33 Implementing streams We can indeed represent streams as lists: Constructor: cons for cons-stream Selectors: car and cdr for resp. stream-car and stream-cdr Recognizer : null? for stream-null? The empty object : () for the-empty-stream This representation is very inefficient both wrt. time and space

34 Example 1: summing prime numbers ( define ( sum-primes1 a b) ( define ( iter count accum ) ( cond ((> count b) accum ) (( prime? count ) ( iter (+ count 1) (+ count accum ))) ( else ( iter (+ count 1) accum )))) ( iter a 0)) ( define ( sum-primes2 a b) ( sum-stream ( stream-filter prime? ( enumerate-interval a b)))) sum-primes1 only stores the sum being calculated (and not the list of integers, etc)

35 Example 2: finding the second prime Find the second prime number between and : ( stream-car ( stream-cdr ( stream-filter prime? ( enumerate-interval ) ))) An incremental computation would be more efficient because it would interleave enumeration and filtering and would stop when it reaches the second prime.

36 Streams implementation Formulate programs elegantly as sequence manipulations, with the efficiency of incremental computation cons-stream constructs streams partially: just a first element and enough information to construct the rest stream-cdr forces the computation of the second element Elements of the stream are only generated if needed. This is transparent to the user: on the surface streams are lists with a different syntax Transparently and automatically interleave the construction of a stream and its use

37 Streams implementation: help functions (delay expr): returns a delayed object with enough information to evaluate expr if needed. A delayed object can be thought of as a promise. delay is a special form and can be implemented as a ( lambda () expr ) (force delayed-object): Evaluates the delayed object ( define ( force delayed- object ) ( delayed-object ))

38 Streams implementation ;; cons- stream is a " special form " ;; ( cons- stream a b) corresponds to ( cons a ( delay b)) ( define ( stream- car stream ) ( car stream )) ( define ( stream- cdr stream ) ( force ( cdr stream ))) ( define the- empty- stream ()) ( define stream- null? null?)

39 Example2: finding primes revisited Streams allow for a demand-driven evaluation model. Only the needed cells for finding the result are evaluated. ( stream- car ( stream- cdr ( stream- filter prime? ( enumerate- interval ) ))) In the primes example, the numbers 10000, 10001, 10002, 10003, 10004, and will be generated and rejected by stream-filter. Then is a prime number followed by a non-prime and finally the second prime is generated and returned as the result. Intermediary results that are not needed are not generated!

40 Memoization: an important optimization The same evaluations will be forced several times if we were to reuse the beginning of a stream. ( define s ( cons- stream 1 ( cons- stream 2 ( cons- stream 3 the-empty-stream )))) result in s = (1 "lambda () (cons-stream 2...)"). Now evaluating: ( stream-car ( stream-cdr s)) ( stream-cdr ( stream-cdr s)) (lambda () (cons-stream 2...)) is evaluated twice.

41 Memoization: an important optimization (cont.) Instead of re-evaluating, remember the result of the first evaluation of the delayed argument by replacing (delay expr) with (memo-proc (lambda () expr)) where : ( define ( memo- proc proc ) ( let (( already-run? false ) ( result false )) ( lambda () (if ( not already-run?) ( begin ( set! result ( proc )) ( set! already-run? true ) result ) result )))) Compare : call-by-name: evaluate each time call-by-need: evaluate once, remember the result

42 Infinite streams! ( define ( integers-starting-from n) ( cons- stream n ( integers-starting-from (+ n 1)))) A recursive function without a base case! ( define integers ( integers-starting-from 1)) ( define ( stream-ref stream n) (if (= n 0) ( stream-car stream ) ( stream-ref ( stream-cdr stream ) (- n 1)))) ( define ( nth-prime n) ( stream-ref ( stream-filter prime? integers ) (- n 1)))

43 Infinite streams ;; Fibonacci numbers ( define ( fib-gen a b) ( cons-stream a ( fib-gen b (+ a b)))) ( define fibs ( fib-gen 0 1)) ;; add streams ( define ( add-streams s1 s2) ( cond (( stream-null? s1) s2) (( stream-null? s2) s1) ( else ( cons-stream (+ ( stream-car s1) ( stream-car s2)) ( add-streams ( stream-cdr s1) ( stream-cdr s2)))))) ( define ones ( cons-stream 1 ones )) ( define mysterious ( cons-stream 1 ( add-streams ones mysterious )))

44 Infinite streams Yet another definition of the Fibonacci numbers ( define fibs ( cons- stream 0 ( cons- stream 1 ( add-streams ( stream-cdr fibs ) fibs ))))

45 Sieve of Erastothenes Generate all primes by starting from all naturals larger than 1 The first natural (2) is a prime, filter out all its multiples The first element (3) is a prime. Filter out all naturals that are divisible by 3 The first element (5) is a prime. Filter... ( define ( sieve streams ) ( cons- stream ( stream- car stream ) ( sieve ( stream- filter ( lambda (x) ( not ( divisible? x ( stream-car stream )))) ( stream-cdr stream )))))

46 Sieve of Erastothenes (cont.) ( define ( sieve streams ) ( cons- stream ( stream- car stream ) ( sieve ( stream- filter ( lambda (x) ( not ( divisible? x ( stream-car stream )))) ( stream-cdr stream ))))) ( define primes ( sieve ( integers-starting-from 2))) ( stream- ref primes 100) > 547

47 History without mutation We can build sequences that represent the history of the modeled system. We can now model systems with state without using assignment or mutable data (see example 3.5.3) For instance, the random number generator (example 3.5.5): with objects and state: ( define random-init...) ( define ( rand-update x)... ) ( define rand ( let ((x random-init )) ( lambda () ( set! x ( rand-update x)) x))) ( rand )

48 History without mutation We can build sequences that represent the history of the modeled system. We can now model systems with state without using assignment or mutable data (see example 3.5.3) For instance, the random number generator (example 3.5.5): with streams: ( define random- numbers ( cons- stream random- init ( stream- map rand- update random-numbers ))) ( stream- ref random- numbers 10)

49 Outline Streams (SICP sec 3.5) Modeling the Modules Efficient Implementation Delayed evaluation and Parameter passing

50 Evaluation strategies here only cons had delayed arguments why not delay the evaluation of all arguments? when to force the evaluation? Evaluate according to normal order! in Algol 60, it was possible to pass parameters with a call-by-name Functional languages (e.g. lazy-ml, Haskel) use lazy evaluation. Parameters are then evaluated in a call-by-need The usual evaluation strategy is called call-by-value (eager evaluation)

51 Strict and non-strict functions Let? mean an undefined value, e.g. resulting from the evaluation of an inappropriate value or from a non-terminating computation ((sqrt -4), (/10 0), (factorial -1), etc.) A function is strict if it is undefined if any of its arguments is undefined. Otherwise, the function is not strict: (if? 10 20) =? (if false? 20) = 20 (if true 10? ) = 10

52 Delayed evaluation - lazy evaluations Normal order evaluation ( SICP p16 and p350 ). ( define ( sum-of-squares x y) (+ ( square x) ( square x))) ( define ( square x) (* x x)) > ( sum-of-squares (+ 5 1) (* 5 2)) --> (+ ( square (+ 5 1)) ( square (* 5 2))) --> (+ (* (+ 5 1) (* 5 1)) (* (+ 5 1) (* 5 1))) -->... ( define ( foo x y) (if (= x 1) y x)) > (f 2 (/ 10 0)) --> (if (= 2 1) (/ 10 0) 2) --> 2

53 Models of parameter passing call-by-value, call-by-name, call-by-need, call-by-reference,... call-by-name: Copying rule in Algol 60 procedure f(x, y); value x; name y; integer x,y; y:= 10 * x + y; integer inut, in; inut := 3; in := 5; f(in, inut ); print ( inut ); inut is replaced by y (using static binding) also called call-by-name thunks (footnote p324 SICP) inut := 10 * 5 + inut Ada: by-constant-value, by-result, by-copy-restore procedure f(in x, out y, in out z);

54 call-by-name and Jensen s device in Algol procedure Sigma ( start, stop, index, expr, sum ); value start, stop ; integer start, stop, index ; real expr, sum ; comment index, expr, sum are passed by name begin sum := 0; for index := start step 1 until stop do sum := sum + expr end ; comment when called with : Sigma (1, 100, i, 1/i^2, s) comment results in s := 0; for i := 1 step 1 until 100 do s := s + 1/ i^2 We will get back to this in lab 5 and lazy evaluation

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