#+TITLE: 3.5 - Streams
#+STARTUP: indent
#+BEGIN_HTML
#+END_HTML
* Streams are Delayed Lists
** Notes
#+BEGIN_SRC scheme
(define x (delay 5))
;; is equivalent to
(define x (lambda () 5))
;; A "thunk" (a function of no arguments, used to delay evaluation of its return value)
(force x)
;; => 5
#+END_SRC
The MIT implementation of =delay= returns a =promise= value. Force
will evaluate a =promise= if it hasn't yet been computed. The result
of the evaluation will be memoized for future calls to force.
Stream processing explanation: [[https://www.youtube.com/watch?v%3Da2Qt9uxhNSM#t%3D48m00s][MIT OpenCourseWare Lecture 6A]]
A stream a sequence built as a pair of an initial value, and a
procedure to generate the next value and the next procedure to
continue the sequence.
** Code
#+begin_src scheme
(define (cons-stream a b)
(cons a (delay b)))
(define (stream-car s)
(car s))
(define (stream-cdr s)
(force (cdr s)))
(define the-empty-stream '())
(define (stream-ref s n)
(if (= n 0)
(stream-car s)
(stream-ref (stream-cdr s) (- n 1))))
(define (stream-map proc s)
(if (stream-null? s)
the-empty-stream
(cons-stream (proc (stream-car s))
(stream-map proc (stream-cdr s)))))
(define (stream-for-each proc s)
(if (stream-null? s)
'done
(begin (proc (stream-car s))
(stream-for-each proc (stream-cdr s)))))
#+end_src
#+begin_src scheme :tangle yes
;; ===================================================================
;; 3.5.1: Streams are Delayed Lists
;; ===================================================================
(define (display-stream s)
(stream-for-each display-line s))
(define (display-line x)
(newline)
(display x))
(define (stream-enumerate-interval low high)
(if (> low high)
the-empty-stream
(cons-stream
low
(stream-enumerate-interval (+ low 1) high))))
(define (stream-filter pred stream)
(cond ((stream-null? stream) the-empty-stream)
((pred (stream-car stream))
(cons-stream (stream-car stream)
(stream-filter pred
(stream-cdr stream))))
(else (stream-filter pred (stream-cdr stream)))))
#+end_src
** Exercises
*** Exercise 3.50
Complete the following definition, which generalizes `stream-map' to
allow procedures that take multiple arguments, analogous to `map' in
section *Note 2-2-3::, footnote *Note Footnote 12::.
#+begin_src scheme
(define (stream-map proc . argstreams)
(if ( (car argstreams))
the-empty-stream
(
(apply proc (map argstreams))
(apply stream-map
(cons proc (map argstreams))))))
#+end_src
----------------------------------------------------------------------
#+begin_src scheme
;; -------------------------------------------------------------------
;; Exercise 3.50
;; -------------------------------------------------------------------
(define (stream-map proc . argstreams)
(if (stream-null? (car argstreams))
the-empty-stream
(cons-stream
(apply proc (map stream-car argstreams))
(apply stream-map
(cons proc (map stream-cdr argstreams))))))
#+end_src
*** Exercise 3.51
In order to take a closer look at delayed
evaluation, we will use the following procedure, which simply
returns its argument after printing it:
#+begin_src scheme
(define (show x)
(display-line x)
x)
#+end_src
What does the interpreter print in response to evaluating each
expression in the following sequence?(7)
#+begin_src scheme
(define x (stream-map show (stream-enumerate-interval 0 10)))
(stream-ref x 5)
(stream-ref x 7)
#+end_src scheme
----------------------------------------------------------------------
#+begin_src scheme
(define x (stream-map show (stream-enumerate-interval 0 10)))
; 0
;Value: x
(stream-ref x 5)
1
2
3
4
5
;Value: 5
(stream-ref x 7)
6
7
;Value: 7
#+end_src
*** Exercise 3.52
Consider the sequence of expressions
#+begin_src scheme
(define sum 0)
(define (accum x)
(set! sum (+ x sum))
sum)
(define seq (stream-map accum (stream-enumerate-interval 1 20)))
(define y (stream-filter even? seq))
(define z (stream-filter (lambda (x) (= (remainder x 5) 0))
seq))
(stream-ref y 7)
(display-stream z)
#+end_src
What is the value of `sum' after each of the above expressions is
evaluated? What is the printed response to evaluating the
`stream-ref' and `display-stream' expressions? Would these responses
differ if we had implemented `(delay )' simply as `(lambda ()
)' without using the optimization provided by `memo-proc'?
Explain
----------------------------------------------------------------------
#+begin_example
1 ]=> sum
;Value: 210
1 ]=> (stream-head y 10)
;Value 18: (210 204 200 182 174 144 132 90 74 20)
1 ]=> (display-stream z)
210
200
195
165
155
105
90
20
;Value: done
#+end_example
After the definition of =seq=, =sum= is equal to 210. It remains at
210 through the remainder of the operations.This would not be the
case if delay were not memoized, as without being so it would be
recalculated each time the items in the node were resolved, adding to
the value of =sum= each time, and changing the results captured by =y=
and =z=.
* Infinite Streams
** Notes
Streams can continue forever if the promise never returns an empty
stream.
Streams can be combined to model complex sequences.
** Code
#+begin_src scheme :tangle yes
;; ===================================================================
;; 3.5.2: Infinite Streams
;; ===================================================================
(define (integers-starting-from n)
(cons-stream n (integers-starting-from (+ n 1))))
(define integers (integers-starting-from 1))
(define (divisible? x y) (= (remainder x y) 0))
(define no-sevens
(stream-filter (lambda (x) (not (divisible? x 7)))
integers))
(define (fibgen a b)
(cons-stream a (fibgen b (+ a b))))
(define fibs (fibgen 0 1))
(define (sieve stream)
(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)))
#+end_src
*** Defining streams implicitly
#+begin_src scheme :tangle yes
(define ones (cons-stream 1 ones))
(define (add-streams s1 s2)
(stream-map + s1 s2))
(define integers (cons-stream 1 (add-streams ones integers)))
(define fibs
(cons-stream 0
(cons-stream 1
(add-streams (stream-cdr fibs)
fibs))))
(define (scale-stream stream factor)
(stream-map (lambda (x) (* x factor)) stream))
(define double (cons-stream 1 (scale-stream double 2)))
(define primes
(cons-stream
2
(stream-filter prime? (integers-starting-from 3))))
(define (prime? n)
(define (iter ps)
(cond ((> (square (stream-car ps)) n) true)
((divisible? n (stream-car ps)) false)
(else (iter (stream-cdr ps)))))
(iter primes))
#+end_src
** Exercises
*** Exercise 3.53
Without running the program, describe the elements of the stream
defined by
#+begin_src scheme
(define s (cons-stream 1 (add-streams s s)))
#+end_src
----------------------------------------------------------------------
\[
\sum_{i=1}^\infty 2^i
\]
*** Exercise 3.54
Define a procedure `mul-streams', analogous to `add-streams', that
produces the elementwise product of its two input streams. Use this
together with the stream of `integers' to complete the following
definition of the stream whose nth element (counting from 0) is n + 1
factorial:
#+begin_src scheme
(define factorials (cons-stream 1 (mul-streams )))
#+end_src
----------------------------------------------------------------------
#+begin_src scheme :tangle yes
(define (mul-streams s1 s2)
(stream-map * s1 s2))
(define factorials (cons-stream 1 (mul-streams (add-streams ones integers) factorials)))
#+end_src
*** Exercise 3.55
Define a procedure `partial-sums' that takes as argument a stream S
and returns the stream whose elements are S_0, S_0 + S_1, S_0 + S_1 +
S_2, .... For example, `(partial-sums integers)' should be the stream
1, 3, 6, 10, 15, ....
* Exploiting the Stream Paradigm
** Notes
Streams and their property of delayed evaluation can be used to build
abstractions over the sequences and computations used to generate
them. The examples are the square-root stream and the pi streams being
accelerated via a generic stream transformation method.
* Streams and Delayed Evaluation
** Notes
#+BEGIN_QUOTE
...stream models of systems with loops may require uses of delay
beyond the “hidden” delay supplied by cons-stream.
#+END_QUOTE
#+BEGIN_QUOTE
Unfortunately, including delays in procedure calls wreaks havoc with
our ability to design programs that depend on the order of events,
such as programs that use assignment, mutate data, or perform input or
output.
...
As far as anyone knows, mutability and delayed evaluation do not mix
well in programming languages, and devising ways to deal with both of
these at once is an active area of research.
#+END_QUOTE
* Modularity of Functional Programs and Modularity of Objects
** Notes
Random number generation can be implemented as an infinite stream
instantiated with some seed.
*** A functional-programming view of time
#+BEGIN_QUOTE
We can model a changing quantity, such as the local state of some
object, using a stream that represents the time history of successive
states. In essence, we represent time explicitly, using streams, so
that we decouple time in our simulated world from the sequence of
events that take place during evaluation.
#+END_QUOTE
Events over time can be merged / serialized (deterministically?) into
a stream of events.
#+BEGIN_QUOTE
This is precisely the same constraint that we had to deal with in
3.4.1, where we found the need to introduce explicit synchronization
to ensure a “correct” order of events in concurrent processing of
objects with state. Thus, in an attempt to support the functional
style, the need to merge inputs from different agents reintroduces the
same problems that the functional style was meant to eliminate.
#+END_QUOTE
With a working merge solution, a system can be designed in a
functional way, operating on a stream of state and inputs.
The Erlang/OTP generic server, generic fsm and other behaviours are
implemented in such a way that input streams received concurrently are
merged by the vm and combined with the state of the process as a
single stream pairing the current state with the next input to
process, allowing an Erlang developer to build a functional interface
with the complexities of concurrency abstracted away.