3.5.2 Infinite Streams
2024-05-19 Sun

• We can use streams to represent infinite sequences.

• For example, this is the definition of the stream of positive integers:

  (define (integers-starting-from n)
    (cons-streams n (integers-starting-from (+ n 1))))
  
  (define integers (integers-starting-from n))
  

  ``Our programs will never know that the entire infinite stream is not there.''

• We can now define other infinite streams based on this one, e.g.:

  (define (divisible? x y) (= (remainder x y) 0))
  
  (define no-sevens ;; streams of integers that are not divisible by 7
    (stream-filter (lambda (x) (not (divisible? x 7)))
                   integers))
  
  (stream-ref no-sevens 100)
  ;; => 117
  

• Authors show the analog way to define the infinite streams of Fibonacci numbers.
• Authors show the method, known as the ``sieve of Eratosthenes'', to construct the infinite streams of prime numbers.

• integers is

  defined by specifying ``generating'' procedures that explicitly compute the stream elements one by one. An alternative way to specify streams is to take advantage of delayed evaluation to define streams implicitly. (328)

• Here is an infinite streams of ones:

  (define ones (cons-stream 1 ones))
  

• Consider the following operation, which uses the generalized version of stream-map from exercise 3.50:

  (define (add-stream s1 s2)
    (stream-map + s1 s2))
  

• Now integers can be defined as follows:

  (define integers (cons-stream 1 (add-streams ones integers)))